Artin-Tate Motives and Z-Graded-Normed Galois-Equivariant Spectra

Dissertation zur Erlangung des Grades Doktor der Naturwissenschaften (Dr. rer. nat.) des Fachbereichs Mathematik/Informatik/Physik der Universität Osnabrück Artin-Tate Motives and Z-Graded-Normed Galois-Equivariant Spectra Submitted by Daniel Uschogov Supervisor: Prof. Dr. Markus Spitzweck University of Osnabrück September 2025 Examination Information Thesis Reviewers First Reviewer: Prof. Dr. Markus Spitzweck University of Osnabrück Second Reviewer: Prof. Dr. Elden Elmanto University of Toronto (Scarborough) Thesis Defense Date of Defense: February 04, 2026 Abstract We investigate the stable ∞-subcategory of Artin-Tate motives DMAT⊗(S)in the stable ∞-category of Spitzweck motives DM⊗(S)over a quasi-compact and quasi-separated base scheme S. We found our approach upon the Bachmann-Hoyois theory of norms in motivic homotopy theory. We prove a symmetric monoidal equivalence between the stable ∞-category DMAT⊗(S)and the stable ∞-catgeory ModAS(Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z)with respect to some Z-graded E∞-algebra AS. We then construct a Z-graded norm functor Sp⊗(−)Zin stable equivariant homotopy theory with respect to profinite groupoids. We then construct a natural transformation cZ∶Sp⊗(−)Z○ Π´et 1(−)→SH⊗(−)Zto the Z-graded norm functor in stable motivic homotopy theory. We then employ the natural transformation cZand prove the Z-graded E∞-algebra ASto be in particular a Z-graded-normed  Π´et 1(S)- spectrum provided that Sis a Dedekind scheme. We expect this result also to extend to every quasi-compact and quasi-separated base scheme S. We then establish natural stable ∞-subcategories of DMAT⊗(S)and show them separately to be symmetrically monoidally equivalent to stable ∞-categories associated with ModAS(Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z). We then employ these equivalences to investigate the stable ∞-category of Artin-Tate motives DMAT⊗(Spec(R)) over the spectrum of the real numbers. In the end we present several straightforward conclusion and examples if the base scheme Sis the spectrum S=Spec(k)of a field kof characteristic zero or an algebraically closed field. . Zusammenfassung Wir erforschen die stabile ∞-Unterkategorie von Artin-Tate Motiven DMAT⊗(S) in der stabilen ∞-Kategorie von Spitzweck Motiven DM⊗(S)über einem quasikompakten und quasi-separierten Basisschema S. Wir legen unserer Vorgehensweise die Bachmann-Hoyois Theorie von Norm-Funktoren in der Motivischen Homotopietheorie zugrunde. Wir beweisen eine symmetrisch monoidale Äquivalenz zwischen DMAT⊗(S)und der stabilen ∞-Kategorie ModAS(Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z) bezüglich einer Z-graduierten E∞-Algebra AS. Anschließend konstruieren wir einen Z-graduierten Norm-Funktor Sp⊗(−)Zin der stabilen äquivarianten Homotopietheorie bezüglich proendlicher Gruppoide. Anschließend konstruieren wir eine natürliche Transformation cZ∶Sp⊗(−)Z○ Π´et 1(−)→SH⊗(−)Zzu dem Z-graduierten NormFunktor in der stabilen motivischen Homotopietheorie. Anschließend benutzen wir die natürlich transformation cZund beweisen für ein Dedekind-Schema S, dass AS insbesondere ein Z-graduiertes-normiertes  Π´et 1(S)-Spektrum ist. Wir gehen davon aus, dass sich dieses Ergebnis auch auf jedes quasikompakte und quasigetrennte Schema Sübertragen lässt. Anschließend betrachten wir auch natürliche stabile ∞-Unterkategorien von DMAT(S)und beweisen, dass diese separat symmetrisch monoidal äquivalent zu stabilen ∞-Kategorien sind die mit der stabilen ∞-Kategorie ModAS(Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z)einhergehen. Anschließend benutzen wir diese Äquivalenzen, um die stabile ∞-Kategorie von Artin-Tate Motiven DMAT(Spec(R)) über dem Spektrum der reellen Zahlen zu untersuchen. Zum Abschluss stellen wir einige direkte Schlussfolgerungen und Beispiele vor, falls das Basisschema Sdas Spektrum S=Spec(k)eines Körpers kvon Charakteristik Null oder eines algebraisch abgeschlossenen Körpers ist. Acknowledgments I wish to express my appreciation to my supervisor Markus Spitzweck. His continuous support throughout the many adventures during my PhD was of immense help to me. I credit most of what I have learned to his guidance. I am grateful to Elden Elmanto for his thorough engagement and commitment throughout the review process. I am also grateful to my colleagues, who have accompanied me over the past years, viz., Christian Ahring, Holger Brenner, Xiaowen Dong, Andrea Figini, Marco Giustetto, Robin Louis, Federico Mocchetti, Ahina Nandy, Manh Toan Nguyen, Fynn Pörtner, Oliver Röndigs, Thor Wittich. Furthermore, I am thankful to all people in the mathematical department for the shared experiences and the friendly atmosphere. I would like to express my deepest gratitude to my parents, my sister and my friends. Contents Introduction 7 Context........................................... 7 Results of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Related Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1 Voevodsky Motives 20 1.1 The Unstable A1-Motivic Homotopy Category . . . . . . . . . . . . . . . . 20 1.2 The Stable A1-Motivic Homotopy Category . . . . . . . . . . . . . . . . . . 24 1.3 Mixed Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.4 Beilinson Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2 Spitzweck Motives 46 2.1 The Stable ∞-Category of Spitzweck Motives . . . . . . . . . . . . . . . . . 46 2.2 Tate Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3 Norm Functors in Homotopy Theory 54 3.1 Norm Functors in Motivic Homotopy Theory . . . . . . . . . . . . . . . . . 54 3.2 Norm Functors in Stable Equivariant Homotopy Theory . . . . . . . . . . 67 3.3 The Étale Fundamental Group . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.4 The Profinite Étale Fundamental Groupoid . . . . . . . . . . . . . . . . . . 81 3.5 Galois-Equivariant Spectra and Motivic Spectra . . . . . . . . . . . . . . . 87 3.6 Normed ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.7 Normed Motivic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.8 Graded-Normed Motivic Spectra . . . . . . . . . . . . . . . . . . . . . . . . 100 4 Artin-Tate Motives 108 4.1 Z-Graded Galois-Equivariant Spectra . . . . . . . . . . . . . . . . . . . . . . 108 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra . . . . . . . 117 4.3 Graded Norm Functors in Stable Equivariant Homotopy Theory . . . . . 149 4.4 Graded-Normed Equivariant Spectra . . . . . . . . . . . . . . . . . . . . . . 160 List of Symbols 170 References 174 7 Introduction Context Motives - A cohomology theory of schemes is in particular a functor H●∶Schop /S→AbZ. The cohomology H●(X)of a scheme Xover the base scheme Sis a Z-graded abelian group. The cohomology H●(X)of the scheme Xis simpler then the scheme itself but nevertheless complicated enough to reason about X. There exist various cohomology theories of schemes such as the ´ etale cohomology or the l-adic cohomology. - A. Grothendieck proposed the existence of a category reflecting the universal property of all reasonable cohomology theories of schemes. In particular there should exists some abelian category M(S, R)assigned to some scheme Sand some commutative ring Ralong with some functor MS∶Schop /S→M(S, R)such that for every reasonable cohomology theory of schemes there exists also some functor RH●∶M(S, R)→AbZ and an isomorphism H●(X)≅RH●(MS(X)). - A. Grothendieck constructed the pseudo-abelian and additive category of Chow motives Chow(Spec(k)) over the spectrum Spec(k)of some field k. This category was not able to capture the complexity of the reasonable cohomology theories of schemes. P. Deligne proposed to construct a triangulated category of motives DM(S, R)over a scheme Salong with a t-structure whose heart is the hypothetical abelian category of motives M(S, R)proposed by A. Grothendieck. In the following V. Voevodsky constructed the triangulated category DM(S, R)and also the triangulated category SH(S), see among others, [62] and [59]. Later, in [52], M. Spitzweck constructed also a triangulated category DM(S). It is unresolved if there exists a t-structure on any of the established triangulated categories DM(S, R)and SH(S)and DM(S)whose heart is the abelian category M(S, R). - Because of the high complexity of these categories we are investigating small and accessible subcategories such as the triangulated category of Tate motives DMT(S, R) and the triangulated category of Artin motives DMA(S, R)and the triangulated category of Artin-Tate motives DMAT(S, R)over a quasi-compact quasi-separated base scheme Swith the assumption on Rto be the rational numbers Qor the integers Z. Introduction 8 Tate Motives - The triangulated category of Tate motives DMT(S, R)over a base scheme Sis a natural triangulated subcategory of Voevodsky’s triangulatd category of mixed motives DM(S, R)over a base scheme S. The triangulated category DMT(S, R)is small in comparison to the triangulated category DM(S, R)and admits in a wide range of cases a simple cellular structure. - The full triangulated subcategory of compact Tate motives DMT(Spec(k),Q)cover the spectrum Spec(k)of a number field k, as considered in [45], is particularly well studied. The t-structures and the weight structures along with the representation theory of DMT(Spec(k),Q)cand its relation to algebraic K-theory and to the set up of I. Kriz and J. P. May were extensively studied by M. Levine and M. Spitzweck, see [45] and [29] and[30] and [54] and also [53]. - In [52], M. Spitzweck constructed a stable ∞-category of motives DM⊗(S)over the spectrum S=Spec(D)of a Dedekind domain Dof mixed characteristic, see Definition 2.3. M. Spitzweck along with O. Röndigs and P.A. Østvær proved this category to be closely related to DM(S, Z)in a wide range of cases, as shown in the Theorem 2.1 and the Corollary 2.14. In addition M. Spitzweck established a stable ∞-subcategory of Tate motives DMT⊗(S), see Definition 2.23, in the stable ∞-category DM⊗(S) and studied the t-structures and the representation theory of it. In Theorem 2.43, M. Spitzweck also proved DMT⊗(S), see Definition 2.23, to be symmetrically monoidally equivalent to some stable ∞-category of Z-graded chain complexes of abelian groups ModAS(D(Ab)Z), see Definition 2.39, over some Z-graded E∞-algebra ASwith respect to a quasi-compact and quasi-separated base scheme S. Artin Motives - The triangulated category of compact Artin motives DMA(Spec(k),Q)cover the spectrum Spec(k)of a number field kis a natural triangulated subcategory of Voevodsky’s triangulated category of mixed motives DM(Spec(k),Q), see Definition 1.101. It is closely related to the the absolute Galois group Gal(ksep/k), as shown in the Theorem 4.109 and the Theorem 4.112. It is the triangulated category of mixed motives associated to zero-dimensional varieties over Spec(k), see [45]. It is also strongly associated with algebraic number theory. Introduction 9 Artin-Tate Motives - The triangulated category of compact Artin-Tate motives DMAT(Spec(k),Q)cover the spectrum of a number field k, as considered by M. Preisinger in [45], is a natural triangulated subcategory of Voevodsky’s triangulated category of mixed motives DM(Spec(k),Q), see Definition 1.101. The triangulated category contains the triangulated category DMA(Spec(k),Q)cand the triangulated category DMT(Spec(k),Q)c, see Definition 1.105. - The triangulated category DMAT(Spec(k),Q)cover the spectrum Spec(k)of a number field kwas extensively investigated by M. Preisinger in [45] and J. Scholbach in [50] and J. Wildeshaus in [65]. In addition in [50] J. Scholbach constructed a tstructure on DMAT(S, Q)c, if S=Spec(Ok)is the ring of integers of a number field k, see also Theorem 4.123. The Étale Fundamental Group - The ´ etale fundamental group  π´et 1(X, x)of a connected scheme X, see Definition 3.127, as considered in the Section 3.3, was introduced by A. Grothendieck in the 1960s. It is a profinite group in the sense of Definition 3.112. It classifies the finite ´ etale coverings morphisms of the scheme X, see in particular Definition 3.122. The ´ etale fundamental group  π´et 1(Spec(k), x)of the spectrum Spec(k)of a field k, is by Example 3.129 also isomorphic to the absolute Galois group Gal(ksep/k). - The profinite ´ etale fundamental groupoid  Π´et 1(X)of a scheme X, as considered in the Section 3.4 is a profinite groupoid. It does not depend on a basepoint. If Xis also a connected scheme then by Example 3.160, the delooping groupoid, see Definition 3.62, of the ´ etale fundamental group  π´et 1(X, x)is isomorphic to the profinite ´ etale fundamental groupoid  Π´et 1(X), see Definition 3.157. - The triangulated category of compact Artin motives DMA(Spec(k),Q)cover the spectrum Spec(k)of a number field k, as investigated by M. Preisinger in [45] is closely related to the absolute Galois group Gal(ksep/k), as shown in Theorem 4.109 and Theorem 4.112 and Theorem 4.121 and therefore also to the ´ etale fundamental group  π´et 1(Spec(k), x). Introduction 16 - In the Section 1.2, we present explicitly the construction of Voevodsky’s stable A1homotopy category SH(S)over some base scheme Sand provide also its fundamental properties, see in particular Theorem 1.55 and Theorem 1.65 and Theorem 1.66. - In the Section 1.3, we present explicitly the construction of Voevodsky’s category of mixed motives DM(S)over some base scheme Sand provide also its fundamental properties, see in particular Theorem 1.88 and Theorem 1.91 and Theorem 1.92. - In the Section 1.4, we present explicitly the construction of the triangulated category of Beilinson motives DMB(S, Q)over some base scheme Sand provide also its fundamental properties, see in particular Theorem 1.110 and Definition 1.111. - We provide the relations between H(S)and SH(S)and DM(S)and also DMB(S, Q), see in particular Theorem 1.89 and Theorem 1.118 and Theorem 1.90. Chapter 2 - We introduce Spitzweck’s motivic Eilenberg-MacLane spectrum MZSpi Sover the spectrum S=Spec(D)of a Dedekind domain Dof mixed characteristic, see Theorem 2.1. We further employ MZSpi Sfor the construction of the stable ∞-category DM⊗(S)of Spitzweck motives over the base scheme S, see Definition 2.3. - We provide the fundamental properties and relations of Spitzweck’s stable ∞-category DM⊗(S)to Voevodsky’s categories of motives SH(S)and DM(S)and DMB(S, Q), see Theorem 2.1 and Corollary 2.14. - We introduce Spitzweck’s multiplicatively periodized motivic Eilenberg-MacLane spectrum PMZSpi Sover some base scheme S, see Corollary 2.33 and [52, Appendix C], and present in addition its properties, see Remark 2.35. - In the Definition 2.23 we introduce the stable ∞-category of Tate motives DMT⊗(S) over a quasi-compact and quasi-separated base scheme Sin the stable ∞-category DM⊗(S)of Spitzweck motives over S. - We present Spitzweck’s fundamental equivalence of stable ∞-categories which involves the stable ∞-category of Tate motives DMT⊗(S)over a quasi-compact and quasiseparated base scheme S, see Theorem 2.43. Introduction 17 Chapter 3 - We introduce initially the Bachmann-Hoyois norm functors in motivic homotopy theory p⊗∶H⊗ ∗(X)→H⊗ ∗(S)and also p⊗∶SH⊗(X)→SH⊗(S)associated to some morphism of schemes p∶X→S, see Corollary 3.40 and Proposition 3.44. We then provide the fundamental properties of the motivic norm functors p⊗. We then employ the motivic norm functors p⊗for the construction of the norm functors H⊗ ∗(−)and SH⊗(−)in motivic homotopy theory, see Corollary 3.58 and Theorem 3.61. We then also provide the fundamental properties of the functors H⊗ ∗(−)and SH⊗(−). - We introduce the Bachmann-Hoyois norm functors in equivariant homotopy theory with respect to profinite groupoids p⊗∶H⊗ ∗(X)→H⊗ ∗(Z)and p⊗∶Sp⊗(X)→Sp⊗(Z) associated to some morphism of profinite groupoids p∶X→Z, see Remark 3.91 and Remark 3.98. We then employ the norm functors p⊗in equivariant homotopy theory with respect to profinite groupoids for the construction of the norm functors H⊗ ∗(−) and Sp⊗(−)in equivariant homotopy theory with respect to profinite groupoids, see Theorem 3.92 and Theorem 3.99. We then also provide the fundamental properties of the functors H⊗ ∗(−)and Sp⊗(−). - We present the construction and the fundamental properties of the ´ etale fundamental group  π´et 1(X, x)of a connected scheme X, see Definition 3.122. We then present the construction and the fundamental properties of the profinite ´ etale fundamental groupoid  Π´et 1(X)of a scheme X, see Section 3.4. We then present in addition the relation of the ´ etale fundamental group  π´et 1(X, x)and the ´ etale fundamental groupoid  Π´et 1(X), see Example 3.160. - We employ the norm functor SH⊗(−)in stable motivic homotopy theory, see Proposition 3.44, and also the norm functor Sp⊗(−)in equivariant homotopy theory with respect to profinite groupoids, see Theorem 3.99, and also the profinite ´ etale fundamental groupoid functor  Π´et 1(−), see Proposition 3.162, and construct the natural transformation c∶Sp⊗(−)○ Π´et 1(−)→SH⊗(−)of Bachmann-Hoyois, see Theorem 3.168. - We employ the norm functor SH⊗(−)in stable motivic homotopy theory, see Theorem 3.61, to introduce the ∞-category of normed motivic spectra NAlgC(SH⊗(−)) over some category of schemes Cand provide the fundamental properties of this category, see Definition 3.188. We then employ the norm functor Sp⊗(−)in equivariant homotopy theory with respect to profinite groupoids to introduce the ∞-category of Introduction 18 normed X-spectra NAlg(Sp⊗(X))with respect to a profinite groupoid Xand provide the fundamental properties of this category, see Definition 3.203. - We introduce the Bachmann-Hoyois Z-graded-norm functor SH⊗(−)Zin stable motivic homotopy theory, see Theorem 3.217. We then present the fundamental properties of SH⊗(−)Z. We then introduce the Z-graded-norm subfunctor (SH⊗(−)Z)eff and the Z-graded-norm subfunctor (SH⊗(−)Z)veff and present also the relations of these, see Corollary 3.230 and Remark 3.239. - We employ the Z-graded-norm functor SH⊗(−)Zin stable motivic homotopy theory, see Theorem 3.217, to the introduce ∞-category of Z-graded-normed motivic spectra NAlgC(SH⊗(−)Z)over some category of schemes Cand provide the fundamental properties of this category, see Definition 3.223. Chapter 4 - We initially determine the dualizable and compact generators of the stable ∞-category Sp⊗( Π´et 1(S))with respect to a quasi-compact and quasi-separated base scheme S, see Proposition 4.39 and Proposition 4.41. We then also determine the dualizable and compact generators of the stable ∞-category Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z, see Corollary 4.49. - We introduce the stable ∞-category of Artin motives DMA⊗(S), see the Definition 4.51, and also the stable ∞-category of Artin-Tate motives DMAT⊗(S), see Definition 4.53, in the stable ∞-category of Spitzweck motives DM⊗(S)over a quasicompact and quasi-separated base scheme S. We then also provide the relations of the stable ∞-categories DMA⊗(S)and DMT⊗(S)and DMAT⊗(S), see Remark 4.54. We then present explicitly Spitzweck’s fundamental equivalence between the stable ∞- category DMT⊗(S)and the stable ∞-category ModAS(D⊗(Ab)Z), see Theorem 4.67. - We employ the natural transformation c∶Sp⊗(−)○ Π´et 1(−)→SH⊗(−)of BachmannHoyois, see Theorem 3.168, and also a result of H. Heine, see Theorem 4.71, with respect to the dualizable and compact generators of Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z, see Corollary 4.49, to establish a symmetric monoidal equivalence between the stable ∞-category DMAT⊗(S)over a quasi-compact and quasi-separated base scheme S, see Definition 4.53, and the stable ∞-category ModAS(Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z)with respect to some Z-graded E∞-algebra ASassocciated to S, see Theorem 4.73. Introduction 19 - We employ the Z-graded-norm functor SH⊗(−)Zin stable motivic homotopy theory, see Theorem 3.217, and the norm functor Sp⊗(−)in stable equivariant homotopy theory with respect to profinite groupoids, see Theorem 3.99, for the construction of aZ-graded-norm functor Sp⊗(−)Zin stable equivariant homotopy theory with respect to profinite groupoids, see Proposition 4.146. - We employ the Z-graded-norm functor Sp⊗(−)Zin stable equivariant homotopy theory with respect to profinite groupoids, see Proposition 4.146, along with the Zgraded-norm functor SH⊗(−)Zin stable motivic homotopy theory, see Theorem 3.217, along with the profinite ´ etale fundamental groupoid functor  Π´et 1(−), see Proposition 3.162, and also the natural transformation c∶Sp⊗(−)○ Π´et 1(−)→SH⊗(−)of Bachmann-Hoyois, see Theorem 3.168, for the contruction of the following natural transformation of functors cZ∶Sp⊗(−)Z○ Π´et 1(−)→SH⊗(−)Z, see Proposition 4.162. - We prove the multiplicative periodization PMZSpi Sof Spitzweck’s motivic EilenbergMacLane spectrum MZSpi S, see Corollary 2.33, over the spectrum S=Spec(D)of a Dedekind domain of mixed characteristic is unique, see Theorem 4.65. - We employ the natural transformation cZ, see Proposition 4.162, and the uniqueness of the multiplicative periodization PMZSpi Sof Spitzweck’s motivic Eilenberg-MacLane spectrum MZSpi Sover a base scheme S, see Theorem 4.65, to prove that the Z-graded E∞-algebra ASis a Z-graded-normed  Π´et 1(S)-spectrum, provided that Sis a Dedekind scheme, see Theorem 4.177. We expect this result to extend to every quasi-compact and quasi separated base scheme S. - We establish natural stable ∞-subcategories of the stable ∞-category of Artin-Tate motives DMAT⊗(S)over a quasi-compact and quasi-separated base scheme Sand show them separately to be symmetrically monoidally equivalent to stable ∞-categories associated with the stable ∞-category ModAS(Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z), see Theorem 4.73 and following. - We investigate the stable ∞-category of Artin-Tate motives DMAT⊗(Spec(R)) over the spectrum of the real numbers, see Theorem 4.73 and following. - We present a series of straightforward conclusions and examples of the main results if the base scheme is the spectrum S=Spec(k)of a field kof characteristic zero or an algebraically closed field, see Theorem 4.73 and following. 20 1 Voevodsky Motives 1.1 The Unstable A1-Motivic Homotopy Category Definition 1.1 (Universal Property):Let Sbe a scheme. Let (X, Z)be a closed pair of schemes. The ∞-category H∗(S)is the category of functors F∶Smop /S→Spc∗which satisfies the following properties: (1) (Excision):For any pair of smooth schemes Xand Yover Sand any excisive morphism p∶(Y, W )→(X, Z)of closed pairs is p∗∶F(Y, W)→F(X, Z)a weak equivalence in the pointed ∞-category of spaces Spc∗. (2) (A1-Invariance):For any scheme X∈Ob(Sm/S)the morphism πX∶X×SA1 S→X induces an equivalence F(X)≃F(X×SA1 S)in the pointed ∞-category Spc∗. Reference. See [11, Definition 1.2]. Remark 1.2 (Closed Pair of Schemes):A closed pair of schemes (X, Z)is a scheme X with a closed subscheme Z. A morphism of closed pairs of schemes p∶(Y, W )→(X, Z) is a morphism f∶X→Ysuch that f−1(Z)=W. Remark 1.3 (Notation):Let Sbe a scheme. Let (X, Z)be a closed pair of schemes. Let F∶Smop /S→Spc∗be a functor. Let F(X, Z)be the fiber of F(X)→F(X/Z)in the ∞-category of pointed spaces Spc∗. Definition 1.4: A morphism of schemes X→Yis called a Nisnevich morphism if it is an ´etale morphism such that for every x∈Xthere exists an y∈Yin the fiber f−1(x) of xsuch that the induced morphism of residue fields k(x)→k(y)is an isomorphism. Definition 1.5 (Nisnevich Site):Let Sbe a scheme. A Nisnevich cover of X∈Ob(Sm/S) is a finite set of ´etale morphisms {fα∶Uα→X}α∈A⊂Mor(Sm/S)such that for any x∈Xthere is some α∈Aand a point u∈Uαsuch that the induced morphism of residue fields k(x)→k(u)is an isomorphism. The category of smooth schemes over Swith the Nisnevich coverings is called the Nisnevich site (Sm/S)Nis of S. Reference. See [39, Definition 1.2] and [1, Exercise 3.40]. Definition 1.6 (Distinguished Nisnevich Square):Let Sbe a Noetherian scheme of finite Krull dimension. A cartesian square in (Sm/S)Nis is called an distinguished Nisnevich square if i∶U→Xis a Zariski open immersion and if p∶V→Xis ´etale and 1.1 The Unstable A1-Motivic Homotopy Category 21 if p−1(X/U)→(X/U)is an isomorphism of schemes where X/Uadmits the reduced induced scheme structure. Reference. See [39, Definition 1.3]. Definition 1.7 (Nisnevich Excision):Let Sbe a Noetherian scheme of finite Krull dimension. A simplicial presheaf F∈Ob(PSh((Sm/S)Nis,sSetQ)locinj)over Ssatisfies Nisnevich excision if it sends every distinguished Nisnevich square to a homotopy pullback square in the model category of simplicial sets sSetQand if F(∅)=∗. Definition 1.8: Let Sbe a Noetherian scheme of finite Krull dimension and let the homotopy category of simplicial presheaves over Sbe Hs(S)∶=Ho(PSh((Sm/S)Nis ,sSetQ)locinj). Reference. See the exposition following [39, Definition 2.1]. Definition 1.9: Let Sbe a Noetherian scheme of finite Krull dimension. (1) A simpliclal presheaf Fis called A1-local if for all simpliclal presheaves Gthe following morphism induced by the projection πG∶G×SA1 S→Gis a bijection π∗ G∶HomHs(S)(G, F )→HomHs(S)(G×SA1 S, F ). (2) A morphism of simplicial presheaves f∶F→Gis called an A1-weak equivalence if for every A1-local simpliclal presheaf Hthe following morphism is a bijection f∗∶HomHs(S)(G, H)→HomHs(S)(F, H). (3) A morphism of simplicial presheaves f∶F→Gis called an A1-fibration if it admits the right lifting property with respect to all cofibrations in the model category PSh((Sm/S)Nis ,sSetQ)locinj which are also A1-weak equivalences. Reference. See [39, Definition 2.1]. Proposition 1.10: Let Sbe a Noetherian scheme of finite Krull dimension. There exists the following commutative diagram of monoidal Quillen equivalences. PSh((Sm/S)Nis ,sSetQ)locinj Sh((Sm/S)Nis ,sSetQ)locinj PSh((Sm/S)Nis ,sSetQ)locproj Sh((Sm/S)Nis,sSetQ)locproj ⋅ aNis i aNis i Id Id Id Id 1.1 The Unstable A1-Motivic Homotopy Category 22 Proof. See [13] and [7] and [27] and [39]. Proposition 1.11: Let Sbe a Noetherian scheme of finite Krull dimension. The left Bousfield localization of the model categories above at the class of A1-weak equivalences induces the following commutative diagram of symmetric monoidal Quillen equivalences of model categories. PSh((Sm/S)Nis ,sSetQ)A1-locinj Sh((Sm/S)Nis ,sSetQ)A1-locinj PSh((Sm/S)Nis ,sSetQ)A1-locproj Sh((Sm/S)Nis,sSetQ)A1-locproj ⋅ aNis i aNis i Id Id Id Id Proof. See [32, A.3.7.3] and [39]. Proposition 1.12: Let Sbe a Noetherian scheme of finite Krull dimension. There exists a canonical Quillen adjunction of model categories Id ∶PSh((Sm/S)Nis ,sSetQ)locinj ⇆PSh((Sm/S)Nis ,sSetQ)A1-locinj ∶Id that satisfies the following properties: (1) The fibrant objects in the model category on the right are the A1-local and fibrant objects in the model category on the left. (2) The cofibrations in the model category on the right are also the cofibrations in the model category on the left. (3) The weak equivalances in the model category on the right are the A1-weak equiv. Proof. See [32, Proposition A.2.8.2]. Definition 1.13 (A1-Invariance):Let Sbe a Noetherian scheme of finite Krull dimension. A simplicial presheaf F∈Ob(PSh((Sm/S)Nis,sSetQ)A1-locinj)is called A1-invariant if for every X∈Ob((Sm/S)Nis)the canonical morphism πX∶X×SA1 S→Xinduces a weak equivalence F(X)≃F(X×SA1 S)in the model category sSetQ. Definition 1.14 (Motivic Spaces):Let Sbe a Noetherian scheme of finite Krull dimension. The full subcategory PSh((Sm/S)Nis,sSetQ)f A1-locinj of the A1-fibrant simplicial presheaves is called the category of motivic spaces over S. 1.1 The Unstable A1-Motivic Homotopy Category 23 Definition 1.15 (Unstable A1-Homotopy Category):Let Sbe a Noetherian scheme of finite Krull dimension. The (pointed) unstable A1-homotopy category over the base scheme Sis the homotopy category H(S)∶=Ho(PSh((Sm/S)Nis ,sSetQ)A1-locinj), H∗(S)∶=Ho(PSh∗((Sm/S)Nis ,sSetQ)A1-locinj). Reference. See the exposition following [39, Definition 2.1 and Proposition 2.12]. Definition 1.16 (Simplicial Circle and Tate Circle):Let Sbe a Noetherian scheme of finite Krull dimension. Let the simplicial circle and Tate circle over Sbe: (1) Let S1 sbe the canonical pointed constant simplicial presheaf ∆1/∂∆1. (2) Let S1 tbe simplicial presheaf represented by Gm,S ∶=A1 S/{0}and pointed by 1. (3) Let Tbe the canonical pointed simplicial presheaf represented by A1 S/Gm,S. Reference. See the exposition following [39, Proposition 2.14]. Remark 1.17 (Notation): (1) Let Sn s∶=(S1 s)∧⋯∧(S1 s)and Sn t∶=(S1 t)∧⋯∧(S1 t)and Tn∶=(T)∧⋯∧T. (2) Let Sp,q ∶=(S1 s)p−q∧(S1 t)q. Definition 1.18 (Suspension):Let Sbe a Noetherian scheme of finite Krull dimension. Let F∈Ob(PSh((Sm/S)Nis ,sSetQ)A1-locinj)be a simplicial presheaf over S. (1) Let Σs(F)∶=S1 s∧Fand let Σn s(F)∶=Sn s∧F. (2) Let Σt(F)∶=S1 t∧Fand let Σn t(F)∶=Sn t∧F. (3) Let ΣT(F)∶=T∧Fand let Σn T(F)∶=Tn∧F. Definition 1.19 (Motivic Sphere):Let Sbe a Noetherian scheme of finite Krull dimension. A simplicial presheaf F∈Ob(PSh∗((Sm/S)Nis,sSetQ)A1-locinj)is called a motivic sphere if there exist natural numbers m, n ≥1and an A1-weak equivalence Sn s∧Sm t≃F. 1.2 The Stable A1-Motivic Homotopy Category 24 Example 1.20: Let Sbe a Noetherian scheme of finite Krull dimension. The following are motivic spheres over S. (1) S1 s∧S1 t≃T≃(P1 S,{∞}). (2) S2n,n ≃Sn s∧Sn t. (3) T=S1 s∧Gm,S. Proof. See [39, Lemma 2.15 and Corollary 2.18] and [1, Chapter 4.6]. Definition 1.21 (Thom Space):Let Sbe a Noetherian scheme of finite Krull dimension. Let πX∶V→Xbe a vector bundle of smooth schemes over Sand let i∶X→Vbe the zero section. The Thom space of the vector bundle is the following homotopy cofiber ThX(V)∶=V/(V−i(X)). Reference. See [39, Definition 2.16]. Theorem 1.22 (Purity):Let Sbe a quasi-compact and quasi-separated scheme. Let i∶Z→Xbe a closed immersion in (Sm/S)Nis. Let also νZ∶NZ(X)→Zbe the normal bundle. There exists a canonical A1-weak equivalence ThZ(NZ(X))≃X/(X−Z). Proof. See [1, Theorem 7.6]. 1.2 The Stable A1-Motivic Homotopy Category Definition 1.23 (Universal Property):Let Sbe a scheme. Let C⊗∈Ob(CAlg(PrL,⊗)) be a locally presentable symmetric monoidal ∞-category. The stable A1-homotopy category of symmetric motivic spectra SH⊗(S)over Sadmits the universal property: (1) The following functor of ∞-categories is fully faithful Φ∶Fun⊗,L(SH⊗(S),C⊗)→Fun⊗(Sm/S,C⊗) F↦F○Σ∞ P1(−)+ (2) The essential image of the functor Φis the set of symmetric monoidal ∞-functors F∶Sm/S→C⊗which are characterized by the following properties: 1.2 The Stable A1-Motivic Homotopy Category 25 ●A1-invariance. ●Nisnevich excision. ●(ΣP1-Stability):The cofiber cofib(F(Gm,S)→F(A1 S)) is ⊗-invertible. Reference. See [48, Corollary 5.11]. Definition 1.24: Let Sbe a Noetherian scheme of finite Krull dimension. The category of sequential Σs-spectra SpN(PSh∗((Sm/S)Nis,sSet),Σs)over the base scheme Sis the following category: (1) A sequential Σs-spectrum F=(F●, σF● ●)is a sequence {Fn}n∈Nin the following pointed category PSh∗((Sm/S)Nis,sSet)along with basepoint preserving morphisms σF● n∶S1 s∧Fn→Fn+1. (2) A morphism of sequential Σs-spectra h●∶(F●, σF● ●)→(G●, σG● ●)is a sequence {hn}n∈Nof basepoint preserving morphisms hn∶Fn→Gnsuch that the following diagrams commute. S1 s∧Fn→→ σF● n ↓↓ S1 s∧Gn σG● n ↓↓ Fn+1hn+1 →→Gn+1 Definition 1.25: Let Sbe a Noetherian scheme of finite Krull dimension. Let also F∈Ob(PSh∗((Sm/S)Nis ,sSet)) be a pointed simplicial presheaf over Sand let πn(F) be the sheaf associated to the presheaf (Sm/S)Nis →Grp , U ↦πn(F(U)) , n ≥1. Remark 1.26: If n≥2then πn(F)is a Nisnevich sheaf of abelian groups over S. Definition 1.27: Let Sbe a Noetherian scheme of finite Krull dimension. Let also F=(F●, σF● ●)be a Σs-spectrum in SpN(PSh∗((Sm/S)Nis,sSet),Σs). The sheaf of stable homotopy groups of Fis πn(F)∶=colim m>n(πn+m(Fm)) =colim m>n(⋯→πn+m(Fm)→πn+m+1(S1 s∧Fm)→πn+m+1(Fm+1)→⋯). Reference. See the exposition following [14, Example 2.2]. 1.2 The Stable A1-Motivic Homotopy Category 32 ●(Base Change):If fαβ is separated of finite type then any cartesian square Xδ fδγ →→ gδα ↓↓ Xγ gγβ ↓↓ Xαfαβ →→Xβ implies the base change natural isomorphisms g∗ γβ ○fαβ!→fδγ!○g∗ δα , gδα∗○f! δγ →f! αβ ○gγβ∗. ●(Projection Formula):The projection formula is satisfied f!(G∧f∗(F))≃f!(G)∧F. Proof. See [11, Theorem 2.9]. Example 1.56 (Tate Twist):Let Sbe a Noetherian scheme of finite Krull dimension. Let πS∶P1 S→Sbe the canonical projection and let iS∶S→P1 Sbe the inclusion and let 1S(1)∶=i∗ S(π! S(1S))[−2],1S(n)∶=1S(1)∧⋯∧1S(1) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n-times , n ≥1. Remark 1.57 (Notation):Since the spectrum 1S(1)is ⊗-invertible in SH⊗(S)we set 1S(0)∶=1S,1S(1)∧1S(−1)=1S(0),1S(−n)∶=1S(−1)∧⋯∧1S(−1) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n-times , n ≥1. Remark 1.58 (Notation):Let the n-th Tate twist of a symmetric ΣP1-spectrum be F(n)∶=F∧1S(n), F ∈Ob(SH⊗(S)) , n ∈Z. Definition 1.59 (Effective Spectra):Let Sbe a Noetherian scheme of finite Krull dimension. The triangulated category of effective spectra in the A1-homotopy category SH(S)is the following localizing full triangulated subcategory SH(S)eff ∶=LocSH(S)({Σ∞ P1(X)+∣X∈Ob((Sm/S)Nis)}). Reference. See the introduction to [4, Chapter 13]. 1.2 The Stable A1-Motivic Homotopy Category 33 Definition 1.60 (Homology):Let Sbe a Noetherian scheme of finite Krull dimension. Let Ebe a Σs,t-bispectrum in the stable A1-homotopy category SH(S). The bigraded homology theory of Eis the functor E●,●∶(Sm/S)Nis →AbZ×Z, Ep,q(X)∶=HomSH(S)(Sp,q,Σ∞ P1(X)+∧E). Definition 1.61 (Cohomology):Let Sbe a Noetherian scheme of finite Krull dimension. Let Ebe a Σs,t-bispectrum in the stable A1-homotopy category SH(S). The bigraded cohomology theory of Eis is the functor E●,●∶(Sm/S)op Nis →AbZ×Z, Ep,q(X)∶=HomSH(S)(Σ∞ P1(X)+, E ∧Sp,q). Reference. See [14, Chapter 3.2] and [2, Chapter 4.2]. Example 1.62: Let Sbe a Noetherian scheme of finite Krull dimension and let also X∈Ob((Sm/S)Nis). There exists canonical isomorphisms of abelian groups KGLp,q S(X)≅K2q−p(X), p, q ∈Z. Proof. See [10, Equation (13.2.1.2)]. Example 1.63 (Motivic Cohomology Theory):Let kbe a field of characteristic zero and let X∈Ob((Sm/k)Nis). There exists canonical isomorphisms of abelian groups MZp,q k(X)∶=HomSH(Spec(k))(Σ∞ P1(X)+, Sp,q ∧MZk), p, q ∈Z. Proof. See [14, Chapter 3.1]. Corollary 1.64: Let kbe a field of characteristic zero and let X∈Ob((Sm/k)Nis ). There exist canonical isomorphisms of abelian groups MZp,q k(X)≅Hp Zar (X, Zk(q))≅CHq(X, 2q−p), q ≥0,2q−p≥0. Proof. See [14, Chapter 3.1] and [2, Theorem 4.16]. Theorem 1.65 (Naumann-Spitzweck):If Sis a Noetherian scheme of finite Krull dimension such that S=⋃αSpec(Rα)and Rαare countable rings then SH(S)satisfies Brown representability. 1.2 The Stable A1-Motivic Homotopy Category 34 Proof. See [40]. Theorem 1.66 (Naumann-Østvaer-Spitzweck):Let Sbe a Noetherian scheme of finite Krull dimension. (1) Let M●be an Adams-graded Landweber exact MU●-module. There is a Tate spectrum E∈Ob(SH(S)cell)and an isomorphism of homology theories on SH(S)where E●,●(−)=MGL●,●(−)⊗MU●(∗)M●. (2) If M●is a graded MU●-algebra then Eadmits a quasi-multiplication which represents the ring structure on the Landweber exact theory. Proof. See [41]. Definition 1.67 (Complex Betti Realization):The complex realization functor on Sm/C induces the following functors on the unstable and the stable A1-homotopy category (−)(C)∶Sm/C→SmMfd , X ↦X(C), H(Spec(C))→Spc ,ReC∶SH(Spec(C))→Sp. Reference. See [64, Chapter 3.1]. Definition 1.68 (Real Betti Realization):Let C2∶=Gal(C/R)be the absolute Galois group of R. Let every X(C)∈Ob(Sm/R)admit a group action of C2by complex conjugation. Let ΦC2be the geometric fixed point functor. There exists the following functor ReR∶SH(Spec(R))ReC2 →Sp(C2)ΦC2 →Sp. Reference. See [64, Chapter 3.2]. Remark 1.69 (Slice Filtration):Let Sbe a finite dimensional and Noetherian scheme. (1) The inclusion functor inadmits a right adjoint rnand establishes a filtration in∶Σn P1(SH(S)eff)→SH(S)∶rn, fn∶=in○rn∶SH(S)→SH(S), ⋯⊂Σn+1 P1(SH(S)eff)⊂Σn P1(SH(S)eff)⊂⋯⊂SH(S). 1.2 The Stable A1-Motivic Homotopy Category 35 (2) The slice filtration and the n-th slices sn(E)in the distinguished triangles of Eare ⋯→fn+1(E)→fn(E)→⋯→E, fn+1(E)→fn(E)→sn(E)→fn+1(E)[1]. (3) The slice filtration implies the so called slice spectral sequence s−q(E)p+q,n(F)⇒Ep+q,n(F), which is comparable to the Atiyah-Hirzebruch spectral sequence. Proof. See [14, Chapter 4] and the introduction to [4, Chapter 13.1]. Example 1.70 (Effective Algebraic K-Theory Spectrum):Let Sbe a finite dimensional and Noetherian scheme. The effective algebraic K-theory spectrum over Sis kglS∶=f0(KGLS). Proof. See [2, Definition 3.19 and Exercise 3.5]. Example 1.71: Let kbe a field of characteristic zero. Voevodsky’s motivic EilenbergMacLane spectrum over S=Spec(k)satisfies MZS=s0(KGLS). Proof. See [2, Definition 3.19 and Exercise 3.5]. Remark 1.72 (Homotopy t-Structure):Let Sbe finite dimensional and Noetherian. (1) Let SH(S)≥dbe the triangulated subcategory of SH(S)generated by the homotopy colimits and the extensions of {Σp,q s,t (Σ∞ P1(X)+)∣X∈Ob((Sm/S)Nis), p −q≥d}. (2) A spectrum in SH(S)≥dis called d-connective or else connected if d=0. (3) The triangulated subcategory SH(S)≥0is the nonnegative part of a unique t-structure on SH(S)which is called the homotopy t-structure. (4) If kis a prefect field then Morel’s A1-connectivity theorem implies SH(S)≥d={F∈Ob(SH(Spec(k)))∣πp,q(F)=0if p−q<d}, SH(S)≤d={F∈Ob(SH(Spec(k)))∣πp,q(F)=0if p−q>d}. 1.2 The Stable A1-Motivic Homotopy Category 36 Proof. See [38, Chapter 5.2]. Definition 1.73 (Very Effective Spectra):Let Sbe a finite dimensional and Noetherian scheme. The triangulated category of very effective spectra SH(S)veff is the full triangulated subcategory of SH(S)generated by the homotopy colimits and extensions of {Σ∞ P1(X)+∣X∈Ob((Sm/S)Nis)}. Reference. See the exposition following [4, Lemma 13.2]. Example 1.74: Let Sbe a finite dimensional and Noetherian scheme. The motivic Thom spectrum over the base scheme Sis very effective MGLS∈Ob(SH(S)veff ). Proof. See [38, Lemma 13.1]. Definition 1.75 (Effective Homotopy t-Structure):Let Sbe a finite dimensional and Noetherian scheme. The category of very effective spectra is the non negative part of the effective homotopy t-structure SH(S)veff ∶=SH(S)eff ∩SH(S)≥0. Reference. See the introduction to [4, Chapter 13.1]. Remark 1.76 (Motivic Steenrod Algebra):Let kbe a perfect field and let l≠char(k). Let MFlbe Voevodsky’s motivic Eilenberg-MacLane spectrum in SH(Spec(k)) representing motivic cohomology theory with coefficients in Fl. (1) The motivic Steenrod algebra and the dual motivic Steenrod algebra are respectively A●,●∶=(MFl)●,●(MFl),A●,●∶=(MFl)●,●(MFl). (2) Voevodsky constructed reduced power operations on the motivic Steenrod algebra for fields of characteristic zero. These results were extendet by Hoyois-Kelly-Østvaer and by Frankland and Spitzweck. (3) The motivic Steenrod algebra is crucial for the proof of the Bloch-Kato conjecture. Proof. See [61]. 1.3 Mixed Motives 37 1.3 Mixed Motives Definition 1.77 (Group of Algebraic Cycles of X):Let Sbe a smooth separated scheme of finite type over a perfect field. (1) Let X∈Ob(Sm/S)and let zr(X)be the free abelian group generated by the closed integral subschemes of Xof dimension r. (2) The group of algebraic cycles of the scheme Xis the graded free abelian group z●(X)∶=⊕ r∈Z zr(X). Definition 1.78 (Finite S-Correspondences):Let Sbe a smooth separated scheme of finite type over a perfect field and let X, Y ∈Ob(Sm/S). The graded abelian group of finite S-correspondences is the following free abelian subgroup cS(X, Y )⊂z●(X×SY), X, Y ∈Ob(Sm/S) which is generated by the integral closed subschemes W⊂X×SYwith the properties: (1) The canonical projection morphism πX∶W→Xis finite. (2) The image πX(W)⊂Xis an irreducible component of X. Reference. See [45, Chapter 2.1] and [29, Chapter 3.1]. Definition 1.79 (Smooth Finite S-Correspondences):Let Sbe a smooth separated scheme of finite type over a perfect field. (1) Ob(FinCorr(Sm/S))∶=Ob(Sm/S). (2) HomFinCorr(Sm/S)([X],[Y])∶=cS(X, Y ),[X],[Y]∈Ob(FinCorr(Sm/S)). Reference. See [45, Chapter 2.1] and [29, Chapter 3.1]. Remark 1.80: Let Sbe a smooth separated scheme of finite type over a perfect field. Let X, Y ∈Ob(Sm/S)and let (f∶X→Y)∈Mor(Sm/S)and let also Γf⊂X×SYbe the graph of fover S. There exists a canonical inclusion functor iS∶Sm/S→FinCorr(Sm/S), X ↦[X],(f∶X→Y)↦Γf. 1.3 Mixed Motives 38 Definition 1.81 (Presheaves with Transfer):Let Sbe a smooth separated scheme of finite type over a perfect field. The category of presheaves with transfer over Sis the following category of additive functors PShtr(Sm/S,Ab)∶=Fun⊕(FinCorr(Sm/S)op,Ab). Definition 1.82 (Nisnevich Sheaf with Transfer):Let Sbe a smooth separated scheme of finite type over a perfect field. Let iS∶Sm/S→FinCorr(Sm/S)be the inclusion functor. A Nisnevich sheaf with transfer over Sis a presheaf with transfer Fsuch that F○iSis a sheaf on (Sm/S)Nis. The full subcategory of Nisnevich sheaves with transfer over Sis Shtr((Sm/S)Nis ,Ab)⊂PShtr((Sm/S)Nis ,Ab). Reference. See [45, Chapter 2.3] and [29, Chapter 3.1]. Example 1.83: Let Sbe a smooth separated scheme of finite type over a perfect field. Let X∈Ob(Sm/S)and let also the presheaf with transfer over Srepresented by [X]via the fully faithful Yoneda embedding be FinCorr(Sm/S)op →PShtr(Sm/S,Ab),[X]↦Ztr S(X)(−)∶=cS(−, X). Reference. See [45, Chapter 2.3] and [29, Chapter 3.1]. Definition 1.84 (Effective Mixed Motives):Let Sbe a smooth separated scheme of finite type over a perfect field. The symmetric monoidal triagulated category of effective mixed motives over Sis the homotopy category of the following stable model category DMeff(S)∶=D(Shtr((Sm/S)Nis,Ab))[{Ztr S(X×SA1 S)π∗ X →Ztr S(X)}−1 X]. Reference. See [12, Proposition 3.6] and [29, Chapter 3.1]. Definition 1.85 (Big Mixed Motives):Let Sbe a smooth separated scheme of finite type over a perfect field. The symmetric monoidal triagulated category of big mixed motives over Sis the homotopy category of the following stable model category ZS(1)∶=coker(Ztr S(S)→Ztr S(P1 S))[−2],,ZS(n)∶=ZS(1)⊗S⋯⊗SZS(1) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n-times , n ≥1, DM(S)=Ho(SpΣ(PShtr((Sm/S)Nis,Ab)A1-stable,ΣZS(1))A1-stable),ΣZS(1)∶=ZS(1)⊗S(−). Reference. See [29, Definition 3.1.4]. 1.3 Mixed Motives 39 Remark 1.86: Let Sbe a smooth separated scheme of finite type over a perfect field. The Tate motive ZS(1)becomes ⊗-invertible in the triangulated category DM(S)and 1S∶=ZS(0),ZS(1)⊗SZS(−1)=ZS(0),ZS(−n)∶=ZS(−1)⊗S⋯⊗SZS(−1) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n-times , n ≥1. Definition 1.87 (Geometric Mixed Motives):Let Sbe a smooth separated scheme of finite type over a perfect field. The symmetric monoidal triangulated category of geometric mixed motives over Sis the following full subcategory of compact objects DMgm(S)∶=DM(S)c. Theorem 1.88: Let Sbe a smooth separated scheme of finite type over a perfect field. There exists the following commutative diagram of symmetric monoidal locally presentable stable ∞-categories where the horizontal functors are fully faithful. DMeff gm(S)→→ ↓↓ DMeff(S) ↓↓ DMgm(S)→→DM(S) Proof. See [45, Theorem 2.18]. Theorem 1.89 (Röndigs-Østvaer):Let kbe a field of characteristic zero. There exists the following equivalence of symmetric monoidal triangulated categories Ho(ModMZk(SH(Spec(k)))A1-stable)≃DM(Spec(k)). Proof. See [49]. Theorem 1.90: Let Sbe a Noetherian scheme of finite Krull dimension. There exists the following commutative diagram where every functor is symmetric monoidal and admits a right adjoint. Sm/SH⊗ ∗(S)SH⊗(S)eff SH⊗(S) FinCorr(Sm/S)DM⊗(S)eff DM⊗(S) iS (−)+ Reference. See [11, Remarque 2.6 (2)]. 1.3 Mixed Motives 40 Theorem 1.91 (Six-Functor Formalism):Let Sbe a Noetherian scheme of finite Krull dimension. Let also f∶X→Ybe a morphism over S. (1) The functor f∗is symmetric monoidal. If fis seperated of finite type then there exists and adjunction f!⊣f!. The triangulated category DM(X)admits a closed symmetric monoidal structure. The six functors satisfy the following adjunctions HomDM(X)(f∗(L), K)≅HomDM(Y)(L, f∗(K)), HomDM(Y)(f!(L), K)≅HomDM(X)(L, f!(K)), HomDM(X)(M⊗XK, L)≅HomDM(X)(M, HomDM(X)(K, L)). (2) There exists a natural transformation which is also an isomorphism if fis proper αf∶f∗→f!. (3) For any smooth and separated morphism f∶X→Sof relative dimension dthere exists a canonical natural isomorphism of functors βf∶f∗→f!○(−)(−d)[−2d]. (4) If fαβ is a separated morphism of finite type then any cartesian square implies the following base change natural isomorphisms Xδ fδγ →→ gδα ↓↓ Xγ gγβ ↓↓ Xαfαβ →→Xβ g∗ γβ ○fαβ!→fδγ!○g∗ δα , gδα∗○f! δγ →f! αβ ○gγβ∗. (5) If f∶Y→Xis separated and of finite type then there exist natural isomorphisms f!(K)⊗XL→f!(K⊗Yf∗(L)), HomDM(X)(f!(L), K)→f∗(HomDM(Y)(L, f!(K))), f!(HomDM(X)(L, M))→HomDM(Y)(f∗(L), f!(M)). 1.3 Mixed Motives 41 (6) (Localization):For any closed immersion i∶Z→Swith complementary open immersion j∶U→Sthere exists a distinguished triangle of natural transformations j!○j!→Id →i∗○i∗→j!○j!○(−)[1]. Proof. See [10, Introduction A.5.1]. Theorem 1.92 (Duality of the Six-Functor Formalism):Let Sbe a Noetherian scheme of finite Krull dimension and let KSbe ⊗-invertible in DM(S). Let f∶Y→Xbe a morphism of separated S-schemes of finite type. Let f∶X→Sbe a separated morphism of finite type and let KX∶=f!(KS), DX(M)∶=HomDM(X)(M, KX). (1) KXis a dualizing object in DM(X)and there exists a canonical isomorphism M≅DX(DX(M)). (2) There exists the following canonical isomorphism DX(M⊗XDX(N))≅HomDM(X)(M, N). (3) There exist the following canonical isomorphisms DY(f∗(M))≅f!(DX(M)) , f∗(DX(M))≅DY(f!(M)), DX(f!(N))≅f∗(DY(N)) , f!(DY(N))≅DX(f∗(N)). Proof. See [10, Introduction A.5.2]. Example 1.93: Let kbe a field and let S∈Ob((Sm/k)Nis )and let X∈Ob((Sm/S)Nis ). The six-functor formalism implies a natural isomorphism of abelian groups HomDM(S)(MS(X),ZS(q)[p])≅HomDM(Spec(k))(Mk(X),Zk(q)[p]) , p, q ∈Z. Proof. See [45, Theorem 2.17]. Definition 1.94 (Motivic Cohomology):Let kbe a field and let X∈Ob((Sm/k)Zar ) and let A∈Ob(Ab)be an abelian group. The motivic cohomology groups of Xare the hypercohomology groups of the cochain complexes of sheaves Zk(q)on the Zariski site Hp,q(−, A)∶=Hp Zar (−, Ak(q))∶Smop /k→AbZ×Z, 2.1 The Stable ∞-Category of Spitzweck Motives 48 (3) (Localization):For any closed immersion i∶Z→Swith complementary open immersion j∶U→Sthere exists a distinguished triangle of natural transformations j!○j!→Id →i∗○i∗→j!○j!○(−)[1]. Proof. See [52, Capter 9]. Remark 2.11: Let f∶X→Ybe a morphism of Noetherian separated schemes of finite Krull dimension over S=Spec(Z). The functor f∗is symmetric monoidal and admits a left adjoint f#if the morphism fis smooth. Proof. See [52, Capter 9]. Remark 2.12: The canonical forgetful functor U∶Ho(ModMZSpi X(SH(X)))→SH(X)of triangulated categories induces a six-functor formalism on the triangulated category of Spitzweck motives Ho(ModMZSpi X(SH(X))) which includes the base change property and the localization triangle and except possibly absolute purity. Proof. See [52, Capter 9]. Remark 2.13: The six-functor formalism for triangulated categories of Spitzweck motives extends to a six-functor formalism for stable ∞-categories of Spitzweck motives. Corollary 2.14: Let kbe a field of characteristic zero. Let MZkbe Voevodsky’s motivic Eilenberg-MacLane spectrum over Spec(k). The triangulated category of Spitzweck motives over the Spec(k)is symmetric monoidal equivalent to Voevodsky’s triangulated big category of mixed motives over Spec(k)since there exist the following equivalences DM(Spec(k))≃Ho(ModMZk(SH⊗(Spec(k)))A1-stable) ≃Ho(ModMZSpi k(SH⊗(Spec(k)))A1-stable). Proof. This follows from Theorem 2.1 and Theorem 1.89. Remark 2.15: Let kbe a field of characteristic zero. There exists the equivalence Ho(ModMZSpi k∧HQ(SH⊗(Spec(k))Q)A1-stable)≃Ho(ModMZk∧HQ(SH(Spec(k))Q)A1-stable). Proof. This follows from Theorem 1.118 and Corollary 2.14. 2.1 The Stable ∞-Category of Spitzweck Motives 49 Definition 2.16 (Geometric Spitzweck Motives):The symmetric monoidal triangulated category of geometric Spitzweck motives over Xis the full triangulated subcategory of compact objects in the symmetric monoidal triangulated category of Spitzweck motives (DMgm(X),⊗)∶=(DM(X)c,⊗),DMgm(X)⊂DM(X). Conjecture 2.17 (Beilinson-Soul´e):Let Dbe a Dedekind domain of mixed characteristic and let S=Spec(D). If Xis a smooth scheme over Sthen HomSH(S)(Σ∞ P1(X)+,MZSpi S(q)[p])=0if q≥0, p <0or else q>0, p =0. Reference. See the exposition following [54, Theorem 3.3]. Example 2.18: A1 Sand Gm,S over a number field S=Spec(k)and also Spec(Z)and any localization of a number ring satisfies the Beilinson-Soul´econjecture. Smooth curves over finite fields satisfy the Beilinson-Soul´econjecture. Moduli spaces of genus 0curves with nmarked points M0,n and its Deligne-Mumford compactifications  M0,n are satisfying the Beilinson-Soul´econjecture for n≥3. Also P1 S/Xover a number field S=Spec(k) where Xis a finite set S-points of P1 Ssatisfies the Beilinson-Soul´econjecture. Proof. See [54, Example 3.5]. Theorem 2.19 (Beilinson-Lichtenbaum):Let Xbe a smooth scheme over S=Spec(D). Let ε∶X´et →XZar be the canonical functor of sites. Let Levine’s cycle complexes be ΓXZar (n)∈Ob(D(Sh(XZar ,Z))) where n∈Z. Let m>0be invertible on X. There exists a canonical isomorphism in the derived category D(Sh(XZar ,Z/mZ)) where ΓXZar (n)⊗L ZZ/mZ≅τ≤n(Rε∗(µ⊗n m,X´et )). Proof. See [54, Theorem 3.3]. Definition 2.20 (Motivic Cohomology):Let Dbe a Dedekind domain of mixed characteristic and let S=Spec(D). Let A∈Ob(Ab)and let X∈Ob((Sm/S)Nis). (1) Hp,q(X, Z)∶=HomDM(X)(ZX(0),ZX(q)[p]) , p, q ∈Z. (2) Hp,q(X, A)∶=HomDM(X)(ZX(0), AX(q)[p]) , p, q ∈Z. Reference. See the exposition following [54, Definition 3.1]. Example 2.21: Let Xbe a smooth scheme over S=Spec(D)and let A∈Ob(Ab). 2.2 Tate Motives 50 (1) Hp,q(X, A)=0if p∈Zand q<0. (2) Hp,q(X, A)=0if p>qand if Xis the spectrum of a local ring. (3) Hp,q(X, A)=0if p>q+dim(X). Proof. See [54, Theorem 3.2]. Theorem 2.22 (M. Spitzweck):Let Dbe a Dedekind domain of mixed characteristic and let S=Spec(D). Let also X∈Ob((Sm/S)Nis). There exist canonical isomorphisms HomSH(S)(Σ∞ P1(X)+,MZSpi S(q)[p])≅HomDM(X)(ZX(0),ZX(q)[p])≅Hp(X, q), p, q ∈Z. Proof. See [52, Theorem 7.18 and Corollary 7.19 and Corollary 7.20]. 2.2 Tate Motives Definition 2.23 (Tate Motives):Let Xbe a Noetherian separated scheme of finite Krull dimension. The symmetric monoidal triangulated category of Tate motives over Xis the following full localizing triangulated subcategory (DMT(X),⊗)∶=Loc(DM(X),⊗)({ZX(n)}n∈Z),DMT(X)⊂DM(X). Definition 2.24 (Geometric Tate Motives):Let Xbe a Noetherian separated scheme of finite Krull dimension. The symmetric monoidal triangulated category of geometric Tate motives over Xis the following full triangulated subcategory of compact objects (DMTgm(X),⊗)∶=(DMT(X)c,⊗). Example 2.25: Let S=Spec(Z). Let M0,n be the moduli space of genus 0algebraic curves with nmarked points. Let also  M0,n be the Deligne-Mumford compactification of M0,n. The following are Tate motives MS(M0,n),MS( M0,n)∈Ob(DMT(S)) , n ≥3. Proof. See [51, Chapter 3]. Example 2.26: Let S=Spec(C)and let Confn(R2)be the topological space of unordered configurations of npoints in the plane. For every positive natural number n there exists a smooth scheme Cnover Ssuch that the set of complex points Cn(C)with the canonical complex analytic topology is Confn(R2). The following are Tate motives MS(Cn(C))∈Ob(DMT⊗(S)) , n ≥1. 2.2 Tate Motives 51 Proof. See the introduction to [25, Chapter 1]. Remark 2.27: If kis a field of characteristic zero then there exist the following equivalences of triangulated categories DMT(Spec(k))Q≃DMT(Spec(k),Q),DMAT(Spec(k))Q≃DMAT(Spec(k),Q). Proof. This follows from Remark 2.15. See also Definition 4.53 and Definition 4.51. Definition 2.28 (Multiplicative Periodization):Let Xbe any base scheme. A multiplicative (n, m)-periodization of a motivic E∞-algebra Eis a graded motivic E∞-algebra PE whose multiplication morphism satisfies the following canonical equivalence E∈Ob(CAlg(SH⊗(X))) , P E ∈Ob(CAlg(SH⊗(X)Z)) , P E ∶=⊕ k∈Z E∧Snk,mk, (E∧Snk,mk)∧(E∧Snk′,mk′)≃E∧Sn(k+k′),m(k+k′), k, k′∈Z. Reference. See the exposition following [23, Theorem 1.1]. Example 2.29: Let Xbe a Noetherian separated scheme of finite Krull dimension. The following motivic E∞-algebras are multiplicatively (2,1)-periodizable KGLX,MGLX,MZX∈Ob(CAlg(SH⊗(X))). Proof. See [23, Corollary 4.11]. Example 2.30: Let Xbe a Noetherian separated scheme of finite Krull dimension. The E∞-algebra MZSpi Xadmits a unique E∞-orientation. The E∞-algebra MGLXadmits a multiplicative periodization. The E∞-orientation φ∶MGLX→MZSpi Ximplies therefore MZSpi Xalso to admit a multiplicative (2,1)-periodization. Proof. See [52, Remark 10.2]. Remark 2.31: Let Dbe a Dedekind domain of mixed characteristic and let S=Spec(D). An explicit periodization of the E∞-algebra MZSpi Swas constructed by M. Spitzweck. Reference. See [52, Appendix C]. Example 2.32: Let Xbe a Noetherian separated scheme of finite Krull dimension. Any E∈Ob(CAlg(SH⊗(X))) with an E∞-orientation is multiplicatively periodizable. 2.2 Tate Motives 52 Proof. See the exposition following [23, Theorem 1.1]. Corollary 2.33: Let Xbe a Noetherian separated scheme of finite Krull dimension. Spitzweck’s motivic E∞-algebra over Xis multiplicatively (2,1)-periodizable with MZSpi X∈Ob(CAlg(DM⊗(X))) ,PMZSpi X∈Ob(CAlg(DM⊗(X)Z)), PMZSpi X∶=⊕ k∈Z MZSpi X∧S2k,k. Proof. This follows from Example 2.30 and Definition 2.28. Definition 2.34: Let Xbe Noetherian separated scheme of finite Krull dimension and GrZ 0(−)∶DM⊗(X)Z→DM⊗(X),⊕ n∈Z En↦E0. Remark 2.35: Let Xbe a Noetherian separated scheme of finite Krull dimension. There exists an equivalence of motivic E∞-algebras MZSpi X ≃ →GrZ 0(PMZSpi X). Proof. See the exposition following [56, Definition 4.1]. Definition 2.36: Let Xbe a Noetherian separated scheme of finite Krull dimension. Let E∈Ob(CAlg(SH⊗(X))). The stable ∞-category of cellular E-module spectra in SH⊗(X)is the smallest full stable subcategory of the stable ∞-category SH⊗(X)which includes {E∧Sn,1}n∈Zand is closed under small colimits ModE(SH⊗(X))cell ⊂SH⊗(X). Remark 2.37: Let Xbe a Noetherian separated scheme of finite Krull dimension. Let E∈Ob(CAlg(SH⊗(X))). Since the motivic spheres {Sn,1}n∈Zare ⊗-invertible in SH⊗(X), the E-module spectra {E∧Sn,1}n∈Zare ⊗-invertible in ModE(SH⊗(X)). Proof. See [23, Example 2.8]. Example 2.38: Let Xbe a Noetherian separated scheme of finite Krull dimension. There exists a symmetric monoidal equivalence of stable ∞-categories DMT⊗(X)≃DM⊗(X)cell ≃ModMZSpi X(SH⊗(X))cell. 2.2 Tate Motives 53 Definition 2.39: Let the symmetric monoidal stable ∞-category of Z-graded chain complexes in the category of abelian groups Ab with the Day convolution ⊗-product be D⊗(Ab)Z∶=Fun(N(Z)⊗,D⊗(Ab))⊗Day . Definition 2.40 (∞-Category of Z-Graded Spectra):Let the symmetric monoidal stable ∞-category of Z-graded spectra with the Day convolution ⊗-product be (Sp⊗)Z∶=Fun(N(Z)⊗,Sp⊗)⊗Day . Theorem 2.41 (M. Spitzweck and H. Heine):Let Xbe a Noetherian separated scheme of finite Krull dimension. Let Ebe multiplicatively (n, 1)-periodizable. There exists a Z-graded E∞-algebra AXand a symmetric monoidal equivalence of stable ∞-categories E∈Ob(CAlg(SH⊗(X))) , AX∈Ob(CAlg((Sp⊗)Z)), ModE(SH⊗(X))cell ≃ModAX((Sp⊗)Z). Proof. See [23, Theorem 4.4]. Example 2.42: Let Xbe a Noetherian separated scheme of finite Krull dimension. Since the E∞-algebra MZSpi Xis multiplicatively (2,1)-periodizable there exists a Z-graded E∞-algebra AXand a symmetric monoidal equivalence of stable ∞-categories AX∈Ob(CAlg((Sp⊗)Z)), DMT⊗(X)≃ModAX((Sp⊗)Z). Proof. This follows from Theorem 2.41 and Corollary 2.33 and Example 2.38. Theorem 2.43 (M. Spitzweck):Let Xbe a Noetherian separated scheme of finite Krull dimension. There exists a Z-graded E∞-algebra AXand a symmetric monoidal equivalence of stable ∞-categories AX∈Ob(CAlg(D⊗(Ab)Z)), DMT⊗(X)≃ModAX(D⊗(Ab)Z). Proof. See [52, Corollary C.3]. 54 3 Norm Functors in Homotopy Theory 3.1 Norm Functors in Motivic Homotopy Theory Theorem 3.1: There exists the following Quillen equivalence of model categories C[−]∶sSetJ⇆sSet-CatB∶N∆, LC[−]∶Ho(sSetJ)⇆Ho(sSet-CatB)∶RN∆. Proof. See [19, Theorem 1.38]. Corollary 3.2: If C∈Ob(sSet-CatB)is a category enriched in sSet such that all of its mapping spaces are Kan complexes then the simplicial set N∆(C)is an ∞-category. Proof. See [19, Corollary 1.39]. Example 3.3: Let CMbe a simplicially enriched model category. Let X, Y ∈Ob(CM). If Xis cofibrant and Yfibrant then MapCM(X, Y )is a Kan complex. In particular are the mapping spaces of the full subcategory of fibrant and cofibrant objects Cfc MKan complexes. Therefore is the simplicial set N∆(Cfc M)an ∞-category. Proof. See [19, Example 1.40 (1)]. Example 3.4 (∞-Category of ∞-Categories):The category of small ∞-categories qCat is a simplicially enriched category such that all of its mapping spaces are Kan complexes. The large ∞-category of small ∞-categories is therefore Cat∞∶=N∆(qCat). Proof. See [17, Example 2.51]. Definition 3.5 (Localization of ∞-Category):Let Cbe an ∞-category and let Wbe a class of morphisms in C. A functor F∶C→Dexhibits Das the ∞-category obtained from Cby localization at the class Wif for every ∞-category Ethe functor of ∞-categories (−)○F∶Fun(D,E)→Fun(C,E)is a fully faithful embedding, whose essential image is the set of functors which maps each morphism in Wto an equivalence in E. The ∞-category Dis unique up to equivalence and we write D≃C[W−1]. Reference. See [31, Theorem 1.3.4.1]. 3.1 Norm Functors in Motivic Homotopy Theory 55 Definition 3.6 (Underlying ∞-Category of a Model Category):Let CMbe a model category and let Cc Mbe the full subcategory of cofibrant objects in CM. Let Wbe the class of weak equivalences in Cc Mand let Dbe an ∞-category. A functor F∶N(Cc M)→D exhibits Das the underlying ∞-category of CMif it induces an equivalence of ∞-categories D≃N(Cc M)[W−1]. Reference. See [31, Theorem 1.3.4.15]. Theorem 3.7 (Dwyer-Kan):Let CMbe a simplicial model category and let Cfc Mbe the full subcategory of fibrant and cofibrant objects in CM. Let Wbe the class of weak equivalences in Cc M. There exists a functor Θ∶N∆(Cc M)→N∆(Cfc M)which exhibits N∆(Cfc M)as the underlying ∞-category of CMi.e. N∆(Cc M)[W−1]≃N∆(Cfc M). Proof. See [31, Theorem 1.3.4.20]. Example 3.8 (∞-Category of Spaces):The model category of simplicial sets sSetQis a simplicial model category. The full subcategory sSetfc Qof fibrant and cofibrant objects in sSetQis the category of Kan complexes Kan. The underlying ∞-category of the model category of simplicial sets is the locally presentable ∞-category of spaces Spc ∶=N∆(sSetfc Q)=N∆(Kan). Proof. See [19, Example 1.40 (2)]. Example 3.9: Let CMand DNbe simplicial model categories. The following (simplicial) Quillen adjunction of model categories extends to an adjunction of ∞-categories F∶CM⇆DN∶G, N∆○F∶N∆(Cfc M)⇆N∆(Dfc N)∶N∆○G, LF∶Ho(N∆(Cfc M))⇆Ho(N∆(Dfc N))∶RG. Proof. See [32, Proposition 5.2.4.6 and Remark 5.2.4.7]. Definition 3.10 (Localization Functor):A functor F∶C→Dbetween ∞-categories is called a localization functor if it admits a fully faithful right adjoint functor G∶D→C. 3.1 Norm Functors in Motivic Homotopy Theory 56 Example 3.11: Let Cbe an ∞-category and let Dbe a full subcategory of C. Let also the inclusion functor i∶D↪Cadmit a left adjoint functor L. If Wis a class of morphisms in Csuch that for every morphism f∈Wthe morphism L(f)is an equivalence in D then there exists an equivalence of ∞-categories D≃C[W−1]. Proof. See [31, Example 1.3.4.3]. Remark 3.12: Let Cbe some ∞-category with a terminal object ∗and finite colimits. Let C∗/be the coslice category. The following forgetful functor Uadmits a left adjoint x∈Ob(C), x+∶=x∐∗,C∗∶=C∗/,(−)+∶C⇆C∗∶U. Definition 3.13: Let Sbe a qcqs scheme. Let also PShA1(Sm/S,Spc)be the full subcategory of A1-invariant presheaves on the site of smooth schemes over Sand let PShA1(Sm/S,Spc)⊂PSh(Sm/S,Spc), H(S)∶=Sh((Sm/S)Nis ,Spc)∩PShA1(Sm/S,Spc). Reference. See the introduction to [4, Chapter 2.2]. Remark 3.14: Let Sbe a qcqs scheme. The ∞-category H(S)is the pullback in Cat∞ along the following inclusion functors of full subcategories H(S)→→ ↓↓ PShA1(Sm/S,Spc) ↙↖ ↓↓ Sh((Sm/S)Nis ,Spc)↘↙→→PSh(Sm/S,Spc) Definition 3.15: Let Sbe a qcqs scheme. Let PShA1(Sm/S,Spc)be the full subcategory of A1-invariant presheaves over S. The canonical inclusion functors admit a left adjoint LA1∶PSh(Sm/S,Spc)⇆PShA1(Sm/S,Spc)∶i, LNis ∶PSh(Sm/S,Spc)⇆Sh((Sm/S)Nis,Spc)∶i. Remark 3.16: Let Sbe a quasi-compact and quasi-separated scheme and let also write H(S)∶=LNis(PSh(Sm/S,Spc))∩LA1(PSh(Sm/S,Spc)). Reference. [2, Definition 2.18]. 3.1 Norm Functors in Motivic Homotopy Theory 57 Remark 3.17: Let F∈Ob(PSh(Sm/S,Spc)). If Fis A1-invariant then LNis (F)has not to be and if Fis a Nisnevich sheaf on the site (Sm/S)Nis then LA1(F)has not to be. Remark 3.18: Let F∈Ob(PSh(Sm/S,Spc)). Since the ∞-categories PShA1(Sm/S,Spc) and Sh((Sm/S)Nis ,Spc)are stable under filtered colimits it follows LM(F)∶=colim n∈N(LA1○LNis)○n(F), LM(F)∈Ob(H(S)). Definition 3.19 (Motivic Equivalence):We call a morphism f∈Mor(PSh(Sm/S,Spc)) a Nisnevich equivalence and A1-equivalence and motivic equivalence if the morphism LNis(f)and LA1(f)and LM(f)is an equivalence, respectively. Reference. See the introduction to [4, Chapter 2.2]. Definition 3.20 (Extensive ∞-Category):An ∞-category Cis called extensive if it admits finite coproducts and if for all x, y ∈Ob(C)the coproduct x×x∐yy∈Ob(C)exists and is an initial object and if finite coproduct decompositions are stable under pullbacks. Reference. See [4, Definition 2.3]. Definition 3.21 (Extensive Coverage):Let Cbe some extensive ∞-category. Let also Sh∐(C,Spc)be the ∞-category of sheaves on Cwith respect to the Grothendieck topology generated by the covers {f∶∐α∈Axα→x}such that fis an equivalence and Aa finite set. Reference. See the exposition following [4, Lemma 2.2]. Definition 3.22 (Non-Abelian Derived ∞-Category):Let Cbe a small ∞-category with all finite coproducts. Let the full subcategory of finite products preserving presheaves be PShΣ(C,Spc)⊂PSh(C,Spc). Reference. See the introduction to [4, Chapter 2.1]. Proposition 3.23: Let Cbe a small ∞-category which admits finite coproducts. The ∞-category PShΣ(C,Spc)is compactly generated and stable under sifted colimits. Furthermore preserves the left adjoint functor LΣto the inclusion functor isifted colimits LΣ∶PSh(C,Spc)⇆PShΣ(C,Spc)∶i. Proof. See the introduction to [4, Chapter 2.1]. 3.1 Norm Functors in Motivic Homotopy Theory 64 Definition 3.51: Let Corr(Sch,all,f´et)be the 2-category of correspondences in Sch with: (1) Ob(Corr(Sch,all,f´et))=Ob(Sch). (2) The 1-morphisms are correspondences of schemes Yf ←Xp →Swhere pis a finite ´etale morphism. The 2-morphisms are the isomorphisms of 1-morphisms. Remark 3.52: Let SmQP/Sbe the category of smooth quasi-projective schemes over S. Since every smooth scheme over Sadmits an open cover by quasi-projective schemes over Sthe inclusion of categories SmQP/S⊂Sm/Sinduces an equivalence of ∞-categories Sh((SmQP/S)Nis,Spc)≃Sh((Sm/S)Nis,Spc). Proof. See the introduction to [4, Chapter 6.1]. Remark 3.53: If p∶X→Sis a finite locally free morphism of schemes and Ya smooth scheme over Xthen Rp(Y)is smooth and quasi-projective over S. It also follows p∗∶PSh(SmQP/X,Spc)→PSh(SmQP/S,Spc), p∗(SmQP/X)⊂SmQP/S, p⊗(SmQP/X+)⊂SmQP/S+. Proof. See the introduction to [4, Chapter 6.1]. Remark 3.54: Let SmQP/Sbe the category of smooth quasi-projective schemes over S. (1) Let flf be the class of locally free morphisms of schemes. If p∶X→Sis a finite locally free morphism of schemes and Rpthe Weil restriction functor along the morphism pthen there exists the following pair of adjoint functors p∗∶SmQP/S→SmQP/X∶Rp. Through Barwick’s unfurling construction we obtain the following induced functor Corr(Sch,all,flf)→Cat1, S ↦SmQP/S,(Yf ←Xp →S)↦Rp○f∗. (2) Let clopen be the class of clopen immersions in SmQP/S. The functors Rpand f∗ preserve clopen immersions and there exists a canonical equivalence of categories SmQP/S+≃Corr(SmQP/S,clopen,all). 3.1 Norm Functors in Motivic Homotopy Theory 65 (3) Through the functoriality of 2-categories of correspondences we obtain the functor SmQP⊗ +(−)∶Corr(Sch,all,flf)→Cat1, S ↦SmQP/S+,(Yf ←Xp →S)↦p⊗○f∗. The forgetful functor below induces a unique lift of the functor above to the next U∶CAlg(Cat1)→Cat1, SmQP⊗ +(−)∶Corr(Sch,all,flf)→CAlg(Cat1), S↦SmQP/S+,(Yf ←Xp →S)↦p⊗○f∗. Proof. See the introduction to [4, Chapter 6.1]. Remark 3.55: (1) Let Ube the canonical forgetful functor. There exists an adjunction of ∞-categories such that the value of the functor PShΣon a symmetric monoidal ∞-category is PShΣ(−)∶CAlg(Cat∞)→CAlg(Catsift ∞)∶U , C⊗↦PShΣ(C,Spc)⊗Day . (2) Let ibe the inclusion functor and let the following functor be the composition i∶CAlg(Cat1)↪CAlg(Cat∞), (PShΣ○i○SmQP⊗ +)(−)∶Corr(Sch,all,flf)→CAlg(Catsift ∞), S↦PShΣ∗(SmQP/S,Spc). (3) Let OCat∞be the ∞-category of the ∞-categories that are endowed with a certain set of equivalence classes of objects. Let MCat∞be the ∞-category of the ∞- categories that are endowed with a certain set of equivalence classes of morphisms. Let Mbe the class of motivic equivalences in the ∞-category PShΣ∗(SmQP/S,Spc). Since the functors f∗and p⊗preserve motivic equivalences and the functor p⊗is stable under the smash product the functor above extends to the following functor Corr(Sch,all,flf)→CAlg(MCatsift ∞), S ↦(PShΣ∗(SmQP/S,Spc),M). Proof. See the exposition following [4, Remark 6.1]. Remark 3.56: 3.1 Norm Functors in Motivic Homotopy Theory 66 (1) Let Ebe the class of equivalences in a prescribed ∞-category C. By the universal property of localization of symmetric monoidal ∞-categories the ∞-category H⊗ ∗(S) is the image of (PShΣ∗(SmQP/S,Spc),M)by a partial left adjoint to the functor CAlg(Catsift ∞)→CAlg(MCatsift ∞),C→(C,E). (2) We therefore obtain a functor H⊗ ∗(−)∶Corr(Sch,all,flf)→CAlg(Catsift ∞), S ↦H⊗ ∗(S),(Yf ←Xp →S)↦p⊗○f∗. (3) There is f∗(SV)≃Sf∗(V)and if pis finite ´etale then p⊗(SV)≃SRp(V)and also Corr(Sch,all,f´et)→CAlg(OCatsift ∞), S ↦(H⊗ ∗(S),{SV}V/S). Proof. See the exposition following [4, Equation (6.2)]. Remark 3.57: (1) SH⊗(S)is the image of (H⊗ ∗(S),{SV}V/S)by a partial left adjoint to the functor CAlg(Catsift ∞)→CAlg(OCatsift ∞),C⊗↦(C⊗, π0(Pic(C⊗))). (2) We therefore obtain a functor SH⊗(−)∶Corr(Sch,all,f´et)→CAlg(Catsift ∞), S ↦SH⊗(S),(Yf ←Xp →S)↦p⊗○f∗. (3) By starting with the category SmQP instead of SmQP+we obtain also the functor H⊗(−)∶Corr(Sch,all,flf)→CAlg(Catsift ∞), S ↦H⊗(S),(Yf ←Xp →S)↦p∗○f∗. Proof. See the exposition following [4, Equation (6.2)]. Corollary 3.58: There exist the following canonical natural transformations of functors H⊗(−)(−)+ →H⊗ ∗(−)Σ∞ →SH⊗(−). Proof. See [4, Remark 6.3]. 3.2 Norm Functors in Stable Equivariant Homotopy Theory 67 Remark 3.59: The natural transformations above are constructed analogously to the functors themselfs. In particular is the natural transformation Σ∞constructed via Corr(Sch,all,f´et)×∆1→CAlg(OCatsift ∞), (S, 0→1)↦((H⊗ ∗(S),{1S})→(H⊗ ∗(S),{SV}V/S)). Proof. See [4, Remark 6.3]. Remark 3.60: Let Sbe a scheme. The functor SH⊗(−)recovers the symmetric monoidal structure of the stable ∞-category SH⊗(S)via the following composition of functors FinSet∗≃ →Corr(FinSet,inj,all)↪Corr(FinSet)→Corr(FEt/S)→Corr(Sch,all,f´et)SH⊗(−) →Cat∞. Proof. See [4, Remark 6.4]. Theorem 3.61 (Bachmann-Hoyois):There exists the following functor SH⊗(−)∶Corr(Sch,all,f´et)→CAlg(Catsift ∞), S↦SH⊗(S),(Yf ←Xp →S)↦p⊗○f∗. Proof. See the introduction to [4, Chapter 6.1]. 3.2 Norm Functors in Stable Equivariant Homotopy Theory Definition 3.62 (Delooping groupoid):Let Gbe a group. The delooping groupoid BG of the group Gis the following category: (1) Ob(BG)={G}. (2) Mor(BG)is the set of elements of Gand the composition is the group operation. Definition 3.63 (Groupoid):Let Cbe a small category. (1) The category Cis called a groupoid if all morphisms of Care isomorphisms. (2) The category Cis called a finite groupoid if it is a groupoid and if it admits a finite set of objects and a finite set of morphisms. Example 3.64: The delooping groupoid BGof a group Gis a groupoid. 3.2 Norm Functors in Stable Equivariant Homotopy Theory 68 Example 3.65: The delooping groupoid BGof a finite group Gis a finite groupoid. Example 3.66: The nerve N(C)of a small category Cis a Kan complex if and only if the category Cis a groupoid. Definition 3.67 (2-Category of Groupoids):Let the 2-category of groupoids Grpd be: (1) Ob(Grpd)is the set of groupoids. (2) A morphism of groupoids is a functor. A 2-morphism of groupoids is a natural transformation of functors and necessarily a natural isomorphism. Remark 3.68: The category of groupoids fits also in the following diagram of categories Cat∞Ho →Cat Core →Grpd ↪Cat N →sSet π0 →Set. Remark 3.69: The homotopy coherent nerve induces an adjunction of ∞-categories Core ∶qCat ⇆Kan ∶i, N∆○Core ∶Cat∞⇆Spc ∶N∆○i. Proof. See [17, Example 3.53]. Definition 3.70 (Fundamental Groupoid of X):The fundamental grupoid Π1(X)of X∈Ob(TopQ)is the following category: (1) Ob(Π1(X))=X. (2) HomΠ1(X)(x, y)is the set of homotopy classes of continuous paths from xto y. Corollary 3.71: Let X∈Ob(TopQ)and let x∈Xbe a point. The fundamental group of Xwith the basepoint xis isomorphic to the following automorphism group π1(X, x)≅AutΠ1(X)(x). Proof. See [19, Example 1.1]. Remark 3.72 (Homotopy Hypothesis): (1) The underlying ∞-category of TopQis the ∞-category Grpd∞of ∞-groupoids. 3.2 Norm Functors in Stable Equivariant Homotopy Theory 69 (2) The following Quillen equivalence of model categories induces an equivalence between the ∞-category of spaces and the ∞-category of ∞-groupoids ∣−∣∶sSetQ⇆TopQ∶Sing, Spc ⇆Grpd∞. Proof. See [17, Commentary 1.40 and Commentary 1.43]. Remark 3.73: Let FinGrpd be the 2-category of finite groupoids. Let Grpd∞be the ∞- category of ∞-groupoids. There exists the following diagram of inclusions of categories FinGrpd↘↙→→ ↙↖ ↓↓ Grpd↘↙→→Grpd∞≃→→Spc ↙↖ ↓↓ Pro(FinGrpd)↘↙→→Pro(Spc) Example 3.74: If X∈Ob(TopQ)then Sing(X)is a Kan complex and therefore also an ∞-category. In particular is the fundamental groupoid of Xis equivalent to Π1(X)≃Ho(Sing(X)). Proof. See [17, Exercise 1.36]. Example 3.75 (Picard ∞-Groupoid):The Picard ∞-groupoid of a symmetric monoidal ∞-category C⊗is the maximal ∞-groupoid in C⊗spanned by the ⊗-invertible objects Pic(C⊗)∈Ob(Grp(Spc)). Definition 3.76 (Finitely Presented Morphism):A morphism h∈Mor(Pro(Spc)) is called finitely presented if it is a pullback h=f∗(g)of a morphism gof spaces X×ZY h=f∗(g) ↓↓ →→Y g∈Mor(Spc) ↓↓ Xf→→Z . Reference. See the introduction to [4, Chapter 9.1]. Example 3.77: The class of finitely presented morphisms in the ∞-category Pro(Spc) is closed under composition and base change and satisfies the 2-out-of-3property. 3.2 Norm Functors in Stable Equivariant Homotopy Theory 70 Proof. See [4, Lemma 9.2]. Definition 3.78 (Finite Covering Morphism):A morphism h∈Mor(Pro(Spc))is called a finite covering if it is finitely presented and if its fibers are (possibly empty) finite sets. Reference. See the exposition following [4, Lemma 9.3]. Example 3.79: Let Gbe a finite group and let i∶H↪Gbe a subgroup. The induced morphism of finite groupoids B(i)∶BH→BGis a finite covering morphism. Proof. See [4, Chapter 1.4]. Remark 3.80: There exist the following inclusions of sets of morphisms fcov ⊂fp ⊂Mor(Pro(FinGrpd)). Definition 3.81 (Category of Finite Covers of X):Let Xbe a profinite groupoid. Let the category of finite covers of Xbe the full subcategory FC/X⊂Pro(FinGrpd)/Xwith: (1) The set of objects in the category of finite covers of the profinite groupoid Xis Ob(FC/X)∶={(f∶Y→X)∣Y∈Ob(Pro(FinGrpd)) , f is a finite covering}. (2) A morphism from (f∶Y→X)to (f′∶Y′→X)is a commutative diagram Y g ↙↙ f ↘↘ Y′ f′→→X Example 3.82: Let Gbe a finite group. The following full subcategory of finite covers of the delooping groupoid BGis equivalent to the category of finite G-sets G-FinSet ≃FC/BG,FC/BG⊂FinGrpd/BG⊂Pro(FinGrpd)/BG. Proof. See [4, Chapter 1.4]. Remark 3.83: Let Cbe some ∞-category. The functor of ∞-categories Fun(−,C)extends uniquely to a colimit preserving functor on the opposite ∞-category of pro-spaces Spcop Fun(−,C)→→ ↓↓ Cat∞ Pro(Spc)op ↗↗ 3.2 Norm Functors in Stable Equivariant Homotopy Theory 71 Proof. See the exposition following [4, Lemma 9.2]. Remark 3.84: Let X∈Ob(Pro(Spc)) be a profinite space. There is the following fully faithful functor whose image consists of the finitely presented morphims and which is ∫∶Fun(X, Spc)→Pro(Spc)/X. Proof. See [4, Lemma 9.3]. Remark 3.85: The following are symmetric monoidal embeddings natural in Xwith Fun(X, Spc)(−)+ →Fun(X, Spc)+↪Corr(Fun(X, Spc)) , X ∈Ob(Pro(Spc)op). Proof. See the exposition following [4, Equation 9.4]. Remark 3.86: Let p∶Y→Xbe a finitely presented map of profinite ∞-groupoids. (1) The functor p∗has a right adjoint p∗functor and a left adjoint p#functor and is p∗∶Fun(X, Spc)⇆Fun(Y, Spc)∶p∗, p#∶Fun(Y, Spc)⇆Fun(X, Spc)∶p∗. (2) The first symmetric monoidal functor restricts to the following one p⊗∶=Corr(p∗)∶Corr(Fun(Y, Spc))→Corr(Fun(X, Spc)), p⊗∶Fun(Y, Spc)+→Fun(X, Spc)+. Proof. See [4, Lemma 9.5]. Definition 3.87: Let Xbe a profinite groupoid. Let the 1-category of finite X-sets be X∈Ob(Pro(FinGrpd)) , X-FinSet ∶=Fun(X, FinSet),Fun(X, FinSet)⊂Fun(X, Spc). Proposition 3.88: Let X, Y ∈Ob(Pro(FinGrpd)) be profinite groupoids. (1) There exists the following equivalence of categories X-FinSet ≃FC/X. (2) If p∶Y→Xis a finitely presented morphism of profinite groupoids then the fibers of the morphism phave finitely many connected components. 3.2 Norm Functors in Stable Equivariant Homotopy Theory 72 Proof. See the exposition following [4, Lemma 9.5]. Proposition 3.89: Let X, Y ∈Ob(Pro(FinGrpd)) be profinite groupoids. (1) If p∶Y→Xis a finitely presented morphism then there exists an adjunction p∗∶X-FinSet ⇆Y-FinSet ∶p∗. (2) If in addition pis a finite covering then there is also the following adjunction p#∶Y-FinSet ⇆X-FinSet ∶p∗. Proof. See the exposition following [4, Lemma 9.5]. Definition 3.90 (∞-Catgory of Left Exact ∞-Categories):Let Catlex ∞be the ∞-category: (1) Ob(Catlex ∞)is the class of the ∞-categories with finite limits. (2) MapCatlex ∞(C,D)is the space of left exact functors between Cand D. Remark 3.91: (1) There exists the following functor FinSet⊗(−)∶Corr(Pro(FinGrpd),all,fp)→Catlex 1, X↦X-FinSet ,(Yf ←Xp →Z)↦p∗○f∗. (2) Since the functors f∗and p∗are left exact the functor above extends to the functor Corr(−)∶Catlex ∞→CAlg(Cat∞), (Corr ○FinSet⊗)(−)∶Corr(Pro(FinGrpd),all,fp)→CAlg(Cat2), X↦Corr(X-FinSet),(Yf ←Xp →Z)↦p⊗○f∗. (3) There exists the following symmetric monoidal subfunctor and we also set FinSet⊗ +(−)⊂(Corr ○FinSet⊗)(−)∶Corr(Pro(FinGrpd),all,fp)→CAlg(Cat1), X↦X-FinSet+,(Yf ←Xp →Z)↦p⊗○f∗. H⊗(−),H⊗ ∗(−)∶Corr(Pro(FinGrpd),all,fp)→CAlg(Catsift ∞), H⊗(−)∶=(PShΣ○FinSet⊗)(−),H⊗(X)∶=PShΣ(X-FinSet,Spc), H⊗ ∗(−)∶=(PShΣ○FinSet⊗ +)(−),H⊗ ∗(X)∶=PShΣ(X-FinSet+,Spc). 3.2 Norm Functors in Stable Equivariant Homotopy Theory 73 Proof. See the introduction to [4, Chapter 9.2]. Theorem 3.92 (Bachmann-Hoyois):There exists the following functor H⊗ ∗(−)∶Corr(Pro(FinGrpd),all,fp)→CAlg(Catsift ∞), X↦H⊗ ∗(X),(Yf ←Xp →Z)↦p⊗○f∗. Proof. See the introduction to [4, Chapter 9.2]. Example 3.93 (Orbit Category):Let Gbe a compact Lie group. The orbit category OGof Gis the full subcategory of the ∞-category of G-equivariant spaces G-Spc where Ob(OG)∶={G/H∣H⊂Gis a closed subgroup , G/His a coset space}. Proof. See [8, Definition 1.3.1]. Definition 3.94 (G-Spaces):Let Gbe a finite group with the discrete topology or else a compact Lie group. The symmetric monoidal ∞-category of G-equivariant spaces is G-Spc ∶=Fun(Oop G,Spc). Reference. See the introduction to [8, Chapter 1.1]. Example 3.95 (Bachmann-Hoyois): (1) If Gis a finite group then there exist the following symmetric monoidal equivalences H⊗(BG)≃G-Spc⊗,H⊗ ∗(BG)≃G-Spc⊗ ∗. (2) If B(f)∶BG→BHis induced by a group homomorphism f∶G→Hwith the kernel Nthen B(f)⊗∶H⊗ ∗(BG)→H⊗ ∗(BH)is the functor NH G/N○ΦN. (3) If p∶∗→BGthen p⊗(S1)∈Ob(H⊗ ∗(BG)) is the regular representation sphere. Proof. See the introduction to [4, Chapter 9.2]. Definition 3.96: Let X∈Ob(Pro(FinGrpd)). The compactly generated symmetric monoidal ∞-category Sp⊗(X)is obtained from the symmetric monoidal ∞-category H⊗ ∗(X)by inverting the compact object p⊗(S1)for each finite covering p∶Y→Xi.e. Sp⊗(X)∶=H⊗ ∗(X)[{p⊗(S1)−1}p]. Reference. See the introduction to [4, Chapter 9.2]. 3.3 The Étale Fundamental Group 80 Proof. See [58, Tag 0BN8]. Definition 3.124 (Finite Galois Cover):A finite ´etale morphism of schemes f∶Y→X is called a Galois cover if the scheme Yis connected and if ∣G∣=deg(f), G ∶=AutFEt/X(Y/X). Reference. See [58, Tag 03SF]. Definition 3.125 (Fiber Functor):Let Xbe a connected scheme and let kbe an algebraically closed field. Let x∶Spec(k)→Xbe a morphism of schemes and let f∶Y→X be a finite ´etale morphism. Let also the following functor be Fx∶FEt/X→Set , Y ↦HomSch/X(Spec(k), Y ). Reference. See the exposition following [58, Tag 0BL7]. Proposition 3.126: Let Xbe a scheme and let kbe an algebraically closed field and let x∶Spec(k)→Xbe a morphism of schemes. (1) The set Fx(Y)is finite for every Y∈Ob(FEt/X). (2) (FEt/X, Fx)is a Galois category. Proof. See [58, Tag 0BNB]. Definition 3.127 (´ Etale Fundamental Group):Let Xbe a connected scheme and let k be an algebraically closed field and let x∶Spec(k)→Xbe a morphism of schemes. The ´etale fundamental group of Xwith the basepoint xis the profinite group  π´et 1(X, x)∶=AutFun(FEt/X,Set)(Fx). Reference. See [58, Tag 0BNC]. Corollary 3.128: Let Xbe a connected scheme and let kbe an algebraically closed field and let x∶Spec(k)→Xbe a morphism. There is a canonical equivalence of categories FEt/X ≃ → π´et 1(X, x)-FinSet , Y ↦(Fx(Y), φFx). Proof. See [58, Tag 0BND]. 3.4 The Profinite Étale Fundamental Groupoid 81 Example 3.129: Let kbe a field and let x∶Spec(k)→Xbe a morphism of schemes. (1) There exists a canonical isomorphism of profinite groups  π´et 1(Spec(k), x)≅Gal(ksep/k). (2) There exists a canonical equivalence of categories FEt/Spec(k)≃ →Gal(ksep/k)-FinSet. (3) If kis an algebraically closed field then its ´etale fundamental group is trivial {e}≅Gal(ksep/k)≅ π´et 1(Spec(k), x). Proof. See [58, Tag 0BNE]. Example 3.130: Let Xbe a connected variety over Spec(C). The GrothendieckRiemann existence theorem implies an isomorphism of profinite groups  π´et 1(X, x)≅ π1(X(C), x). Proof. See [58, Tag 03VD]. Proposition 3.131: Let Xbe a connected scheme and let kbe an algebraically closed field and let x∶Spec(k)→Xbe a morphism. There exists an equivalence of categories Sh(X´et ,Set)flc ≃ π´et 1(X, x)-FinSet. Proof. See [58, Tag 0DV5]. 3.4 The Profinite Étale Fundamental Groupoid Definition 3.132 (∞-Topos):An ∞-category Xis called an ∞-topos if there exists a small ∞-category Cand an accessible left exact localization functor L∶PSh(C,Spc)⇆X∶i. Reference. See [32, Definition 6.1.0.4]. Theorem 3.133 (Lurie):An ∞-category Xis an ∞-topos if and only if it satisfies the ∞-categorical Giraud’s axioms. Reference. See [32, Theorem 6.4.1.5]. 3.4 The Profinite Étale Fundamental Groupoid 82 Example 3.134: The ∞-category of presheaves on a small ∞-category Cis an ∞-topos X∶=PSh(C,Spc). Example 3.135 (Terminal ∞-Topos):The ∞-category of spaces Spc is a terminal object in the ∞-category of ∞-topoi T∞. Proof. See [46, Example 6.3]. Example 3.136 (Slice ∞-Topos):Let Xbe an ∞-topos and let U∈Ob(X). The slice ∞-category X/Uis an ∞-topos. Proof. See [46, Proposition 4.2]. Example 3.137 (´ Etale ∞-Topos):The small ´etale ∞-topos and the big ´etale ∞-topos of a scheme Xis the ∞-category of sheaves on the small ´etale site X´et and the big ´etale site (Sch/X)´et , respectively. Proof. See [46, Chapter 2.3]. Example 3.138 (Grothendieck Topos):Let Xbe an ∞-topos. The full subcategory of 0-truncated objects τ≤0(X)⊂Xis a Grothendieck topos. Proof. See [46, Chapter 2.8]. Remark 3.139: Every ∞-topos Xis powered over the terminal ∞-topos of spaces MapSpc(K, MapX(Y, X))≃MapX(Y, XK). Definition 3.140 (Homotopy Groups in ∞-Topos):Let Xbe an ∞-topos and let also F∈Ob(X). Let Sn=∂∆n+1be the simplicial n-sphere with the basepoint x∶ ∗ →Sn. Let evx∶FSn→Xbe the canonical morphism in the slice ∞-topos X/Fand let πn(F, x)∶=τ≤0(FSn), τ≤0(FSn)∈Ob(τ≤0(X/F)) , n ≥0. Reference. See [32, Definition 6.5.1.1]. Example 3.141: Let Spc be the ∞-topos of spaces. Let Xbe a space. The n-th homotopy group πn(X)of the space Xis precisely its n-th simplicial homotopy group. Proof. This follows from Example 3.135 and Definition 3.140. 3.4 The Profinite Étale Fundamental Groupoid 83 Example 3.142: Let Cbe a small ∞-category. Let PSh(C)be the ∞-topos of presheaves and let and let F∈Ob(PSh(C)) be a presheaf with some basepoint. The powering FSn is the limit of the constant diagram constF∶Sn→PSh(C). Limits in the category of presheaves are computed pointwise. The n-th homotopy group of Fin the ∞-topos PSh(C)is therefore the ordinary n-th homotopy group of the presheaf Fon Cand is πn(F(x))≅τ≤0(FSn(x))≅τ≤0(F(x)Sn), n ≥0. Proof. This follows from Example 3.134 and Definition 3.140. Definition 3.143 (Geometric Morphism):A geometric morphism f∶X→Yof ∞-topoi is a pair of adjoint functors f∗∶X→Y∶f∗. Definition 3.144: Let Cbe an accessible ∞-category which admits finite limits. Let the following be the full subcategory spanned by the accessible left exact functors Pro(C)⊂Fun(C,Spc)op. Example 3.145 (∞-Category of Shapes):The ∞-category of shapes is the following full subcategory spanned by the accessible left exact functors Pro(Spc)⊂Fun(Spc,Spc)op. Proof. See [32, Definition 7.1.6.1]. Definition 3.146 (Shape of ∞-Topos):Let Xbe an ∞-topos and let q∶X→Spc be the unique up to homotopy geometric morphism to the terminal ∞-topos of spaces. The shape Sh(X)of the ∞-topos Xis the accessible left exact functor (q∗○q∗∶Spc →Spc)∈Ob(Pro(Spc)). Reference. See [32, Definition 6.5.1.1]. Example 3.147 (Trivial Shape):We say an ∞-topos Xis of trivial shape if Sh(X)is equivalent to the identity functor on the ∞-category of spaces Spc. Proof. See [32, Definition 7.1.6.11]. Definition 3.148 (π-Finite Space):A space X∈Ob(Spc)is called π-finite if it satisfies: 3.4 The Profinite Étale Fundamental Groupoid 84 (1) The space Xis n-truncated for some integer n. (2) The set of connected components π0(X)is finite. (3) For each vertex x∈Xand each integer m≥1is the group πm(X, x)finite. Reference. See [33, Definition E.0.7.8]. Definition 3.149 (∞-Category of π-Finite Spaces):The ∞-category of π-finite spaces is the full subcategory of the ∞-category of spaces spanned by the π-finite spaces Spcπ⊂Spc. Definition 3.150 (∞-Category of Profinite Spaces):The ∞-category of profinite spaces is the Pro-completion of the ∞-category of π-finite spaces  Spc ∶=Pro(Spcπ),Pro(Spcπ)⊂Fun(Spcπ,Spc)op. Reference. See [33, Definition E.0.7.11]. Definition 3.151 (Profinite Completion):The profinite completion of a space Xis X∈Ob(Spc), Xπ∈Ob( Spc), Xπ∶Spcπ→Spc , T ↦ Xπ(T)∶=MapSpc(X, T ). Reference. See [33, Example E.0.7.12]. Definition 3.152 (Profinite Shape of ∞-Topos):Let T∞be the ∞-category of ∞-topoi and let Ube the forgetful functor. The profinite shape of an ∞-topos Xis  Shπ(X)∶=U(Sh(X)) ,T∞Sh →Pro(Spc)U →Pro(Spcπ). Reference. See [33, Variant E.2.2.2]. Definition 3.153: Let Xbe an ∞-topos. We call F∈Ob(X)finite locally constant if there exists an effective epimorphism ∐α∈AUα→∗along with {Lα}α∈A⊂Ob(Spcπ)such that for every α∈Athere is an equivalence Uα×F≃Uα×Lαin the slice ∞-topos X/Uα. If the set Ais finite then Fis called locally constant constructible. Reference. See the introduction to [4, Chapter 10.1]. 3.4 The Profinite Étale Fundamental Groupoid 85 Definition 3.154: Let Xbe an ∞-topos. Let the full subcategories of Xspanned by the finite locally constant objects and the locally constant constructible objects be Xlcc ⊂Xflc ⊂X. Reference. See the introduction to [4, Chapter 10.1]. Proposition 3.155: Let Xbe an ∞-topos. There exist equivalences of ∞-categories Xflc ≃Fun(Sh(X),Spcπ),Xlcc ≃Fun( Shπ(X),Spcπ). Proof. See [4, Proposition 10.1]. Definition 3.156 (´ Etale Fundamental Groupoid):The ´etale fundamental groupid of a scheme Xis the 1-truncation of the shape of the small ´etale ∞-topos of Xand is Π´et 1(X)∶=τ≤1(Sh(Sh(X´et ,Spc))) ,Π´et 1(X)∈Ob(Pro(Grpd)). Reference. See the exposition following [4, Proposition 10.1]. Definition 3.157 (Profinite ´ Etale Fundamental Groupoid):The profinite ´etale fundamental groupid of a scheme Xis the 1-truncation of the profinite shape of the small ´etale ∞-topos of Xand is  Π´et 1(X)∶=τ≤1( Shπ(Sh(X´et ,Spc))) , Π´et 1(X)∈Ob(Pro(FinGrpd)). Reference. See the exposition following [4, Proposition 10.1]. Corollary 3.158: (1) If Xis an arbitrary scheme then there exists a natural equivalence of categories FEt/X≃Fun(Π´et 1(X),FinSet). (2) If Xis quasi-compact in the clopen topology then there exists a natural equivalence FEt/X≃Fun( Π´et 1(X),FinSet). Proof. See [4, Corollary 10.2]. 3.4 The Profinite Étale Fundamental Groupoid 86 Proposition 3.159: Let Xbe a scheme. (1) The following square is commutative FEt/X ≃→→ ↙↖ ↓↓ Fun( Π´et 1(X),FinSet) ∫ ↓↓ Sch  Π´et 1(−)→→Pro(FinGrpd) (2) The functor  Π´et 1(−)sends finite ´etale morphisms of schemes to finite covering morphisms of profinite groupoids. (3) The functor  Π´et 1(−)preserves pullbacks along finite ´etale morphisms of schemes. Proof. See [4, Lemma 10.4]. Example 3.160: Let Xbe a connected scheme and let kbe an algebraically closed field and let x∶Spec(k)→Xbe a morphism. Let ˆπ´et 1(X, x)be the ´etale fundamental group of Xwith basepoint x. There exists a canonical equivalence of profinite groupoids  Π´et 1(X)≃Bˆπ´et 1(X, x). Proof. See [4, Remark 10.3]. Example 3.161: Let Sbe a scheme. The following inclusion functor induces a symmetric monoidal equivalence of ∞-categories  Π´et 1(S)-FinSet+↪Corr( Π´et 1(S)-FinSet), Sp⊗( Π´et 1(S))≃Sp⊗(PShΣ(Corr( Π´et 1(S)-FinSet),Spc)). Proof. This follows from Proposition 3.107. Proposition 3.162: The profinite ´etale fundamental groupoid functor extends to  Π´et 1(−)∶Corr(Sch,all,f´et)→Corr(Pro(FinGrpd),all,fcov). Proof. See [4, Lemma 10.4]. 3.5 Galois-Equivariant Spectra and Motivic Spectra 87 3.5 Galois-Equivariant Spectra and Motivic Spectra Remark 3.163: In the following we present the construction of the natural transformation c, see Theorem 3.168. We first collect the relevant functors for the construction of the functor c, see in particular Remark 3.164. We then employ the following strategy. (1) In Remark 3.165 we first construct the following natural transformation of functors FinSet⊗ +(−)○ Π´et 1(−)↪SmQP⊗ +(−). (2) In Remark 3.166 we then promote the natural transformation of functors in (1) to c∶H⊗ ∗(−)○ Π´et 1(−)→H⊗ ∗(−)∶Corr(Sch,all,f´et)→CAlg(Catsift ∞). (3) In Remark 3.167 we then promote the natural transformation of functors in (2) to c∶Sp⊗(−)○ Π´et 1(−)→SH⊗(−)∶Corr(Sch,all,f´et)→CAlg(Catsift ∞). Remark 3.164: (1) There exists the following functor SmQP⊗ +(−)∶Corr(Sch,all,flf)→CAlg(Cat1), S↦SmQP/S+,(Yf ←Xp →S)↦p⊗○f∗. (2) There exists the following functor FinSet⊗(−)∶Corr(Pro(FinGrpd),all,fp)→Catlex 1, X↦X-FinSet ,(Yf ←Xp →Z)↦p∗○f∗. (3) There exists the following symmetric monoidal subfunctor FinSet⊗ +(−)⊂(Corr ○FinSet⊗)(−)∶Corr(Pro(FinGrpd),all,fp)→CAlg(Cat1), X↦X-FinSet+,(Yf ←Xp →Z)↦p⊗○f∗ Proof. See the exposition following [4, Equation (10.5)]. Remark 3.165: 3.5 Galois-Equivariant Spectra and Motivic Spectra 88 (1) The equivalence of categories below is natural in S∈Ob(Schop)and admits a canonical lift to the following natural equivalence of functors FEt/S ≃ →Fun( Π´et 1(S),FinSet) FinSet⊗(−)○ Π´et 1(−)≃ →FEt⊗ /(−)∶Corr(Sch,all,f´et)→Catlex 1. (2) Applying the functor of correspondences and restricting to pointed objects we obtain Corr(−)∶Catlex 1→CAlg(Cat2), FinSet⊗ +(−)○ Π´et 1(−)≃ →FEt⊗ /(−)+∶Corr(Sch,all,f´et)→CAlg(Cat1). (3) The inclusion of categories below is natural in S∈Ob(Schop)and lifts uniquely to a natural transformation of functors which also promotes to the following one FEt⊗ /S+⊂SmQP/S+, FEt⊗ +(−)↪SmQP⊗ +(−)∶Corr(Sch,all,f´et)→CAlg(Cat1), FinSet⊗ +(−)○ Π´et 1(−)↪SmQP⊗ +(−). Proof. See the exposition following [4, Equation (10.5)]. Remark 3.166: (1) The natural transformation above is a functor Corr(Sch,all,f´et)×∆1→CAlg(Cat∞). (2) By composition with the functor PShΣthe functor above lifts to the following one PShΣ∶CAlg(Cat∞)→CAlg(Catsift ∞), Corr(Sch,all,f´et)×∆1→CAlg(MCatsift ∞), (S, 0→1)↦((H⊗ ∗( Π´et 1(S)),E)→(PShΣ(SmQP/S+,Spc),M)). (3) Through composition with a partial left adjoint functor to the functor below we obtain the following natural transformation of functors CAlg(Catsift ∞)→CAlg(MCatsift ∞),C→(C,E), c∶H⊗ ∗(−)○ Π´et 1(−)→H⊗ ∗(−)∶Corr(Sch,all,f´et)→CAlg(Catsift ∞). 3.5 Galois-Equivariant Spectra and Motivic Spectra 89 Proof. See the exposition following [4, Equation (10.5)]. Remark 3.167: (1) If p∶Y→ Π´et 1(S)is a finite covering morpism then the following is ⊗-invertible Σ∞(cS(p⊗(S1)))∈Ob(SH⊗(S)). (2) The functor Σ∞○cadmits therefore a lift to the following functor of categories Corr(Sch,all,f´et)×∆1→CAlg(OCatsift ∞), (S, 0→1)↦((H⊗ ∗( Π´et 1(S)),{p⊗(S1)}p)→(SH⊗(S), π0(Pic(SH⊗(S)))). (3) Through composition with a partial left adjoint functor to the functor below we obtain the following natural transformation of functors CAlg(Catsift ∞)→CAlg(OCatsift ∞),C⊗↦(C⊗, π0(Pic(C⊗))), c∶Sp⊗(−)○ Π´et 1(−)→SH⊗(−)∶Corr(Sch,all,f´et)→CAlg(Catsift ∞). Proof. See the exposition following [4, Equation (10.5)]. Theorem 3.168 (Bachmann-Hoyois):There exists a natural transformation of functors c∶Sp⊗(−)○ Π´et 1(−)→SH⊗(−)∶Corr(Sch,all,f´et)→CAlg(Catsift ∞). Proof. See the exposition following [4, Equation (10.5)]. Theorem 3.169 (Bachmann-Hoyois): (1) There exists the following adjunction of symmetric monoidal ∞-categories where cSis a left adjoint symmetric monoidal functor natural in Swith cS∶Sp⊗( Π´et 1(S))⇆SH⊗(S)∶uS, S ∈Ob(Corr(Sch,all,f´et)). (2) If Sis a connected scheme with a finite Galois cover S′/Sand with the automorphism group Gthen there exists a left adjoint symmetric monoidal functor G∶=AutFEt/S(S′/S), cS′/S∶Sp⊗(BG)⇆SH⊗(S)∶uS′/S. 3.7 Normed Motivic Spectra 96 (1) A normed motivic spectrum over Cis a section of SH⊗(−)over the category Corr(C,all,f´et)that is cocartesian over Cop. (2) Let the ∞-category of normed motivic spectra over Cbe the full subcategory NAlgC(SH⊗(−))⊂Sect(SH⊗(−)∣Corr(C,all,f´et)). Reference. See [4, Definition 7.1]. Remark 3.189 (Notation):We also write NAlgFEt (SH⊗(S))∶=NAlgFEt/S(SH⊗(−)). Proposition 3.190: Let Sbe a scheme. Let C⊂f´et Sch/Sbe a full subcategory that contains Sand is closed under finite sums and finite ´etale extensions. (1) The ∞-category NAlgC(SH⊗(−)) admits all colimits and finite limits. If Cis small then it is locally presentable and therefore admits all limits. (2) The canonical forgetful functor U∶NAlgC(SH⊗(−))→SH⊗(S)is conservative and preserves sifted colimits and finite limits. If C⊂Sm/Sthen it preserves limits and is therefore monadic. (3) The canonical forgetful functor U∶NAlgC(SH⊗(−))→CAlg(SH⊗(S)) is conservative and preserves colimits and finite limits. If C⊂Sm/Sthen it preserves limits and is therefore monadic and comonadic. (4) There exist the following functor along with the equivalence of ∞-categories ModE(SH⊗(−))∶Corr(C,all,f´et)→CAlg(Catsift ∞), E ∈Ob(NAlgC(SH⊗(−))), NAlgC(SH⊗(−))E/≃NAlgC(ModE(SH⊗(−))). Proof. See [4, Proposition 7.6]. Example 3.191: Let Sbe any scheme. The monoidal unit 1Sis canonically a normed motivic spectrum over the category Sch/Si.e. 1S∈Ob(NAlgSch/S(SH⊗(−))). Proof. See [4, Example 7.11]. 3.7 Normed Motivic Spectra 97 Example 3.192: Voevodsky’s motivic Eilenberg-MacLane spectrum MZSover a Noetherian scheme Sis a normed motivic spectrum over the category Sm/Si.e. MZS∈Ob(NAlgSm/S(SH⊗(−))). Proof. See [4, Example 7.12]. Remark 3.193: Let Sbe a Noetherian scheme. Let p∶X→Sbe finite ´etale. Since MZSis normed and oriented there exists some E∞-multiplicative transfer morphism νp∶⊕ n∈Z MapSH(X)(1X,Σn P1(MZX))→⊕ m∈Z MapSH(S)(1S,Σm P1(MZS)). If Sis a smooth scheme over a field then νpis a morphism of Bloch’s cycle complexes νp∶z●(X, ●)→z●(S, ●). Proof. See the exposition following [4, Remark 14.9]. Remark 3.194: Let kbe a field and let Sbe a smooth quasi-projective scheme over Spec(k). Let p∶X→Sbe a finite ´etale morphism. Let z●(S, ●)be Bloch’s cycle complex of S. Since MZSis normed and oriented there exists some E∞-multiplicative transfer morphism νpthat induces the Fulton-MacPherson norm νFM pon following Chow groups MapSH⊗(S)(1S,Σn P1(MZS))≃zn(S, ●), n ∈Z, νp∶z●(X, ●)→z●(S, ●), CH●(−)∶Corr(SmQP/k,all,f´et)→Set , S ↦CH●(S),(Yf ←Xp →S)↦νFM p○f∗. Proof. See [4, Proposition 14.12]. Example 3.195: The algebraic K-theory spectrum KGLSover any scheme Sis a normed motivic spectrum over the category Sch/Si.e. KGLS∈Ob(NAlgSch/S(SH⊗(−))). Proof. See [4, Example 7.13]. Example 3.196: The algebraic cobordism spectrum MGLSover any scheme Sis a normed motivic spectrum over the category Sch/Si.e. MGLS∈Ob(NAlgSch/S(SH⊗(−))). 3.7 Normed Motivic Spectra 98 Proof. See [4, Example 7.14]. Example 3.197: Let Dbe a Dedekind domain of mixed characteristic and S=Spec(D). The following are normed motivic spectra over the category Sm/Si.e. MZSpi S/m∈Ob(NAlgSm/S(SH⊗(−))) , m ≥0. Proof. See [4, Theorem 13.9]. Remark 3.198 (Power Operations):Let Sbe a scheme. Let Tbe a finite ´etale scheme over S. Let Gbe a finite ´etale group scheme over S. Let B´et Gbe the ´etale classifying space of G. Let Tbe G-equivariant. Let Ebe a normed motivic spectrum over Sm/S. There exists the following total T-power operation PTin the ∞-category H(S)where E∈Ob(NAlgSm/S(SH⊗(−))) , PT∶Ω∞ P1(E)→MapH(S)(B´et G, Ω∞ P1(E)). Proof. See [4, Example 7.25]. Example 3.199: Let kbe a field. Let Sbe a smooth scheme over Spec(k). Let Tbe a finite ´etale scheme over S. Let Gbe a finite ´etale group scheme over S. Let Tbe also Gequivariant. The following total T-power operation PTnatural in X∈Ob(Sm/S)recovers on homogeneous elements Voevedosky’s total power operation in motivic cohomology ⊕ n∈Z Σn P1(MZS)∈Ob(NAlgSm/S(SH⊗(−))) , PT∶z●(X, ●)→z●(X×B´et G, ●). Proof. See [4, Example 7.25]. Theorem 3.200: Let f∶S′→Sbe a finite ´etale morphism of schemes. Let C⊂f´et Sch/S and let C′=C/S′. The norm functor f⊗lifts to the following adjunction of ∞-categories f⊗∶SH⊗(S′)→SH⊗(S), f⊗∶NAlgC′(SH⊗(−))⇆NAlgC(SH⊗(−))∶f∗. Proof. See [4, Theorem 8.1]. Theorem 3.201: Let f∶S′→Sbe a finite ´etale morphism of schemes. Let C⊂f´et Sch/S and let C′⊂f´et Sch/S′be subcategories such that f#(C′)⊂Cand f∗(C)⊂C′. Let also f∗∶SH⊗(S′)→SH⊗(S)be compatible with any base change (g∶T→S)∈Mor(C). There exists the following adjunction of ∞-categories f∗∶NAlgC(SH⊗(−))⇆NAlgC′(SH⊗(−))∶f∗. 3.7 Normed Motivic Spectra 99 Proof. See [4, Theorem 8.2]. Example 3.202: If f∶S′→Sis a pro-smooth morphism of schemes then there exists f∗∶NAlgSm/S(SH⊗(−))⇆NAlgSm/S′(SH⊗(−))∶f∗. Proof. See [4, Example 8.4]. Definition 3.203 (Normed X-Spectrum):Let Xbe a profinite groupoid. (1) Let a normed X-spectrum be a section of the functor Sp⊗(−)over the category Corr(X-FinSet)that is cocartesian over backward morphisms. (2) Let the ∞-category of normed X-spectra be NAlg(Sp⊗(X)). Reference. See [4, Definition 9.14]. Example 3.204: If Xis a profinite set then there is an equivalence of ∞-categories NAlg(Sp⊗(X))≃CAlg(Sp⊗(X)) , X ∈Ob(Pro(FinSet)). Proof. See [4, Example 9.15]. Proposition 3.205 (Bachmann-Hoyois):The first adjunction of symmetric monoidal ∞-categories where cSis a symmetric monoidal functor natural in Slifts to the following S∈Ob(Corr(Sch,all,f´et)), cS∶Sp⊗( Π´et 1(S))⇆SH⊗(S)∶uS, NAlg(Sp⊗( Π´et 1(S)))⇆NAlgFEt(SH⊗(S)). Proof. See [4, Proposition 10.8]. Example 3.206: The C2-equivariant Betti realization functor lifts to the adjunction ReC2∶SH⊗(Spec(R))⇆Sp⊗(BC2)∶SingC2, NAlgFEt (SH⊗(R))⇆NAlg(Sp⊗(BC2)). Proof. See the exposition following [4, Lemma 11. 4]. 3.8 Graded-Normed Motivic Spectra 100 Example 3.207: Let Sbe an essentially smooth scheme over a field. Let MZSbe Voevodsky’s motivic Eilenberg-MacLane spectrum. Since ⊕n∈ZΣn P1(MZS)is a normed motivic spectrum over the category Sm/S, it follows that Bloch’s cycle complex z●(S, ●) of Spromotes to a normed  Π´et 1(S)-spectrum. Proof. See [4, Example 10.9]. Example 3.208: Since the algebraic K-theory spectrum KGLSover any scheme Sis a normed motivic spectrum over Sch/S, it follows that Weibel’s homotopy invariant Ktheory spectrum KHSover any scheme Spromotes to a normed  Π´et 1(S)-spectrum. Proof. See [4, Example 10.10]. Example 3.209: Let Sbe a smooth scheme over a field of characteristic zero. The algebraic cobordism cohomology theory of Levine-Morel satisfies Ωn(S)≃MGL2n,n S(S). Since ⊕n∈ZΣn P1(MGLS)is a normed motivic spectrum over the category Sch/S, there exists a normed  Π´et 1(S)-spectrum whose 0-th homotopy group is ⊕n∈ZΩn(S). Proof. See [4, Example 10.11]. 3.8 Graded-Normed Motivic Spectra Definition 3.210: Let Sbe a scheme. Let Γbe a discrete commutative monoid. Let ΓSmQP/S+be the following category: (1) Let (X, γ)∈Ob(ΓSmQP/S+)be X∈Ob(SmQP/S)along with a locally constant function γ∶X→Γ. (2) Let (f∶(X, γ)→(Y, δ)) ∈Mor(ΓSmQP/S+)be (f∶X+→Y+)∈Mor(SmQP/S+) such that the locally constant functions γand δcoincide on f−1(Y)⊂X. Reference. See the exposition following [4, Equation (13.12)]. Remark 3.211: Let Sbe a scheme. Let Γbe a discrete commutative monoid. (1) The category ΓSmQP/S+admits all finite coproducts. The following canonical functor is the initial functor that preserves finite coproducts in its second argument and there exists an equivalence of ∞-categories Γ×SmQP/S+→ΓSmQP/S+, PShΣ(ΓSmQP/S+,Spc)≃PShΣ∗(SmQP/S,Spc)Γ. 3.8 Graded-Normed Motivic Spectra 101 (2) For every morphism of discrete commutative monoids φthe functor φ!is the left Kan extension of the following one φ∶Γ→Γ′, φ!∶PShΣ∗(SmQP/S,Spc)Γ→PShΣ∗(SmQP/S,Spc)Γ′, ΓSmQP/S+→Γ′SmQP/S+,(X, γ)↦(X, φ ○γ). (3) The commutative monoid structure of Γinduces a symmetric monoidal structure on the category ΓSmQP/S+. The Day convolution induces a symmetric monoidal structure on the ∞-category PShΣ∗(SmQP/S,Spc)Γ. Proof. See the exposition following [4, Equation (13.12)]. Definition 3.212: Let Sbe a scheme. Let Γbe a discrete commutative monoid. (1) Let (ΓS,+)be the commutative monoid of locally constant functions and let γ∶S→Γ,(f∶X→Y)∈Mor(SmQP/S), f∗∶ΓY→ΓX, γ ↦γ○f. (2) Let p∶X→Sbe finite locally free and γ∶X→Γa locally constant function and p∗(γ)∶S→Γ, p∗(γ)(s)∶=∑ x∈p−1(s) degx(p)⋅γ(x),degx(p)=dimκ(s)(OXs,x). (3) If pis ´etale at x∈Xthen OXs,x =κ(x)and therefore degx(p)=[κ(x)∶κ(s)]. Reference. See the exposition following [4, Equation (13.12)]. Remark 3.213: Let Sbe a scheme. Let Γbe a discrete commutative monoid. (1) Let flf be the class of finite locally free morphisms in Sch. There exists a functor Corr(Sch,all,flf)→CAlg(Set), S ↦ΓS,(Yf ←Xp →S)↦p∗○f∗. (2) The functor above extends to the following functor ΓSmQP⊗ +(−)∶Corr(Sch,all,flf)→CAlg(Cat1), S↦ΓSmQP/S+,(Yf ←Xp →S)↦p⊗○f∗, f∗(U, γ)∶=(U×YX, f∗ U(γ)) , p⊗(U, γ)∶=(Rp(U), q∗(e∗(γ))), Ue ←Rp(U)×SXq →Rp(U). 3.8 Graded-Normed Motivic Spectra 102 (3) The values of the functors f∗and p⊗on morphisms as well as their symmetric monoidal structures are uniquely determined by the requirement that the faithful functor ΓSmQP/S+→SmQP/S+is natural in S∈Ob(Corr(Sch,all,flf)). Proof. See the exposition following [4, Lemma 13.13]. Remark 3.214: Let Sbe a scheme. Let Γbe a discrete commutative monoid. (1) For every morphism φthe following functor is natural in Swith φ∶Γ→Γ′, ΓSmQP/S+→Γ′SmQP/S+. (2) We obtain therefore the follwing functor (−)SmQP⊗ +(−)∶CAlg(Set)×Corr(Sch,all,flf)→CAlg(Cat1). (3) By repeating the construction of the functor SH⊗(−)we obtain also the functor SH⊗(−)(−)∶CAlg(Set)×Corr(Sch,all,f´et)→CAlg(Catsift ∞),(Γ, S)↦SH⊗(S)Γ. Proof. See the exposition following [4, Lemma 13.13]. Remark 3.215: Let Sbe any scheme. Let Γbe a discrete commutative monoid. The symmetric monoidal ∞-category H⊗ ∗(S)Γis obtained from the symmetric monoidal ∞- category PShΣ∗(SmQP/S,Spc)Γby inverting the families of motivic equivalences which are generated by the Nisnevich sieves and the A1-homotopy equivalences in ΓSmQP/S+. Proof. See the exposition following [4, Lemma 13.13]. Definition 3.216: Let Sbe scheme. Let Γbe a discrete commutative monoid. The compactly generated symmetric monoidal ∞-category SH⊗(S)Γis obtained from the symmetric monoidal ∞-category H⊗ ∗(S)Γby inverting the motivic spheres in degree 0∈Γi.e. SH⊗(S)Γ∶=H⊗ ∗(S)Γ[{GrΓ 0(Sp,q)−1}p,q∈Z]. Theorem 3.217 (Bachmann-Hoyois):There exists the following functor SH⊗(−)(−)∶CAlg(Set)×Corr(Sch,all,f´et)→CAlg(Catsift ∞),(Γ, S)↦SH⊗(S)Γ. 3.8 Graded-Normed Motivic Spectra 103 Proof. See the exposition following [4, Lemma 13.13]. Lemma 3.218: Let Γbe a discrete commutative monoid. The following canonical morphism of commutative monoids yields a natural transformation Γ→0, ⊕ Γ∶SH⊗(−)Γ→SH⊗(−). Proof. See [4, Equation (13.12)]. Lemma 3.219: Let Γbe a discrete commutative monoid and let δ∈Γ. Let f∶X→Y be a morphism of schemes. The graded pullback functor f∗is computed degreewise. If the morphism fis smooth f∗admits a left adjoint f#which is also computed degreewise f#∶SH⊗(X)Γ⇆SH⊗(Y)Γ∶f∗,{Eγ}γ∈Γ∈Ob(SH⊗(Y)Γ), f∗({Eγ}γ∈Γ)δ=f∗(Eδ). Proof. See [4, Remark 13.15]. Lemma 3.220: Let Γbe a discrete commutative monoid and let δ∈Γ. Let Sbe a scheme. The functor δ∗admits a left adjoint δ!with δ!∶SH⊗(S)⇆SH⊗(S)Γ∶δ∗, δ∗({Eγ}γ∈Γ)∶=Eδ, δ!(F)ξ∶=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Fif ξ=δ 0else Proof. See the exposition following [4, Remark 13.16]. Lemma 3.221: Let Γbe a discrete commutative monoid and let γ∈Γ. If p∶X→Sis finite ´etale of constant degree dthen there is an equivalence of SH⊗(X)-module functors p⊗○γ! ≃ →(d⋅γ)!○p⊗∶SH⊗(X)→SH⊗(S)Γ. Proof. See [4, Lemma 13.17]. Corollary 3.222: Let Γbe a discrete commutative monoid. The following is also a presentably normed ∞-category over the category of schemes SH⊗(−)Γ∶Corr(Sch,all,f´et)→CAlg(Catsift ∞), S ↦SH⊗(S)Γ,(Yf ←Xp →S)↦p⊗○f∗. Proof. See the exposition following [4, Remark 13.16]. 3.8 Graded-Normed Motivic Spectra 104 Definition 3.223 (Γ-Graded-Normed Motivic Spectrum):Let Sbe a scheme. Let C⊂f´et Sch/Sbe a full subcategory that contains Sand is closed under finite sums and finite ´etale extensions. (1) A Γ-graded-normed motivic spectrum over Cis a section of SH⊗(−)Γover the category Corr(C,all,f´et)that is cocartesian over Cop. (2) The ∞-category of Γ-graded-normed motivic spectra over Cbe the full subcategory NAlgC(SH⊗(−)Γ)⊂Sect(SH⊗(−)Γ∣Corr(C,all,f´et)). Reference. See the exposition following [4, Lemma 13.17]. Remark 3.224: Let Sbe a scheme. Let C⊂f´et Sch/Sbe a full subcategory that contains Sand is closed under finite sums and finite ´etale extensions. (1) Let φbe a morphism of discrete commutative monoids. There exists an adjunction φ∶Γ→Γ′, φ!∶Sect(SH⊗(−)Γ∣Corr(C,all,f´et))⇆Sect(SH⊗(−)Γ′∣Corr(C,all,f´et))∶φ∗, (2) Since φ!and φ∗preserve Γ-graded-normed spectra we obtain the adjunction φ!∶NAlgC(SH⊗(−)Γ)⇆NAlgC(SH⊗(−)Γ′)∶φ∗. (3) The following trivial morphism of commutative groupoids yields the adjunction Γ→0, ⊕ Γ∶NAlgC(SH⊗(−)Γ)⇆NAlgC(SH⊗(−))∶constΓ, ⊕ γ∈Γ({Eγ}γ∈Γ)∶=⊕ γ∈Γ Eγ, constΓ(E)γ∶=E , ∀γ∈Γ. Proof. See the exposition following [4, Lemma 13.17]. 3.8 Graded-Normed Motivic Spectra 105 Example 3.225: Let Sbe a scheme. Let C⊂f´et Sch/Sbe a full subcategory that contains Sand is closed under finite sums and finite ´etale extensions. Let NSymC(−)be the free normed spectrum functor over C. Let Ube the forgetful functor. The free normed motivic spectra over Care canonically N-graded and there exists an adjunction of ∞-categories NSymC(−)∶SH⊗(S)⇆NAlgC(SH⊗(−))∶U. Proof. See [4, Example 13.19]. Definition 3.226: Let Sbe a scheme. Let C⊂f´et Sch/Sbe a full subcategory that contains Sand is closed under finite sums and finite ´etale extensions. Let the full subcategory of Z-graded effective spectra over Sbe (SH⊗(S)Z)eff ⊂SH⊗(S)Z,{En}n∈Z∈Ob((SH⊗(S)Z)eff), En∈Ob(Σn P1(SH⊗(S)eff)). Remark 3.227: Let Sbe a scheme. The ∞-category of Z-graded effective spectra over Sis a symmetric monoidal subcategory since Σn P1(SH⊗(S)eff)∧Σm P1(SH⊗(S)eff)⊂Σn+m P1(SH⊗(S)eff). Proof. See the introduction to [4, Chapter 13.4]. Lemma 3.228: Let Sbe a scheme. The ∞-category of Z-graded effective spectra over Sis generated under colimits by the set {n!(Σn P1(E))∈Ob((SH⊗(S)Z)eff)∣E∈Ob(SH⊗(S)eff)and n∈Z}. Proof. See the introduction to [4, Chapter 13.4]. Lemma 3.229: If p∶X→Sis a finite ´etale morphism of constant degree dthen p⊗(n!(Σn P1(E)))≃(n⋅d)!(p⊗(S2n,n)∧p⊗(E)) , E ∈Ob(SH⊗(X)eff ), n ∈Z. Proof. See the introduction to [4, Chapter 13.4]. Corollary 3.230: There exists a subfunctor (SH⊗(−)Z)eff ⊂SH⊗(−)Z∶Corr(Sch,all,f´et)→CAlg(Catsift ∞). Proof. See the introduction to [4, Chapter 13.4]. 4.1 Z-Graded Galois-Equivariant Spectra 112 Proof. See [16, Lemma 9.1.5]. Lemma 4.25: Let C⊗be a symmetric monoidal locally presentable stable ∞-category. Let C⊗be generated under arbitrary shifts and small colimits by a set of compact objects {Gα}α∈A⊂Ob(Cc)and let A∈Ob(CAlg(C⊗)). The symmetric monoidal locally presentable stable ∞-category ModA(C⊗)is generated under arbitrary shifts and small colimits by the set of compact objects {A⊗Gα}α∈A⊂Ob(ModA(C⊗)c). Proof. See [31, Chapter 4.6.2]. Lemma 4.26: Let C⊗be a symmetric monoidal locally presentable stable ∞-category. Let A∈Ob(CAlg(C⊗)) be some E∞-algebra. If X∈Ob(C⊗)is dualizable and admits the dual X∗then A⊗X∈Ob(ModA(C⊗)) is dualizable and admits the dual A⊗X∗. In particular is Adualizable and self-dual as a module over itself. Proof. See [31, Chapter 4.6.2]. Definition 4.27: The ∞-category PrLis the subcategory of the ∞-category of not necessarily small ∞-categories  Cat∞with the following properties: (1) Ob(PrL)is the class of locally presentable ∞-categories in  Cat∞. (2) MapPrL(C,D)is the space of colimit preserving functors of ∞-categories FunL(C,D). Theorem 4.28: There exists a closed symmetric monoidal ∞-category PrL,⊗such that: (1) Let FunL,L(C×D,E)be the ∞-category of functors which preserve small colimits independently in each argument. There exists an equivalence of ∞-categories FunL,L(C×D,E)≃FunL(C⊗D,E). (2) The ∞-category Spc is the ⊗-unit of the symmetric monoidal ∞-category PrL,⊗. (3) There exists the following equivalence of ∞-categories MapPrL(C,FunL(D,E))≃MapPrL(C⊗D,E). Proof. See [19, Theorem 5.31]. Theorem 4.29: Let PrL St be the full subcategory of PrLspanned by the locally presentable stable ∞-categories. There is a closed symmetric monoidal ∞-category PrL,⊗such that: 4.1 Z-Graded Galois-Equivariant Spectra 113 (1) Let FunL,L(C×D,E)be the ∞-category of functors which preserve small colimits independently in each argument. There exists an equivalence of ∞-categories FunL,L(C×D,E)≃FunL(C⊗D,E). (2) The stable ∞-category of spectra Sp is also the ⊗-unit of the ∞-category PrL,⊗ St . (3) There exists the following equivalence of ∞-categories MapPrL St (C,FunL(D,E))≃MapPrL St (C⊗D,E). Proof. See [19, Theorem 5.34]. Remark 4.30: The following Proposition 4.31 is used in Corollary 4.49 to identify the dualizable and compact objects of the stable ∞-category Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z. The Corollary 4.49 is then employed in the proof of Theorem 4.73, see also Remark 4.72. Proposition 4.31: Let C,D∈Ob(CAlg(PrL,⊗ St )) be locally presentable and stable ∞- categories. Let Cand Dbe compactly generated. Let Cc⊂Cand Dc⊂Dbe the full subcategories of compact objects. Let c0∈Ob(Cc)and d0∈Ob(Dc). The locally presentable stable ∞-category C⊗Dis compactly generated by the following set {c⊗d∈Ob(C⊗D)∣c∈Ob(Cc), d ∈Ob(Dc)}. Proof. See [16, Proposition 7.4.2]. Remark 4.32: In the following Remark 4.33 and Remark 4.34 and Definition 4.35 and Definition 4.36 we present the fundamental components of a bootstrapping argument in Proposition 4.39, applied to the dualizable and self-dual and compact generators {Σ∞ G(G/H)+}H≤G, see Proposition 4.38 and Proposition 4.40, of the symmetric monoidal stable ∞-category of genuine G-equivariant spectra Sp⊗(G)with respect to a finite group G, see Definition 3.103. Remark 4.33: Let G∈Ob(Pro(FinGrpd)) be a profinite groupoid with the cofiltered limit G=lim α∈A(Gα). For every α∈Athe finite groupoid Gα∈Ob(FinGrpd)satisfies Gα=∐β∈BαBGα,β such that Gα,β ∈Ob(FinGrp)is a finite group and BGα,β is the delooping groupoid of Gα,β and Bαis a finite set. 4.1 Z-Graded Galois-Equivariant Spectra 114 Remark 4.34: Let G∈Ob(Pro(FinGrpd)) be a profinite groupoid with the cofiltered limit G=lim α∈A(Gα). Let Gα=∐β∈BαBGα,β. Let H⊂Gα,β be a finite subgroup and let Fun(Gα,Set)≃Fun(∐ β∈Bα BGα,β,Set)≃∏ β∈Bα Fun(BGα,β,Set), Gα,β/H∈Ob(Fun(BGα,β,Set)), Sα,β,H ∶=(∅,⋯,∅, Gα,β/H, ∅,⋯,∅), Sα,β,H ∈Ob(∏ β∈Bα Fun(BGα,β,Set)). Definition 4.35: Let G∈Ob(Pro(FinGrpd)) be a profinite groupoid with the cofiltered limit G=lim α∈A(Gα). Let the category G-Set be the filtered colimit in the category PrLwith G-Set ∶=colim α∈A(Fun(Gα,Set)). Definition 4.36: Let G∈Ob(Pro(FinGrpd)) be a profinite groupoid with the cofiltered limit G=lim α∈A(Gα). Let α∈Awith Gα=∐β∈BαBGα,β such that Gα,β ∈Ob(FinGrp)is a finite group and BGα,β is the delooping groupoid of Gα,β and Bαis a finite set. Let also H⊂Gα,β be a finite subgroup. Let Φα∶Fun(Gα,Set)→colim α∈A(Fun(Gα,Set)) be the canonical morphism in the ordinary category PrL. Let Sα,β,H be as above and let also Xα,β,H ∶=Φα(Sα,β,H), Sα,β,H ∈Ob(Fun(Gα,Set)), {Xα,β,H ∣α∈A , β ∈Bα, H ⊂Gα,β}⊂Ob(G-Set). Proposition 4.37: Let Gis a compact Lie group and let H⊂Gbe a closed subgroup. The suspension spectra {Σ∞ G(G/H)+}H≤Gare dualizable in the symmetric monoidal stable ∞-category of genuine G-equivariant spectra Sp⊗(G). Proof. See [34, Chapter 5.3]. Proposition 4.38: Let Gbe a group of finite order and let H⊂Gbe a subgroup. The suspension spectra {Σ∞ G(G/H)+}H≤Gare dualizable and self-dual in the symmetric monoidal stable ∞-category of genuine G-equivariant spectra Sp⊗(G). Proof. See [8, Example 3.2.6 (1)]. Proposition 4.39: Let G∈Ob(Pro(FinGrpd)). Let the Bachmann-Hoyois functor of spectra over profinite groupoids be Sp⊗(−)∶Corr(Pro(FinGrpd),all,fp)→CAlg(Catsift ∞). 4.1 Z-Graded Galois-Equivariant Spectra 115 The following G-spectra in the symmetric monoidal ∞-category Sp⊗(G)are dualizable and self-dual {Σ∞ Gα,β (Xα,β,H )∣α∈A , β ∈Bα, H ⊂Gα,β}⊂Ob(Sp⊗(G)). Proof. This follows from Proposition 4.38 and Definition 4.36. Proposition 4.40: Let Gbe a group of finite order. The symmetric monoidal stable ∞-category of genuine G-equivariant spectra Sp⊗(G)is compactly generated by the set of dualizable and self-dual spectra {Σ∞ G(G/H)+}H≤G. Proof. See [22, Proposition 2.1]. Proposition 4.41: Let G∈Ob(Pro(FinGrpd)). Let the Bachmann-Hoyois functor of spectra over profinite groupoids be Sp⊗(−)∶Corr(Pro(FinGrpd),all,fp)→CAlg(Catsift ∞). The symmetric monoidal ∞-category Sp⊗(G)is compactly generated by the set of dualizable and self-dual G-spectra. Proof. This follows from Proposition 4.40 and Definition 4.36. Example 4.42: Let Sbe a quasi-compact and quasi-separated scheme. Let  Π´et 1(S) be the profinite ´etale fundamental groupid of S. The symmetric monoidal locally presentable stable ∞-category Sp⊗( Π´et 1(S)) is compactly generated by the set of of dualizable and self-dual  Π´et 1(S)-spectra. Proof. This follows from Proposition 4.39 and Proposition 4.41. Corollary 4.43: Let Sbe a quasi-compact and quasi-separated scheme and let  Π´et 1(S) be the profinite ´etale fundamental groupid of S. The full subcategory of the compact  Π´et 1(S)-spectra Sp⊗( Π´et 1(S))cis generated under arbitrary shifts and finite colimits and retracts by the set of dualizable and self-dual  Π´et 1(S)-spectra. Proof. This follows from Example 4.42. Definition 4.44 (Unbounded ∞-Category of Chain Complexes):Let Abe a Grothendieck abelian category. The unbounded ∞-category of chain complexes is the following symmetric monoidal locally presentable stable ∞-category D(A)∶=Ndg(Ch●(A)fc inj). Reference. See [31, Definition 1.3.5.8]. 4.1 Z-Graded Galois-Equivariant Spectra 116 Example 4.45: Let Rbe a commutative ring. There exists the following equivalence of symmetric monoidal triangulated categories Ho(D(ModR))≃D(ModR). Example 4.46 (∞-Category of Connected Chain Complexes):Through the use of N∆(−)and the Dold-Kan correspondence N∶sAbproj ⇆Ch≥0(Ab)proj ∶Γ, we recover the ∞-category D≥0(Ab), of connected chain complexes. Proof. See [19, Example 1.40 (3)]. Lemma 4.47: Let D⊗(Ab)Zbe the symmetric monoidal stable ∞-category of Z-graded chain complexes in Ab. The full subcategory (D⊗(Ab)Z)cof compact unbounded Z-graded chain complexes is D⊗(Ab)c=Perf⊗(Ab), A●,●∈Ob(D⊗(Ab)Z), A●,●∶=(⋯, A●,m, A●,m+1, A●,m+2,⋯), Ob((D⊗(Ab)Z)c)={A●,●∈Ob(Perf⊗(Ab)Z)∣A●,m =0almost all m∈Z}. Proof. This follows from Example 4.8. Corollary 4.48: Let D⊗(Ab)Zbe the symmetric monoidal ∞-category of unbounded Z-graded chain complexes in Ab. The full subcategory of dualizable unbounded Z-graded chain complexes in Ab is equivalent to (D⊗(Ab)Z)c. Proof. This follows from Example 4.8 and Lemma 4.47. Corollary 4.49: Let Sbe a quasi-compact and quasi-separated scheme. Let  Π´et 1(S)be the profinite ´etale fundamental groupid of S. Let D⊗(Ab)Zbe the locally presentable stable ∞-category of unbounded Z-graded chain complexes in the category Ab. Let Sp⊗(−)∶Corr(Pro(FinGrpd),all,fp)→CAlg(Catsift ∞)be the Bachmann-Hoyois functor of spectra over profinite groupoids. Let Sp⊗( Π´et 1(S))⊗D(Ab)Zbe the ⊗-product of symmetric monoidal locally presentable stable ∞-categories in the symmetric monoidal ∞- category CAlg(PrL,⊗ St ). Let AS∈Ob(CAlg(Sp⊗( Π´et 1(S))⊗D(Ab)Z))be some E∞-algebra. The following symmetric monoidal locally presentable stable ∞-category ModAS(Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z)∈Ob(CAlg(PrL,⊗ St )) is generated under shifts and small colimits by the set of dualizable compact AS-modules {AS⊗(P●⊗Q●,●)∣P●∈Ob(Sp⊗( Π´et 1(S))c), Q●,●∈Ob(D⊗(Ab)Z)c)}. 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 117 Proof. This follows from Proposition 4.31 and Corollary 4.43 and Corollary 4.48. 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra Definition 4.50 (Artin Spectra):Let Sbe a quasi-compact and quasi-separated scheme. The category of Artin spectra over Sin the triangulated category SH(S)is the full localizing triangulated subcategory A(SH(S))∶=LocSH(S)({Σ∞ P1(X)+∣X∈Ob((Sm/S)Nis), f ∶X→Sis finite ´etale}). Definition 4.51 (Artin Motives):Let Sbe a quasi-compact and quasi-separated scheme. The triangulated category of Artin motives DMA(S)over Sin the triangulated category of Spitzweck motives DM(S)is the full localizing triangulated subcategory DMA(S)∶=LocDM(S)({MS(X)∣X∈Ob((Sm/S)Nis), f ∶X→Sis finite ´etale}). Remark 4.52: Let Sbe a quasi-compact and quasi-separated scheme and let also be X∈Ob((Sm/S)Nis)and f∶X→Sfinite ´etale. The following Artin spectra over Sand Artin motives over Sare dualizable and self-dual and compact Σ∞ P1(X)+∈Ob(A(SH(S))c),MS(X)∈Ob(DMA(S)c). Proof. This follows from Lemma 4.14, by applying the symmetric monoidal functor c, see Theorem 3.168, to the dualizable and self-dual generators of Sp⊗( Π´et 1(S)), see Example 4.42. The Corollary 3.158, implies every finite étale covering of Sto be hit by the functor c. A variant of the proof is possible via Atiyah duality. Definition 4.53 (Artin-Tate Motives):Let Sbe a quasi-compact and quasi-separated scheme. The triangulated category of Artin-Tate motives DMA(S)over Sin the triangulated category of Spitzweck motives DM(S)is the following full localizing triangulated subcategory closed under ⊗-products DMAT(S)∶=Loc⊗ DM(S)(Ob(DMA(S))∪Ob(DMT(S))). Remark 4.54: Let Sbe a quasi-compact and quasi-separated scheme. There exist canonical inclusions of symmetric monoidal locally presentable stable ∞-categories Loc⊗ DM⊗(S)({MZSpi S})↘↙→→ ↙↖ ↓↓ DMA⊗(S) ↙↖ ↓↓ DMT⊗(S)↘↙→→DMAT⊗(S) 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 118 Remark 4.55: Since Sp⊗∈Ob(CAlg(PrL,⊗ St )) is initial there exists a unique up to homotopy morphism (α∶Sp⊗→SH⊗(S)) ∈Mor(CAlg(PrL,⊗ St )). Let HZ∈Ob(Sp⊗)be the Eilenberg-MacLane spectrum. There exists a canonical morphism φof E∞-algebras in SH⊗(S)along with the following functors of symmetric monoidal stable ∞-categories (φ∶α(HZ)→MZSpi S)∈Mor(CAlg(SH⊗(S))), MZSpi S∧α(HZ)(−)∶Modα(HZ)(SH⊗(S))→ModMZSpi S(SH⊗(S)), α(HZ)∧α(HZ)α(−)∶ModHZ(Sp⊗)→Modα(HZ)(SH⊗(S)), DK ∶D⊗(Ab)≃ →ModHZ(Sp⊗), MZSpi S∧α(HZ)(−)○α(HZ)∧α(HZ)α(−)○DK ∶D⊗(Ab)→ModMZSpi S(SH⊗(S)). Remark 4.56: The following natural diagram of stable ∞-categories is commutative LocModα(HZ)(SH⊗(S))({α(HZ)}) →→ ↓↓ Modα(HZ)(SH⊗(S)) ↓↓ LocModMZSpi S (SH⊗(S))({MZSpi S})↘↙→→DMA⊗(S)↘↙→→DM⊗(S) Definition 4.57 (Day Convolution):Let C⊗be a small symetric monoidal ∞-category. Let D⊗be a symetric monoidal ∞-category which admits small colimits and in which the ⊗-product bifunctor preserves small colimits in each variable. The Day convolution ⊗-product F⊗Day Gof the functors F, G ∈Ob(Fun(C⊗,D⊗)) is the left Kan extension of the functor C⊗×C⊗F×G →D⊗×D⊗⊗D →D⊗along the functor C⊗×C⊗⊗C →C⊗. Reference. See the introduction to [18, Chapter 1]. Proposition 4.58: Let C⊗be a symetric monoidal ∞-category. Let D⊗be a symetric monoidal ∞-category which admits small colimits and in which the ⊗-product bifunctor preserves small colimits in each variable. There exists an equivalence of ∞-categories CAlg(Fun(C⊗,D⊗)⊗Day )≃Fun⊗,lax(C⊗,D⊗). Proof. See [18, Proposition 2.12]. Example 4.59: Let Sbe a quasi-compact and quasi-separated scheme. Let DM⊗(S)be the symmetric monoidal locally presentable stable ∞-category of Spitzweck motives over 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 119 S. Let D⊗(Ab)be the symmetric monoidal stable ∞-category of chain complexes in the category of abelian groups. There exists a canonical equivalence of ∞-categories DM⊗(S)Z∶=Fun(N(Z)⊗,DM⊗(S))⊗Day , CAlg(Fun(N(Z)⊗,DM⊗(S))⊗Day )≃Fun⊗,lax(N(Z)⊗,DM⊗(S)). Proof. This follows from Proposition 4.58. Remark 4.60: Let Sbe a quasi-compact and quasi-separated scheme. The following symmetric monoidal left adjoint functor is induced by the functor of ∞-categories below L∶D⊗(Ab)Z⇆DM⊗(S)∶R, MZSpi S∧α(HZ)(−)○α(HZ)∧α(HZ)α(−)○DK ∶D⊗(Ab)→DM⊗(S), D⊗(Ab)Z∶=Fun(N(Z)⊗,D⊗(Ab))⊗Day =D⊗(Ab)⊗Fun(N(Z)op,Sp)⊗Day , R(MZSpi S)∈Ob(CAlg(D⊗(Ab)Z)), AS∶=R(MZSpi S). Remark 4.61: The left adjoint functor of the adjunction above admits the factorization D⊗(Ab)Z→DMT⊗(S)↪DM⊗(S), A●,●∈Ob(D⊗(Ab)Z), A●,●∶=(⋯,0,0, A●,n,0,0,⋯), A●,n ∶=Z, n ∈Z, A●,n ↦ZS(n)[2n],ZS(n)[2n]∈Ob(DMT⊗(S)). Reference. See also Theorem 4.73. Remark 4.62: Let Sbe a quasi-compact and quasi-separated scheme. Let DM⊗(S)be the symmetric monoidal locally presentable stable ∞-category of Spitzweck motives over S. There exist the following adjunctions where each left adjoint is symmetric monoidal L∶D⊗(Ab)⇆DM⊗(S)∶R , LZ∶D⊗(Ab)Z⇆DM⊗(S)Z∶RZ, MZSpi S∈Ob(CAlg(DM⊗(S))) ,PMZSpi S∈Ob(CAlg(DM⊗(S)Z)), RZ(PMZSpi S)∈Ob(CAlg(D⊗(Ab)Z)), AS∶=RZ(PMZSpi S). 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 120 Proof. See [56, Theorem 4.3]. Remark 4.63: There exists the following diagram of symmetric monoidal functors. Loc⊗ DM⊗(S)({MZSpi S}) DMA⊗(S)Sp⊗( Π´et 1(S)) DMT⊗(S)DMAT⊗(S) D(Ab)ZSp⊗( Π´et 1(S))⊗D(Ab)Z MS(−)○Θ L Proof. See Remark 4.54 and Theorem 4.73. See also Remark 4.83. Remark 4.64: The following Theorem 4.65 is used in Theorem 4.177. An explicit periodization PMZSpi Sof the E∞-algebra MZSpi Swith respect to the spectrum S=Spec(D) of a Dedekind domain Dof mixed characteristic was constructed by M. Spitzweck, see also Remark 2.31. Theorem 4.65 (M. Spitzweck):Let Dbe a Dedekind domain of mixed characteristic and S=Spec(D). The multiplicative periodization PMZSpi Sof the E∞-algebra MZSpi Sis unique up to equivalence. Proof. (1) There exists the following symmetric monoidal ∞-category with the equivalences DM⊗(S)Z∶=Fun(N(Z)⊗,DM⊗(S))⊗Day , CAlg(DM⊗(S)Z)≃Fun⊗,lax(N(Z)⊗,DM⊗(S)), MZSpi S∈Ob(CAlg(DM⊗(S))) ,PMZSpi S∈Ob(CAlg(DM⊗(S)Z)). Let φbe the following lax symmetric monoidal functor of Picard ∞-groupoids with essential image Im(φ). The essential image Im(φ)is a full Picard ∞-subgroupoid φ∶N(Z)⊗→Pic(DM⊗(S)) , n ↦ZS(n)[2n], Im(φ)={ZS(n)[2n]∣n∈Z},Im(φ)⊂Pic(DM⊗(S)), 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 121 Im(φ)≃N(Z×BC2),Im(φ)∈Ob(Grp(Spc)) , π0(Im(φ))∈Ob(Grp(Spc)). Let Sp≥0be the ∞-category of connective spectra. There exists also the equivalence Grp(Spc)≃Sp≥0,Grp(Spc)⊂CAlg(Spc), π0(N(Z×BC2))=π0(Im(φ))=Z, π1(N(Z×BC2))=C2, πn(N(Z×BC2))=0, n ≥2. (2) The functor φinduces also the following sequence of spectra where Im(φ)splits N(Z)→Im(φ)→π0(Im(φ)) ≃ →N(Z). Motivic cohomology in weight zero implies the following isomorphisms of groups AutDM(S)(ZS(0))=C2,EndDM(S)(ZS(0))=Z,AutDM(S)(ZS(0))∈Ob(Grp(Spc)). The essential image of the functor φsatisfies therefore also the following property π0(AutDM(S)(ZS(0)))=C2, πn(AutDM(S)(ZS(0)))=0, n ≥1. Im(φ)=π0(Im(φ))⊕(π1(Im(φ)))[1], (3) Let Sp⊗be the stable ∞-category of spectra. Let HZ∈Ob(Sp⊗)be the EilenbergMacLane spectrum. The first sequence where Im(φ)splits implies the following N(Z)→Im(φ)→π0(Im(φ)) ≃ →N(Z), HZ→Im(φ)≃ →HZ×HF2[1]. Let Sp⊗be admit the canonical t-structure. There is the following distinguished triangle in the triangulated category of spectra placed in the following diagram τ≥1(S)→→S→→ ↘↘ HZ→→ ↓↓ τ≥1(S)[1] ↙↙ HF2[1] It follows by properties of the t-structure that there exists only the zero morphism HomHo(Sp)(S,HF2[1])=π0(HF2[1])=0,HomHo(Sp)(τ≥1(S)[1],HF2[1])=0. It follows therefore that there exists only the zero morphism of following spectra HomHo(Sp)(HZ,HF2[1])=0. 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 128 The universal property of the coproduct implies the existence of the dashed functor DMA⊗(S) Sp⊗( Π´et 1(S)) MS(−)○Θ→→ →→Sp⊗( Π´et 1(S))⊗D⊗(Ab) Ψ ↑↑ D⊗(Ab) Λ ←← ←← It follows there is the following symmetric monoidal functor of stable ∞-categories Sp⊗( Π´et 1(S))⊗D⊗(Ab)Ψ →DMA⊗(S)↪DMAT⊗(S). (3) According to Remark 4.54, there exists the following diagram of canonical inclusion functors of symmetric monoidal stable ∞-categories Loc⊗ DM⊗(S)({MZSpi S})↘↙→→ ↙↖ ↓↓ DMA⊗(S) ↙↖ ↓↓ DMT⊗(S)↘↙→→DMAT⊗(S) According to Remark 4.60, there exists the following adjunction of stable ∞- categories where the left adjoint is symmetric monoidal L∶D⊗(Ab)Z→DM⊗(S)∶R. The left adjoint functor of the adjunction above admits the following factorization D⊗(Ab)Z→DMT⊗(S)↪DM⊗(S), A●,●∈Ob(D⊗(Ab)Z), A●,●∶=(⋯,0,0, A●,n,0,0,⋯), A●,n ∶=Z, n ∈Z, A●,n ↦ZS(n)[2n],ZS(n)[2n]∈Ob(DMT⊗(S)). The symmetric monoidal functors Ψand Λinduce the following adjunction of stable ∞-categories where the left adjoint functor Lis also symmetric monoidal L∶(Sp⊗( Π´et 1(S))⊗D⊗(Ab))⊗D⊗(Ab)D⊗(Ab)Z⇆DMAT⊗(S)∶R. Properties of the relative ⊗-product imply the adjunction to be equivalent to L∶Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z⇆DMAT⊗(S)∶R 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 129 The adjunction of stable ∞-categories above implies also the following relations ZS(0)∶=MZSpi S, ZS(0)∈Ob(CAlg(DMAT⊗(S))), R(ZS(0))∈Ob(CAlg(Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z)), M∈Ob(ModR(ZS(0))(Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z)), L(M)∈Ob(ModL(R(ZS(0)))(DMAT⊗(S))). The counit morphism below induces the following symmetric monoidal functor ModZS(0)(DMAT⊗(S))≃DMAT⊗(S), εZS(0)∶L(R(ZS(0)))→ZS(0), ZS(0)∧L(R(ZS(0))) (−)∶ModL(R(ZS(0)))(DMAT⊗(S))→DMAT⊗(S). (4) Let C⊗and D⊗be the following cocomplete stable symmetric monoidal ∞-categories where the ⊗-products preserve small colimits. Let L∶C⊗⇆D⊗∶Rbe the adjunction of ∞-categories above where the left adjoint functor Lis symmetric monoidal C⊗∶=Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z,D⊗∶=DMAT⊗(S). Let Aand Bbe the following E∞-algebras in the ∞-categories C⊗and D⊗resp. A∶=R(ZS(0)) , B ∶=ZS(0), A ∈Ob(CAlg(C⊗)) , B ∈Ob(CAlg(D⊗)). Let also fbe the following morphism whose adjoint morphism gis an equivalence (f∶=εZS(0)∶L(A)→B)∈Mor(CAlg(D⊗)), (g∶=idZS(0)∶A→R(B))∈Mor(CAlg(C⊗)), According to Corollary 4.49, the locally presentable symmetric monoidal ∞-category Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z∈Ob(CAlg(PrL,⊗ St )) generated under shifts and small colimits by the set of dualizable compact objects {P●⊗Q●,●∣P●∈Ob(Sp⊗( Π´et 1(S))c), Q●,●∈Ob(D⊗(Ab)Z)c)}. 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 130 According to Definition 4.36 and also Proposition 4.39 and Proposition 4.41 and also Example 4.42, the dualizable and self-dual and compact  Π´et 1(S)-spectra are {Xα,β,H ∣α∈A , β ∈Bα, H ⊂Gα,β}⊂Ob( Π´et 1(S)-Set), Ob(Sp⊗( Π´et 1(S))c)={P●∈Ob(Sp⊗( Π´et 1(S))∣P●=Σ∞ Gα,β (Xα,β,H )}. According to Lemma 4.47, the dualizable and compact Z-graded chain complexes in the category of abelian groups are D⊗(Ab)c=Perf⊗(Ab), Q●,●∈Ob(D⊗(Ab)Z), Q●,●∶=(⋯, Q●,m, Q●,m+1, Q●,m+2,⋯), Ob((D⊗(Ab)Z)c)={Q●,●∈Ob(Perf⊗(Ab)Z)∣Q●,m =0almost all m∈Z}. Since the monoidal unit ZS(0)is compact in the ∞-category DMAT⊗(S), it follows from Proposition 4.24, that every dualizable Artin-Tate motive over Sis also compact. According to Lemma 4.14 and also to Remark 4.52, the left adjoint functor Lsends every dualizable and compact Z-graded  Π´et 1(S)-spectrum P●⊗Q●,● to a compact Artin-Tate motive over S, that is to say L(P●⊗Q●,●)∈Ob(DMAT⊗(S)c). Since the conditions of Theorem 4.71 are satisfied, we obtain the implications: ●The following symmetric monoidal functor of ∞-categories is fully faithful ModR(ZS(0))(Sp⊗( Π´et 1(S))⊗D(Ab)Z) L(R(ZS(0)))⊗L(R(ZS(0)))L(−) ↓↓ ModL(R(ZS(0)))(DMAT⊗(S)) ZS(0)∧L(R(ZS(0)))(−) ↓↓ DMAT⊗(S). ●The essential image of the functor above is compactly generated by the set {ZS(0)∧L(R(ZS(0))) L(P●⊗Q●,●)} ⊂Ob(DMAT⊗(S)). 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 131 It follows there exists some E∞-algebra ASdepending on the base scheme Sand also a symmetric monoidal equivalence of locally presentable stable ∞-categories AS∶=R(ZS(0)), AS∈Ob(CAlg(Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z)), ModAS(Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z)≃DMAT⊗(S). Remark 4.74: Throughout the remainder of Section 4.2 we present straightforward corollaries and examples of Theorem 4.73, as mentioned in the Introduction, while also employing Theorem 4.177, if possible. The proofs of the corollaries and examples of Theorem 4.73, which are referring to Theorem 4.73 and Theorem 4.177 are themself analogous to the proofs of Theorem 4.73 and Theorem 4.177 and are thus suppressed. Remark 4.75: In the following remainder of Section 4.2 we establish natural stable ∞- subcategories of DMAT⊗(S)with respect to a quasi-compact and quasi-separated scheme S. We then show them separately, to be symmetrically monoidally equivalent to stable ∞-categories associated with the stable ∞-category ModAS(Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z), see Theorem 4.73. We then exploit these equivalences to examine the stable ∞-category of Artin-Tate motives DMAT⊗(Spec(R)) over the spectrum of the real numbers. We then also establish some straightforward conclusion and examples if the base scheme S is the spectrum S=Spec(k)of a field kof characteristic zero or an algebraically closed field, see in particular Corollary 4.110 and Corollary 4.113 and also Corollary 4.122. Example 4.76: Let C2∶=Gal(C/R)be the absolute Galois group of the real numbers. (1) The following Artin-Tate E∞-algebra ARis also a Z-graded normed BC2-spectrum L∶Sp⊗(BC2)⊗D⊗(Ab)Z⇆DMAT⊗(Spec(R))∶R, AR∈Ob(NAlg(Sp⊗(BC2)⊗D⊗(Ab)Z)), AR∶=R(MZSpi R). (2) There exists the following symmetric monoidal equivalence of stable ∞-categories ModAR(Sp⊗(BC2)⊗D⊗(Ab)Z)≃DMAT⊗(Spec(R)). 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 132 Proof. This follows from Theorem 4.73 and Theorem 4.177 and Example 3.160. Corollary 4.77: Let Sbe a quasi-compact and quasi-separated scheme. (1) The following Tate E∞-algebra ASis a Z-graded chain complex in abelian groups L∶D⊗(Ab)Z⇆DMT⊗(S)∶R, AS∈Ob(CAlg(D⊗(Ab)Z)), AS∶=R(MZSpi S). (2) There exists the following symmetric monoidal equivalence of stable ∞-categories ModAS(D⊗(Ab)Z)≃DMT⊗(S). Proof. This follows from Theorem 4.73. Corollary 4.78: Let Sbe a quasi-compact and quasi-separated scheme. (1) The following Artin E∞-algebra ASis in particular a normed  Π´et 1(S)-spectrum L∶Sp⊗( Π´et 1(S))⇆DMA⊗(S)∶R, AS∈Ob(NAlg(Sp⊗( Π´et 1(S)))), AS∶=R(MZSpi S). (2) There exists the following symmetric monoidal equivalence of stable ∞-categories ModAS(Sp⊗( Π´et 1(S)))≃DMA⊗(S). Proof. This follows from Theorem 4.73 and Theorem 4.177. Example 4.79: Let C2∶=Gal(C/R)be the absolute Galois group of the real numbers. (1) The following Artin E∞-algebra ARis in particular also a normed BC2-spectrum L∶Sp⊗(BC2)⇆DMA⊗(Spec(R))∶R, AR∈Ob(NAlg(Sp⊗(BC2))), AR∶=R(MZSpi R). 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 133 (2) There exists the following symmetric monoidal equivalence of stable ∞-categories ModAR(Sp⊗(BC2))≃DMA⊗(Spec(R)). Proof. This follows from Corollary 4.78 and Example 3.160 and Example 3.129. Corollary 4.80: Let Sbe a quasi-compact and quasi-separated scheme. (1) The following Artin E∞-algebra ASis in particular a normed  Π´et 1(S)-spectrum L∶Sp⊗( Π´et 1(S))⊗D⊗(Ab)⇆DMA⊗(S)∶R, AS∈Ob(NAlg(Sp⊗( Π´et 1(S))⊗D⊗(Ab))), AS∶=R(MZSpi S). (2) There exists the following symmetric monoidal equivalence of stable ∞-categories ModAS(Sp⊗( Π´et 1(S))⊗D⊗(Ab))≃DMA⊗(S). Proof. This follows from Theorem 4.73 and Theorem 4.177 and Corollary 4.78. Corollary 4.81: Let Sbe a connected scheme with a finite Galois cover S′/Sand trivial automorphism group {e}. Let also cS′/Sbe the left adjoint symmetric monoidal functor of the following adjunction of locally presentable stable ∞-categories. (1) The following Artin-Tate E∞-algebra AS′/Sis a Z-graded-normed B({e})-spectrum {e}∶=AutFEt/S(S′/S), cS′/S∶Sp⊗(B({e}))⇆SH⊗(S)∶uS′/S, cS′/S(Σ∞ {e}({e})+)=Σ∞ P1(S)+, L∶Sp⊗(B({e}))⊗D⊗(Ab)Z⇆DMAT⊗(S)∶R, AS′/S∈Ob(NAlg(Sp⊗(B({e}))⊗D⊗(Ab)Z)), AS′/S∶=R(MZSpi S). 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 134 (2) There exist the following symmetric monoidal equivalences of stable ∞-categories ModAS′/S(Sp⊗(B({e}))⊗D⊗(Ab)Z) ≃ ↓↓ Loc⊗ DM(S)({cS′/S(Σ∞ {e}({e})+)∧MZSpi S}∪Ob(DMT⊗(S))) ≃ ↓↓ Loc⊗ DM(S)({Σ∞ P1(S)+∧MZSpi S}∪Ob(DMT⊗(S))) ≃ ↓↓ Loc⊗ DM(S)({MS(S)}∪Ob(DMT⊗(S))) ≃ ↓↓ DMT⊗(S). (3) Since AutFEt/S(S′/S)is the trivial group {e}the scheme S′is isomorphic to S. We obtain therefore the symmetric monoidal equivalence established by M. Spitzweck AS∶=R(MZSpi S), AS∈Ob(CAlg(D⊗(Ab)Z)), ModAS(D⊗(Ab)Z)≃DMT⊗(S). Proof. This follows from Theorem 4.73 and Theorem 4.177 and Theorem 3.169. Example 4.82: Let kbe an algebraically closed field and let S=Spec(k). Since the unique up to isomorphism finite Galois cover S′/Sis the identity morphism the automorphism group of the finite Galois cover is the trivial group {e}. Therefore is every Artin-Tate E∞-algebra AS′/SaZ-graded-normed B({e})-spectrum. Proof. This follows from Corollary 4.81 and Example 3.160 and Example 3.129. Remark 4.83: There exists the following diagram of symmetric monoidal functors, 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 135 Loc⊗ DM⊗(S)({MZSpi S}) DMA⊗(S)ModAβ S(Sp⊗( Π´et 1(S))⊗D(Ab)) DMT⊗(S)DMAT⊗(S) ModAα S(D(Ab)Z)ModAγ S(Sp⊗( Π´et 1(S))⊗D(Ab)Z) ≃ ≃≃ along with the following E∞-algebras Aα S∈Ob(CAlg(Sp⊗(D⊗(Ab)Z)), Aβ S∈Ob(NAlg(Sp⊗( Π´et 1(S))⊗D⊗(Ab))), Aγ S∈Ob(CAlg(Sp⊗( Π´et 1(S))⊗D⊗(Ab)Z)). Proof. See Theorem 4.73 and Corollary 4.77 and Corollary 4.80 and Cf. Remark 4.63. Corollary 4.84: Let Sis a connected scheme with a finite Galois cover S′/Sand automorphism group G. Let H⊂Gbe a subgroup and let T=S′/Hbe the quotient in the category of schemes. Let also cS′/Sbe the left adjoint symmetric monoidal functor of the following adjunction of of stable ∞-categories. (1) The following Artin E∞-algebra AS′/Sis in particular a normed BG-spectrum G∶=AutFEt/S(S′/S), cS′/S∶Sp⊗(BG)⇆SH⊗(S)∶uS′/S, cS′/S(Σ∞ G(G/H)+)=Σ∞ P1(T)+, L∶Sp⊗(BG)⇆DMA⊗(S)∶R, AS′/S∈Ob(NAlg(Sp⊗(BG))), AS′/S∶=R(MZSpi S). 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 136 (2) There exist the following symmetric monoidal equivalences of stable ∞-categories ModAS′/S(Sp⊗(BG)) ≃ ↓↓ LocDM(S)({cS′/S(Σ∞ G(G/H)+)∧MZSpi S}H≤G) ≃ ↓↓ LocDM(S)({Σ∞ P1(T)+∧MZSpi S}T=S′/H) ≃ ↓↓ LocDM(S)({MS(T)}T=S′/H). Proof. This follows from Theorem 4.73 and Theorem 4.177 and Theorem 3.169. Example 4.85: Let kbe an algebraically closed field and let S=Spec(k). Since the unique up to isomorphism finite Galois cover S′/Sis the identity morphism the automorphism group of the finite Galois cover is the trivial group {e}. Therefore there exists the following symmetric monoidal equivalence of stable ∞-categories ModAS′/S(Sp⊗(B{e}))≃LocDM(S)({MS(S)}). Proof. This follows from Corollary 4.84 and Example 3.160 and Example 3.129. Example 4.86: Let Spec(C)/Spec(R)be the unique up to isomorphism non trivial finite Galois cover of the real numbers. The automorphism group of the finite Galois cover is C2. Let H⊂C2be a subgroup and let T=Spec(C)/Hbe the quotient in the category of schemes. Let also cC/Rbe the left adjoint symmetric monoidal functor of the following adjunction of of stable ∞-categories. (1) The following Artin E∞-algebra AC/Ris in particular a normed BC2-spectrum C2≅AutFEt/Spec(R)(Spec(C)/Spec(R)), cC/R∶Sp⊗(BC2)⇆SH⊗(Spec(R))∶uC/R, cC/R(Σ∞ C2(C2/H)+)=Σ∞ P1(T)+, L∶Sp⊗(BC2)⇆DMA⊗(Spec(R))∶R, AC/R∈Ob(NAlg(Sp⊗(BC2))), AC/R∶=R(MZSpi R). 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 137 (2) There exist the following symmetric monoidal equivalences of stable ∞-categories ModAC/R(Sp⊗(BC2)) ≃ ↓↓ LocDM(Spec(R))({cC/R(Σ∞ C2(C2/H)+)∧MZSpi R}H≤C2) ≃ ↓↓ LocDM(Spec(R))({Σ∞ P1(T)+∧MZSpi R}T=Spec(C)/H) ≃ ↓↓ LocDM(Spec(R))({MR(T)}T=Spec(C)/H). (3) There exists the following symmetric monoidal equivalence of stable ∞-categories Spec(C)/C2≅Spec(R), ModAC/R(Sp⊗(BC2))≃LocDM(Spec(R))({MR(Spec(R)),MR(Spec(C))}). Proof. This follows from Corollary 4.84. Example 4.87: Let Spec(R)/Spec(R)be the trivial finite Galois cover of the real numbers. The automorphism group of the finite Galois cover is {e}. Let also cR/Rbe the left adjoint symmetric monoidal functor of the following adjunction of of stable ∞-categories. (1) The following Artin E∞-algebra AR/Ris in particular a normed B({e})-spectrum {e}≅AutFEt/Spec(R)(Spec(R)/Spec(R)), cR/R∶Sp⊗(B({e}))⇆SH⊗(Spec(R))∶uR/R, cR/R(Σ∞ {e}({e})+)=Σ∞ P1(Spec(R))+, L∶Sp⊗(B({e}))⇆DMA⊗(Spec(R))∶R, AR/R∈Ob(NAlg(Sp⊗(B({e})))), AR/R∶=R(MZSpi R). (2) There exist the following symmetric monoidal equivalence of stable ∞-categories ModAR/R(Sp⊗(B({e})))≃LocDM(Spec(R))({MR(Spec(R))}). Proof. This follows from Corollary 4.84. 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 144 (2) There exist the following symmetric monoidal equivalences of stable ∞-categories ModA{Sσ/S}σ∈Σ((⊗σ∈ΣSp⊗(BGσ))⊗D⊗(Ab)Z) ≃ ↓↓ ModA{Sσ/S}σ∈Σ((∐σ∈ΣSp⊗(BGσ))∐D⊗(Ab)Z) ≃ ↓↓ Loc⊗ DM(S)(⋃σ∈ΣOb(DMA(S)∣Sσ/S)∪Ob(DMT(S))). Proof. This follows from Corollary 4.88. Definition 4.101 (Permutation Modules):Let Rbe a commutative and unital ring. Let Gbe a profinite group. Let Mod(G;R)be the category of free modules over Ron every basis X∈Ob(G-Set)with the linearly extended G-action. The categoy of (G;R)- permutation modules is the essential image of the following symmetric monoidal functor R(−)∶G-Set →Mod(G;R), R(X)∶=⊕ x∈X R, Perm(G;R)∶=Im(R(−)). Reference. See [5, Notation 2.4]. Remark 4.102: Let Rbe a commutative and unital ring. Let Gbe a profinite group. The category Perm(G;R)is Grothendieck abelian with the finitely presented generators {R(G/H)∣H⊂Gis open}⊂Ob(Perm(G;R)). Proof. See [5, Proposition 2.6]. Definition 4.103: Let Rbe a commutative and unital ring. Let Gbe a profinite group. A(G;R)-chain complex A●∈Ob(K(Perm(G;R))) is called G-acyclic if for every open subgroup H⊂Gthe (G;R)-chain complex AH ●of H-fixed points is acyclic. Reference. See [5, Definition 3.1]. Definition 4.104: Let Rbe a commutative unital ring. Let Gbe a profinite group. A morphism f∈Ob(K(Perm(G;R))) is called a G-quasi-isomorphism if the cone Cone(f) is G-acyclic. Let the full subcategory of the G-acyclic (G;R)-chain complexes be denoted Ac(Perm(G;R)). Let the associated class of G-quasi-isomorphisms be WG. 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 145 Definition 4.105: Let Rbe a commutative and unital ring. Let Gbe a profinite group. The derived category of (G;R)-permutation modules is D(Perm(G;R))∶=K(Perm(G;R))[W−1 G] =K(Perm(G;R))/Ac(Perm(G;R)). Reference. See [5, Definition 3.6]. Corollary 4.106: Let Rbe a commutative and unital ring. Let Gbe a profinite group. The derived category of (G;R)-permutation modules D(Perm(G;R)) is a symmetric monoidal and compactly generated triangulated category. Reference. See [5, Corollary 3.10]. Definition 4.107 (Cohomological (G;R)-Mackey Functor):Let Rbe a commutative and unital ring. Let Gbe a profinite group. An R-linear G-Mackey functor is an additive functor F∶Bop G→ModR. The R-linear G-Mackey functor is called cohomological if for all closed subgroups K⊂H⊂Gwith the index [H∶K]and with the canonical projection morphism π∶G/K→G/Hthe value of Fon (G/Hπ ←G/Kπ →G/H)is the multiplication morphism [H∶K]⋅(−). Reference. See [5, Definition 4.5]. Definition 4.108 (Mixed Artin Motives):Let Rbe a commutative and unital ring. Let kbe a field. Let the trinagulated category of mixed Artin motives over the base scheme Spec(k)with R-coefficients be the following full localizing triangulated subcatgory in the triangulated category of Voevodsky’s mixed motives DMA(Spec(k), R)⊂DM(Spec(k), R), LocDM(Spec(k),R)({MS(X)∣X∈Ob((Sm/k)Nis), f ∶X→Spec(k)is finite ´etale}). Theorem 4.109 (Grothendieck-Voevodsky-Yoshida):Let Rbe a commutative and unital ring. Let kbe a field and let Gal(ksep/k)be the absolute Galois group of k. Let DMA(Spec(k), R)be the triangulated category of Voevodsky’s mixed Artin motives over the base scheme Spec(k)with R-coefficients. Let Fun⊕ coh(Bop Gal(ksep/k),ModR)be the abelian category of cohomological (Gal(ksep/k);R)-Mackey functors. There exist the following symmetric monoidal equivalences of triangulated categories D(Perm(Gal(ksep/k);R))≃DMA(Spec(k), R)≃D(Fun⊕ coh(Bop Gal(ksep/k),ModR)). 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 146 Proof. See [5, Equation (1.2)]. Corollary 4.110: Let kbe a field of characteristic zero and let Gal(ksep/k)be the absolute Galois group of k. Let DMA⊗(Spec(k),Z)be the stable ∞-category of Voevodsky’s mixed Artin motives over Spec(k)with Z-coefficients. Let Fun⊕ coh(Bop Gal(ksep/k),Ab)be the abelian category of the cohomological (Gal(ksep/k);Z)-Mackey functors. Let also Ak∈Ob(CAlg(Sp⊗(BGal(ksep/k)))) be the Artin E∞-algebra of Spec(k). There exist the following symmetric monoidal equivalences of stable ∞-categories ModAk(Sp⊗(BGal(ksep/k))) ≃→→ ≃ ↓↓ D⊗(Perm(Gal(ksep/k);Z)) ≃ ↓↓ D⊗(Fun⊕ coh(Bop Gal(ksep/k),Ab)) ≃→→DMA⊗(Spec(k),Z). Proof. This follows from Theorem 4.109 and Corollary 4.78 and Corollary 2.14. Example 4.111: Let kbe an algebraically closed field. (1) There exists the following symmetric monoidal equivalences of abelian categories {e}≅Gal(ksep/k)≅ π´et 1(Spec(k), x), Ab ≃Fun⊕ coh(Bop Gal(ksep/k),Ab)≃Perm(Gal(ksep/k);Z). (2) There exists the following symmetric monoidal equivalence of stable ∞-categories ModAk(Sp⊗)≃LocDM(Spec(k))({Mk(Spec(k))}). (3) If kis an algebraically closed field of characteristic zero then there exist also the following symmetric monoidal equivalences of stable ∞-categories ModAk(Sp⊗)Ak↦Z→→ Ak↦HZ ↓↓ D⊗(Ab) Z↦Mk(Spec(k)) ↓↓ ModHZ(Sp⊗)HZ↦Mk(Spec(k)) →→DMA⊗(Spec(k),Z). Proof. This follows from Corollary 4.110 and Example 4.85 and Example 3.129. 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 147 Theorem 4.112: Let kbe a perfect field. Let also DMA(Spec(k),Q)cbe the full triangulated subcategory of compact rational mixed Artin motives over the base scheme Spec(k). Let Vecc Qbe the category of finite dimensional Q-vector spaces. There exists the following symmetric monoidal equivalence of triangulated categories DMA(Spec(k),Q)c≃Db(Fun(BGal(ksep/k),Vecc Q)). Proof. See [65, Proposition 1.4]. Corollary 4.113: Let kbe a field of characteristic zero. Let also DMA(Spec(k),Q)cbe the full triangulated subcategory of compact rational mixed Artin motives over Spec(k). Let Vecc Qbe the category of finite dimensional Q-vector spaces. There exists the following symmetric monoidal equivalences of triangulated categories DMA(Spec(k),Q)c≃Ho(ModAk(Sp⊗(BGal(ksep/k)))c Q)≃Db(Fun(BGal(ksep/k),Vecc Q)). Proof. This follows from Theorem 4.112 and Corollary 4.78 and Corollary 2.14. Example 4.114: If kis an algebraically closed field of characteristic zero then there exist also the following symmetric monoidal equivalences of triangulated categories DMA(Spec(k),Q)c≃Ho(ModAk(Sp⊗)c Q)≃Db(Vecc Q). Proof. This follows from Corollary 4.113 and Example 4.111. Definition 4.115 (Fiber Functor):Let kbe a field. Let (A,⊗,1A)be a rigid symmetric monoidal abelian category enriched over the category Veck. Let also Vecc kbe the category of finite dimensional k-vector spaces. A fiber functor ωis a faithful exact symmetric monoidal k-linear functor ω∶A→Vecc k. Reference. See [45, Definition 1.10]. Definition 4.116 (Neutral Tannakian Category):Let kbe a field. A neutral Tannakian category (A, ω)over kis a rigid symmetric monoidal abelian category (A,⊗,1A) enriched over the category Veckalong with a fiber functor ω∶A→Vecc k. Reference. See [45, Definition 1.10]. Theorem 4.117 (M. Levine):If Sis a base scheme satisfying the rational BeilinsonSoule conjecture then the triangulated category of geometric rational mixed Tate motives 4.2 Artin-Tate Motives and Z-Graded Galois-Equivariant Spectra 148 DMTgm(S, Q)over Sadmits a natural t-structure whose heart includes {Q(n)}n∈Z. If Sis also connected then this heart MTgm(S, Q)is a neutral Tannakian category over Q with a canonical fiber functor ωSand Tannakian fundamental group scheme GS. If S satisfies the K(π, 1)-conjecture then there exists also the following symmetric monoidal equivalence of triangulated catgories ωS∶MTgm(S, Q)→Vecc Q,MTgm(S, Q)≃RepQ(GS),DMTgm(S, Q)≃Db(RepQ(GS)). Proof. See [54, Theorem 5.1]. Corollary 4.118: If kis a number field then there exist the following symmetric monoidal equivalences of triangulated categories DMTgm(Spec(k),Q)≃Ho(ModAk(D⊗(Ab)Z)c Q)≃Db(RepQ(Gk)). Proof. This follows from Theorem 4.117 and Corollary 4.77 and Corollary 2.14. Theorem 4.119 (M. Spitzweck):Let kbe a perfect field. Let Nkbe the Adams-graded commutative differential graded algebra of algebraic cycles with respect to k. The derived category Df Nkof finite Adams-graded differential graded modules over Nkis symmetrically monoidally equivalent to the triangulated category of rational geometric mixed Tate motives over Spec(k). If the base scheme Spec(k)satisfies the Beilinson-Soule conjecture then there exist canonical t-structures on both sides of the equivalence and an induced equivalence of Tannakian categories Df Nk≃DMTgm(Spec(k),Q),Hf Nk≃MTgm(Spec(k),Q). Proof. See [45, Theorem 3.31]. Corollary 4.120: If kis a number field then there exist the following symmetric monoidal equivalences of triangulated categories along with canonical t-structures on the triangulated categories of the equivalences and an induced equivalence of Tannakian categories DMTgm(Spec(k),Q)≃Ho(ModAk(D⊗(Ab)Z)c Q)≃Df Nk. Proof. This follows from Theorem 4.119 and Corollary 4.77 and Corollary 2.14. Theorem 4.121 (M. Preisinger):Let kbe a number field. The triangulated homotopy category Df Gal(ksep/k)of Gal(ksep/k)-equivariant finite cell modules with respect to kis symmetrically monoidally equivalent to the triangulated category of compact rational 4.3 Graded Norm Functors in Stable Equivariant Homotopy Theory 149 mixed Artin-Tate motives over Spec(k). Moreover there exist canonical t-structures on both sides of the equivalence along with an induced equivalence of Tannakian categories Df Gal(ksep/k)≃DMAT(Spec(k),Q)c,Hf Gal(ksep/k)≃(DMAT(Spec(k),Q)c)♡. Proof. See [45, Theorem 4.41]. Corollary 4.122: If kis a number field then there exist the following symmetric monoidal equivalences of triangulated categories along with canonical t-structures on the triangulated categories of the equivalences and induced equivalences of Tannakian categories DMAT(Spec(k),Q)c≃Ho(ModAk(Sp⊗(BGal(ksep/k))⊗D⊗(Ab)Z)c Q)≃Df Gal(ksep/k). Proof. This follows from Theorem 4.121 and Theorem 4.73 and Corollary 2.14. Theorem 4.123 (J. Scholbach):Let kbe a number field and let S⊂Ok. The triangulated category DMAT(S, Q)cadmits a non-degenerate t-structure. In particular, there exists a symmetric monoidal equivalence between the triangulated category of compact rational mixed Artin-Tate motives DMAT(S, Q)cover Sand the bounded derived category of the heart of this t-structure. Proof. See [50, Theorem 0.1]. 4.3 Graded Norm Functors in Stable Equivariant Homotopy Theory Remark 4.124: The following Section 4.3 is a sine qua non of Section 4.4. We construct the functor Sp⊗(−)(−), see also Proposition 4.146, whereupon we construct the natural transformation cΓ, see Corollary 4.160, via Remark 4.154 and Remark 4.155. The construction of the functor Sp⊗(−)(−)is established by way of the Remark 4.136. Remark 4.125: The following Definition 4.126 and Definition 4.127 are the starting point of the construction of the functor Sp⊗(−)(−), see also Proposition 4.146. The functor is essential for our second main result, see Theorem 4.177. The last one sharpens our main result, see Theorem 4.73. Definition 4.126: Let Z∈Ob(Pro(FinGrpd)). Let Γbe a discrete commutative monoid. Let also ΓZ-FinSet be the following category: (1) Let (X, γ)∈Ob(ΓZ-FinSet)be X∈Ob(Z-FinSet)along with a locally constant function on the following profinite set Z-FinSet ≃FC/Z, X ∈Ob(FC/Z), π0(X)∈Ob(Pro(FinSet)) , γ ∶π0(X)→Γ. 4.3 Graded Norm Functors in Stable Equivariant Homotopy Theory 150 (2) Let (f∶(X, γ)→(Y, δ)) ∈Mor(ΓZ-FinSet)be (f∶X→Y)∈Mor(Z-FinSet)such that the locally constant functions γand δcoincide on X. Reference. See also Definition 3.87. Definition 4.127: Let Z∈Ob(Pro(FinGrpd)). Let Γbe a discrete commutative monoid. Let also ΓZ-FinSet+be the following category: (1) Let (X, γ)∈Ob(ΓZ-FinSet+)be X∈Ob(Z-FinSet)along with a locally constant function on the following profinite set Z-FinSet ≃FC/Z, X ∈Ob(FC/Z), π0(X)∈Ob(Pro(FinSet)) , γ ∶π0(X)→Γ. (2) Let (f∶(X, γ)→(Y, δ)) ∈Mor(ΓZ-FinSet+)be (f∶X+→Y+)∈Mor(Z-FinSet+) such that the locally constant functions γand δcoincide on f−1(Y)⊂X. Reference. Cf. Definition 3.210. Remark 4.128: The following Definition 4.129 is used in Remark 4.134, which itself is used in Remark 4.136, where the functor Sp⊗(−)(−)is constructed, see also Proposition 4.146. The functor Sp⊗(−)(−)is essential for Proposition 4.170 and Theorem 4.177. Definition 4.129: Let Z∈Ob(Pro(FinGrpd)). Let Γbe a discrete commutative monoid. (1) Let (ΓZ,+)be the commutative monoid of locally constant functions and let γ∶π0(Z)→Γ,(f∶X→Y)∈Mor(Z-FinSet), f∗∶ΓY→ΓX, γ ↦γ○f. (2) Let p∶X→Zbe a finite covering of profinite groupoids and γlocally constant and γ∶π0(X)→Γ, p∗(γ)∶π0(Z)→Γ, p∗(γ)(z)∶=∑ x∈p−1(z) γ(x). Reference. Cf. Definition 3.212. Remark 4.130: The following Definition 4.131 is used in Remark 4.134, which is then used in Remark 4.136 for the construction of then functor in Proposition 4.146, i.e. Sp⊗(−)(−)∶CAlg(Set)×Corr(Pro(FinGrpd),all,fcov)→CAlg(Catsift ∞), (Γ, X)↦Sp⊗(X)Γ. 4.3 Graded Norm Functors in Stable Equivariant Homotopy Theory 151 Definition 4.131 (Weil Restriction):Let p∶X→Zbe a finite covering morphism of profinite groupoids. Let Y∈Ob(FC/X)and let hY∈Ob(PSh(FC/X,Spc))be the presheaf represented by Y. Let p∗∶PSh(FC/X,Spc)→PSh(FC/Z,Spc). If the presheaf p∗(hY) is representable by a profinite groupoid over Zthen this profinite groupoid is called the Weil restriction Rp(Y)of Yalong p. Reference. Cf. Definition 3.30. Remark 4.132: Let p∶X→Zbe a finite covering morphism in the category of profinite groupoids and let also Y∈Ob(FC/X). The Weil restriction Rp(Y)of Yalong pexists. This follows from the existence of the following adjunction, see Proposition 3.89, where p∗∶X-FinSet ⇆Y-FinSet ∶p∗. Remark 4.133 (Notation):Let p∶X→Zbe a finite covering morphism of profinite groupoids. Let h∶U→Xbe a be a finite covering morphism of profinite groupoids. Let qand gbe the canonical projection morphisms. Let ebe the counit of the adjunction. Let f=Rp(h). Let also the following diagram in the category of profinite groupoids be U h ↘↘ Rp(U)×ZX e ←← g ↓↓ q→→Rp(U) f ↓↓ Xp→→Z Proof. Cf. Definition 3.174. Remark 4.134: Let Z∈Ob(Pro(FinGrpd)). Let Γbe a discrete commutative monoid. (1) There exists the following functor Corr(Pro(FinGrpd),all,fcov)→CAlg(Set), Z ↦ΓZ,(Yf ←Xp →Z)↦p∗○f∗. (2) The functor above extends to the following functor ΓFinSet⊗ +(−)∶Corr(Pro(FinGrpd),all,fcov)→CAlg(Cat1), Z↦ΓZ-FinSet+,(Yf ←Xp →Z)↦p⊗○f∗, f∗(U, γ)∶=(U×YX, f∗ U(γ)) , p⊗(U, γ)∶=(Rp(U), q∗(e∗(γ))), Ue ←Rp(U)×ZXq →Rp(U). 4.3 Graded Norm Functors in Stable Equivariant Homotopy Theory 152 (3) The values of the functors f∗and p⊗on morphisms as well as their symmetric monoidal structures are uniquely determined by the requirement that the faithful functor ΓZ-FinSet+→Z-FinSet+is natural in Z∈Ob(Corr(Pro(FinGrpd),all,fcov)). Proof. Cf. Remark 3.213. Remark 4.135: The following Remark 4.136 outlines the construction of the functor Sp⊗(−)(−), see also Proposition 4.146. In particular, in Remark 4.141 we expose the construction of the functor H⊗ ∗(−)Γ, see Proposition 4.143. Furthermore, in Remark 4.145 we expose the construction of the functor Sp⊗(−)Γ, with respect to a discrete commutative monoid Γ. Remark 4.136: Let Z∈Ob(Pro(FinGrpd)). Let Γbe a discrete commutative monoid. (1) For every morphism φthe following functor is natural in Zwith φ∶Γ→Γ′, ΓZ-FinSet+→Γ′Z-FinSet+. (2) We obtain therefore the follwing functor (−)FinSet⊗ +(−)∶CAlg(Set)×Corr(Pro(FinGrpd,all,fcov)→CAlg(Cat1). (3) By repeating the construction of the functor Sp⊗(−)we obtain also the functor Sp⊗(−)(−)∶CAlg(Set)×Corr(Pro(FinGrpd),all,fcov)→CAlg(Catsift ∞), (Γ, X)↦Sp⊗(X)Γ. Proof. Cf. Remark 3.214. Remark 4.137: In the following we expose the construction of the functor Sp⊗(−)(−) in the Remark 4.136 (3), see also Proposition 4.146. The functor is essential for Proposition 4.162 and Proposition 4.170 and also for the Theorem 4.177. Remark 4.138: Let Z∈Ob(Pro(FinGrpd)). Let Γbe a discrete commutative monoid. (1) The category ΓZ-FinSet+admits all finite coproducts. The following canonical functor is the initial functor that preserves finite coproducts in its second argument and there exists an equivalence of ∞-categories Γ×Z-FinSet+→ΓZ-FinSet+, PShΣ(ΓZ-FinSet+,Spc)≃PShΣ∗(Z-FinSet,Spc)Γ. 4.3 Graded Norm Functors in Stable Equivariant Homotopy Theory 153 (2) For every morphism of discrete commutative monoids φthe functor φ!is the left Kan extension of the following one φ∶Γ→Γ′, φ!∶PShΣ∗(Z-FinSet,Spc)Γ→PShΣ∗(Z-FinSet,Spc)Γ′, ΓZ-FinSet+→Γ′Z-FinSet+,(X, γ)↦(X, φ ○γ). (3) The commutative monoid structure of Γinduces a symmetric monoidal structure on the category ΓZ-FinSet+. The Day convolution induces therefore a symmetric monoidal structure on the ∞-category PShΣ∗(Z-FinSet,Spc)Γ. Proof. Cf. Remark 3.211. Remark 4.139: The following Proposition 4.140 expands Proposition 3.89, which itself relies on Remark 3.84 and Remark 3.86 and Definition 3.87 and Proposition 3.88. These results are rooted on general considerations within ∞-Category Theory, as explained in the original source of the reference in Proposition 3.89. By following the argumentation of the original source in the context of Definition 4.126 we obtain the Proposition 4.140. Proposition 4.140: Let X, Y ∈Ob(Pro(FinGrpd)) be profinite groupoids. Let Γbe a discrete commutative monoid. (1) If p∶Y→Xis a finitely presented morphism then there exists an adjunction p∗∶ΓX-FinSet ⇆ΓY-FinSet ∶p∗. (2) If in addition pis a finite covering then there is also the following adjunction p#∶ΓY-FinSet ⇆ΓX-FinSet ∶p∗. Proof. Cf. Proposition 3.89. Remark 4.141 (Construction):Let Γbe a discrete commutative monoid. (1) There exists the following functor ΓFinSet⊗(−)∶Corr(Pro(FinGrpd),all,fcov)→Catlex 1, X↦ΓX-FinSet ,(Yf ←Xp →Z)↦p∗○f∗.