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

Uschogov, Daniel · 2026

Abstract

We investigate the stable infinity-subcategory of Artin-Tate motives DMAT(S) in the stable infinity-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 infinity-category DMAT(S) and the stable infinity-category of modules over a Z-graded E-infinity-algebra A(S) in some stable infinity-category of Z-graded Galois-equivariant spectra with respect to S. Provided S a Dedekind scheme, we prove A(S) to be in particular a Z-graded-normed Galois-equivariant spectrum with respect to S. We expect this result also to extend to every quasi-compact and quasi-separated base scheme S. We then establish natural stable infinity-subcategories of DMAT(S) and show them separately to be symmetrically monoidally equivalent to stable infinity-categories associated with the aforementioned stable infinity-catgeory of A(S)-modules. We then employ these equivalences to investigate the stable infinity-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 S is the spectrum S = Spec(k) of a field k of characteristic zero or an algebraically closed field.

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

Author: Uschogov, Daniel
Year: 2026
DOI: 10.48693/894
Source: https://osnadocs.ub.uni-osnabrueck.de/bitstream/ds-2026031114722/1/thesis_uschogov.pdf
Disse a ion
zu E langung des G ades
Dok o de Na u wissenscha en (D . e . na .)
des Fachbe eichs Ma hema ik/In o ma ik/Physik
de Uni e si ä Osnab ück
A in-Ta e Mo i es and Z-G aded-No med
Galois-Equi a ian Spec a
Submi ed by
Daniel Uschogo
Supe iso : P o . D . Ma kus Spi zweck
Uni e si y o Osnab ück
Sep embe 2025
Examina ion In o ma ion
Thesis Re iewe s
Fi s Re iewe :
P o . D . Ma kus Spi zweck
Uni e si y o Osnab ück
Second Re iewe :
P o . D . Elden Elman o
Uni e si y o To on o (Sca bo ough)
Thesis De ense
Da e o De ense:
Feb ua y 04, 2026
Abs ac
We in es iga e he s able ∞-subca ego y o A in-Ta e mo i es DMAT⊗(S)in
he s able ∞-ca ego y o Spi zweck mo i es DM⊗(S)o e a quasi-compac and
quasi-sepa a ed base scheme S. We ound ou app oach upon he Bachmann-Hoyois
heo y o no ms in mo i ic homo opy heo y. We p o e a symme ic monoidal
equi alence be ween he s able ∞-ca ego y DMAT⊗(S)and he s able ∞-ca geo y
ModAS(Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z)wi h espec o some Z-g aded E∞-algeb a AS.
We hen cons uc a Z-g aded no m unc o Sp⊗(−)Zin s able equi a ian homo opy
heo y wi h espec o p o ini e g oupoids. We hen cons uc a na u al ans o -
ma ion cZ∶Sp⊗(−)Z○
Π´e
1(−)→SH⊗(−)Z o he Z-g aded no m unc o in s a-
ble mo i ic homo opy heo y. We hen employ he na u al ans o ma ion cZand
p o e he Z-g aded E∞-algeb a AS o be in pa icula a Z-g aded-no med 
Π´e
1(S)-
spec um p o ided ha Sis a Dedekind scheme. We expec his esul also o
ex end o e e y quasi-compac and quasi-sepa a ed base scheme S. We hen es-
ablish na u al s able ∞-subca ego ies o DMAT⊗(S)and show hem sepa a ely
o be symme ically monoidally equi alen o s able ∞-ca ego ies associa ed wi h
ModAS(Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z). We hen employ hese equi alences o in es iga e
he s able ∞-ca ego y o A in-Ta e mo i es DMAT⊗(Spec(R)) o e he spec um
o he eal numbe s. In he end we p esen se e al s aigh o wa d conclusion and
examples i he base scheme Sis he spec um S=Spec(k)o a ield ko cha ac-
e is ic ze o o an algeb aically closed ield.
.
Zusammen assung
Wi e o schen die s abile ∞-Un e ka ego ie on A in-Ta e Mo i en DMAT⊗(S)
in de s abilen ∞-Ka ego ie on Spi zweck Mo i en DM⊗(S)übe einem quasikom-
pak en und quasi-sepa ie en Basisschema S. Wi legen unse e Vo gehensweise
die Bachmann-Hoyois Theo ie on No m-Funk o en in de Mo i ischen Homo opi-
e heo ie zug unde. Wi beweisen eine symme isch monoidale Äqui alenz zwis-
chen DMAT⊗(S)und de s abilen ∞-Ka ego ie ModAS(Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z)
bezüglich eine Z-g aduie en E∞-Algeb a AS. Anschließend kons uie en wi einen
Z-g aduie en No m-Funk o Sp⊗(−)Zin de s abilen äqui a ian en Homo opie he-
o ie bezüglich p oendliche G uppoide. Anschließend kons uie en wi eine na ü -
liche T ans o ma ion cZ∶Sp⊗(−)Z○
Π´e
1(−)→SH⊗(−)Zzu dem Z-g aduie en No m-
Funk o in de s abilen mo i ischen Homo opie heo ie. Anschließend benu zen wi
die na ü lich ans o ma ion cZund beweisen ü ein Dedekind-Schema S, dass AS
insbesonde e ein Z-g aduie es-no mie es 
Π´e
1(S)-Spek um is . Wi gehen da on
aus, dass sich dieses E gebnis auch au jedes quasikompak e und quasige enn e
Schema Sübe agen läss . Anschließend be ach en wi auch na ü liche s abile
∞-Un e ka ego ien on DMAT(S)und beweisen, dass diese sepa a symme isch
monoidal äqui alen zu s abilen ∞-Ka ego ien sind die mi de s abilen ∞-Ka ego ie
ModAS(Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z)einhe gehen. Anschließend benu zen wi diese
Äqui alenzen, um die s abile ∞-Ka ego ie on A in-Ta e Mo i en DMAT(Spec(R))
übe dem Spek um de eellen Zahlen zu un e suchen. Zum Abschluss s ellen wi
einige di ek e Schluss olge ungen und Beispiele o , alls das Basisschema Sdas
Spek um S=Spec(k)eines Kö pe s k on Cha ak e is ik Null ode eines alge-
b aisch abgeschlossenen Kö pe s is .
Acknowledgmen s
I wish o exp ess my app ecia ion o my supe iso Ma kus Spi zweck. His con inuous
suppo h oughou he many ad en u es du ing my PhD was o immense help o me.
I c edi mos o wha I ha e lea ned o his guidance.
I am g a e ul o Elden Elman o o his ho ough engagemen and commi men h ough-
ou he e iew p ocess.
I am also g a e ul o my colleagues, who ha e accompanied me o e he pas yea s, iz.,
Ch is ian Ah ing, Holge B enne , Xiaowen Dong, And ea Figini, Ma co Gius e o,
Robin Louis, Fede ico Mocche i, Ahina Nandy, Manh Toan Nguyen, Fynn Pö ne ,
Oli e Röndigs, Tho Wi ich.
Fu he mo e, I am hank ul o all people in he ma hema ical depa men o he sha ed
expe iences and he iendly a mosphe e.
I would like o exp ess my deepes g a i ude o my pa en s, my sis e and my iends.

Con en s
In oduc ion 7
Con ex ........................................... 7
Resul s o he Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Rela ed Resul s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
S uc u e o he Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1 Voe odsky Mo i es 20
1.1 The Uns able A1-Mo i ic Homo opy Ca ego y . . . . . . . . . . . . . . . . 20
1.2 The S able A1-Mo i ic Homo opy Ca ego y . . . . . . . . . . . . . . . . . . 24
1.3 Mixed Mo i es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.4 Beilinson Mo i es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 Spi zweck Mo i es 46
2.1 The S able ∞-Ca ego y o Spi zweck Mo i es . . . . . . . . . . . . . . . . . 46
2.2 Ta e Mo i es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 No m Func o s in Homo opy Theo y 54
3.1 No m Func o s in Mo i ic Homo opy Theo y . . . . . . . . . . . . . . . . . 54
3.2 No m Func o s in S able Equi a ian Homo opy Theo y . . . . . . . . . . 67
3.3 The É ale Fundamen al G oup . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4 The P o ini e É ale Fundamen al G oupoid . . . . . . . . . . . . . . . . . . 81
3.5 Galois-Equi a ian Spec a and Mo i ic Spec a . . . . . . . . . . . . . . . 87
3.6 No med ∞-Ca ego ies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.7 No med Mo i ic Spec a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.8 G aded-No med Mo i ic Spec a . . . . . . . . . . . . . . . . . . . . . . . . 100
4 A in-Ta e Mo i es 108
4.1 Z-G aded Galois-Equi a ian Spec a . . . . . . . . . . . . . . . . . . . . . . 108
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a . . . . . . . 117
4.3 G aded No m Func o s in S able Equi a ian Homo opy Theo y . . . . . 149
4.4 G aded-No med Equi a ian Spec a . . . . . . . . . . . . . . . . . . . . . . 160
Lis o Symbols 170
Re e ences 174
7
In oduc ion
Con ex
Mo i es
- A cohomology heo y o schemes is in pa icula a unc o H●∶Schop
/S→AbZ. The
cohomology H●(X)o a scheme Xo e he base scheme Sis a Z-g aded abelian
g oup. The cohomology H●(X)o he scheme Xis simple hen he scheme i sel bu
ne e heless complica ed enough o eason abou X. The e exis a ious cohomology
heo ies o schemes such as he ´
e ale cohomology o he l-adic cohomology.
- A. G o hendieck p oposed he exis ence o a ca ego y e lec ing he uni e sal p ope y
o all easonable cohomology heo ies o schemes. In pa icula he e should exis s
some abelian ca ego y M(S, R)assigned o some scheme Sand some commu a i e
ing Ralong wi h some unc o MS∶Schop
/S→M(S, R)such ha o e e y easonable
cohomology heo y o schemes he e exis s also some unc o RH●∶M(S, R)→AbZ
and an isomo phism H●(X)≅RH●(MS(X)).
- A. G o hendieck cons uc ed he pseudo-abelian and addi i e ca ego y o Chow mo-
i es Chow(Spec(k)) o e he spec um Spec(k)o some ield k. This ca ego y was
no able o cap u e he complexi y o he easonable cohomology heo ies o schemes.
P. Deligne p oposed o cons uc a iangula ed ca ego y o mo i es DM(S, R)o e
a scheme Salong wi h a -s uc u e whose hea is he hypo he ical abelian ca ego y
o mo i es M(S, R)p oposed by A. G o hendieck. In he ollowing V. Voe odsky
cons uc ed he iangula ed ca ego y DM(S, R)and also he iangula ed ca ego y
SH(S), see among o he s, [62] and [59]. La e , in [52], M. Spi zweck cons uc ed also
a iangula ed ca ego y DM(S). I is un esol ed i he e exis s a -s uc u e on any
o he es ablished iangula ed ca ego ies DM(S, R)and SH(S)and DM(S)whose
hea is he abelian ca ego y M(S, R).
- Because o he high complexi y o hese ca ego ies we a e in es iga ing small and
accessible subca ego ies such as he iangula ed ca ego y o Ta e mo i es DMT(S, R)
and he iangula ed ca ego y o A in mo i es DMA(S, R)and he iangula ed
ca ego y o A in-Ta e mo i es DMAT(S, R)o e a quasi-compac quasi-sepa a ed
base scheme Swi h he assump ion on R o be he a ional numbe s Qo he in ege s
Z.
In oduc ion 8
Ta e Mo i es
- The iangula ed ca ego y o Ta e mo i es DMT(S, R)o e a base scheme Sis a na -
u al iangula ed subca ego y o Voe odsky’s iangula d ca ego y o mixed mo i es
DM(S, R)o e a base scheme S. The iangula ed ca ego y DMT(S, R)is small in
compa ison o he iangula ed ca ego y DM(S, R)and admi s in a wide ange o
cases a simple cellula s uc u e.
- The ull iangula ed subca ego y o compac Ta e mo i es DMT(Spec(k),Q)co e
he spec um Spec(k)o a numbe ield k, as conside ed in [45], is pa icula ly well
s udied. The -s uc u es and he weigh s uc u es along wi h he ep esen a ion
heo y o DMT(Spec(k),Q)cand i s ela ion o algeb aic K- heo y and o he se up
o I. K iz and J. P. May we e ex ensi ely s udied by M. Le ine and M. Spi zweck,
see [45] and [29] and[30] and [54] and also [53].
- In [52], M. Spi zweck cons uc ed a s able ∞-ca ego y o mo i es DM⊗(S)o e he
spec um S=Spec(D)o a Dedekind domain Do mixed cha ac e is ic, see De ini-
ion 2.3. M. Spi zweck along wi h O. Röndigs and P.A. Øs æ p o ed his ca ego y
o be closely ela ed o DM(S, Z)in a wide ange o cases, as shown in he Theo em 2.1
and he Co olla y 2.14. In addi ion M. Spi zweck es ablished a s able ∞-subca ego y
o Ta e mo i es DMT⊗(S), see De ini ion 2.23, in he s able ∞-ca ego y DM⊗(S)
and s udied he -s uc u es and he ep esen a ion heo y o i . In Theo em 2.43, M.
Spi zweck also p o ed DMT⊗(S), see De ini ion 2.23, o be symme ically monoidally
equi alen o some s able ∞-ca ego y o Z-g aded chain complexes o abelian g oups
ModAS(D(Ab)Z), see De ini ion 2.39, o e some Z-g aded E∞-algeb a ASwi h e-
spec o a quasi-compac and quasi-sepa a ed base scheme S.
A in Mo i es
- The iangula ed ca ego y o compac A in mo i es DMA(Spec(k),Q)co e he
spec um Spec(k)o a numbe ield kis a na u al iangula ed subca ego y o Voe od-
sky’s iangula ed ca ego y o mixed mo i es DM(Spec(k),Q), see De ini ion 1.101.
I is closely ela ed o he he absolu e Galois g oup Gal(ksep/k), as shown in he
Theo em 4.109 and he Theo em 4.112. I is he iangula ed ca ego y o mixed mo-
i es associa ed o ze o-dimensional a ie ies o e Spec(k), see [45]. I is also s ongly
associa ed wi h algeb aic numbe heo y.
In oduc ion 9
A in-Ta e Mo i es
- The iangula ed ca ego y o compac A in-Ta e mo i es DMAT(Spec(k),Q)co e
he spec um o a numbe ield k, as conside ed by M. P eisinge in [45], is a na u-
al iangula ed subca ego y o Voe odsky’s iangula ed ca ego y o mixed mo i es
DM(Spec(k),Q), see De ini ion 1.101. The iangula ed ca ego y con ains he ian-
gula ed ca ego y DMA(Spec(k),Q)cand he iangula ed ca ego y DMT(Spec(k),Q)c,
see De ini ion 1.105.
- The iangula ed ca ego y DMAT(Spec(k),Q)co e he spec um Spec(k)o a num-
be ield kwas ex ensi ely in es iga ed by M. P eisinge in [45] and J. Scholbach in
[50] and J. Wildeshaus in [65]. In addi ion in [50] J. Scholbach cons uc ed a -
s uc u e on DMAT(S, Q)c, i S=Spec(Ok)is he ing o in ege s o a numbe ield
k, see also Theo em 4.123.
The É ale Fundamen al G oup
- The ´
e ale undamen al g oup 
π´e
1(X, x)o a connec ed scheme X, see De ini ion 3.127,
as conside ed in he Sec ion 3.3, was in oduced by A. G o hendieck in he 1960s. I is
a p o ini e g oup in he sense o De ini ion 3.112. I classi ies he ini e ´
e ale co e ings
mo phisms o he scheme X, see in pa icula De ini ion 3.122. The ´
e ale undamen al
g oup 
π´e
1(Spec(k), x)o he spec um Spec(k)o a ield k, is by Example 3.129 also
isomo phic o he absolu e Galois g oup Gal(ksep/k).
- The p o ini e ´
e ale undamen al g oupoid 
Π´e
1(X)o a scheme X, as conside ed in
he Sec ion 3.4 is a p o ini e g oupoid. I does no depend on a basepoin . I Xis
also a connec ed scheme hen by Example 3.160, he delooping g oupoid, see De i-
ni ion 3.62, o he ´
e ale undamen al g oup 
π´e
1(X, x)is isomo phic o he p o ini e
´
e ale undamen al g oupoid 
Π´e
1(X), see De ini ion 3.157.
- The iangula ed ca ego y o compac A in mo i es DMA(Spec(k),Q)co e he
spec um Spec(k)o a numbe ield k, as in es iga ed by M. P eisinge in [45] is
closely ela ed o he absolu e Galois g oup Gal(ksep/k), as shown in Theo em 4.109
and Theo em 4.112 and Theo em 4.121 and he e o e also o he ´
e ale undamen al
g oup 
π´e
1(Spec(k), x).
In oduc ion 16
- In he Sec ion 1.2, we p esen explici ly he cons uc ion o Voe odsky’s s able A1-
homo opy ca ego y SH(S)o e some base scheme Sand p o ide also i s undamen al
p ope ies, see in pa icula Theo em 1.55 and Theo em 1.65 and Theo em 1.66.
- In he Sec ion 1.3, we p esen explici ly he cons uc ion o Voe odsky’s ca ego y o
mixed mo i es DM(S)o e some base scheme Sand p o ide also i s undamen al
p ope ies, see in pa icula Theo em 1.88 and Theo em 1.91 and Theo em 1.92.
- In he Sec ion 1.4, we p esen explici ly he cons uc ion o he iangula ed ca e-
go y o Beilinson mo i es DMB(S, Q)o e some base scheme Sand p o ide also i s
undamen al p ope ies, see in pa icula Theo em 1.110 and De ini ion 1.111.
- We p o ide he ela ions be ween H(S)and SH(S)and DM(S)and also DMB(S, Q),
see in pa icula Theo em 1.89 and Theo em 1.118 and Theo em 1.90.
Chap e 2
- We in oduce Spi zweck’s mo i ic Eilenbe g-MacLane spec um MZSpi
So e he spec-
um S=Spec(D)o a Dedekind domain Do mixed cha ac e is ic, see Theo em 2.1.
We u he employ MZSpi
S o he cons uc ion o he s able ∞-ca ego y DM⊗(S)o
Spi zweck mo i es o e he base scheme S, see De ini ion 2.3.
- We p o ide he undamen al p ope ies and ela ions o Spi zweck’s s able ∞-ca ego y
DM⊗(S) o Voe odsky’s ca ego ies o mo i es SH(S)and DM(S)and DMB(S, Q),
see Theo em 2.1 and Co olla y 2.14.
- We in oduce Spi zweck’s mul iplica i ely pe iodized mo i ic Eilenbe g-MacLane spec-
um PMZSpi
So e some base scheme S, see Co olla y 2.33 and [52, Appendix C], and
p esen in addi ion i s p ope ies, see Rema k 2.35.
- In he De ini ion 2.23 we in oduce he s able ∞-ca ego y o Ta e mo i es DMT⊗(S)
o e a quasi-compac and quasi-sepa a ed base scheme Sin he s able ∞-ca ego y
DM⊗(S)o Spi zweck mo i es o e S.
- We p esen Spi zweck’s undamen al equi alence o s able ∞-ca ego ies which in ol es
he s able ∞-ca ego y o Ta e mo i es DMT⊗(S)o e a quasi-compac and quasi-
sepa a ed base scheme S, see Theo em 2.43.

In oduc ion 17
Chap e 3
- We in oduce ini ially he Bachmann-Hoyois no m unc o s in mo i ic homo opy
heo y p⊗∶H⊗
∗(X)→H⊗
∗(S)and also p⊗∶SH⊗(X)→SH⊗(S)associa ed o some
mo phism o schemes p∶X→S, see Co olla y 3.40 and P oposi ion 3.44. We hen
p o ide he undamen al p ope ies o he mo i ic no m unc o s p⊗. We hen employ
he mo i ic no m unc o s p⊗ o he cons uc ion o he no m unc o s H⊗
∗(−)and
SH⊗(−)in mo i ic homo opy heo y, see Co olla y 3.58 and Theo em 3.61. We hen
also p o ide he undamen al p ope ies o he unc o s H⊗
∗(−)and SH⊗(−).
- We in oduce he Bachmann-Hoyois no m unc o s in equi a ian homo opy heo y
wi h espec o p o ini e g oupoids p⊗∶H⊗
∗(X)→H⊗
∗(Z)and p⊗∶Sp⊗(X)→Sp⊗(Z)
associa ed o some mo phism o p o ini e g oupoids p∶X→Z, see Rema k 3.91 and
Rema k 3.98. We hen employ he no m unc o s p⊗in equi a ian homo opy heo y
wi h espec o p o ini e g oupoids o he cons uc ion o he no m unc o s H⊗
∗(−)
and Sp⊗(−)in equi a ian homo opy heo y wi h espec o p o ini e g oupoids, see
Theo em 3.92 and Theo em 3.99. We hen also p o ide he undamen al p ope ies
o he unc o s H⊗
∗(−)and Sp⊗(−).
- We p esen he cons uc ion and he undamen al p ope ies o he ´
e ale undamen al
g oup 
π´e
1(X, x)o a connec ed scheme X, see De ini ion 3.122. We hen p esen
he cons uc ion and he undamen al p ope ies o he p o ini e ´
e ale undamen al
g oupoid 
Π´e
1(X)o a scheme X, see Sec ion 3.4. We hen p esen in addi ion he
ela ion o he ´
e ale undamen al g oup 
π´e
1(X, x)and he ´
e ale undamen al g oupoid

Π´e
1(X), see Example 3.160.
- We employ he no m unc o SH⊗(−)in s able mo i ic homo opy heo y, see P opo-
si ion 3.44, and also he no m unc o Sp⊗(−)in equi a ian homo opy heo y wi h
espec o p o ini e g oupoids, see Theo em 3.99, and also he p o ini e ´
e ale un-
damen al g oupoid unc o 
Π´e
1(−), see P oposi ion 3.162, and cons uc he na u-
al ans o ma ion c∶Sp⊗(−)○
Π´e
1(−)→SH⊗(−)o Bachmann-Hoyois, see Theo-
em 3.168.
- We employ he no m unc o SH⊗(−)in s able mo i ic homo opy heo y, see The-
o em 3.61, o in oduce he ∞-ca ego y o no med mo i ic spec a NAlgC(SH⊗(−))
o e some ca ego y o schemes Cand p o ide he undamen al p ope ies o his ca -
ego y, see De ini ion 3.188. We hen employ he no m unc o Sp⊗(−)in equi a ian
homo opy heo y wi h espec o p o ini e g oupoids o in oduce he ∞-ca ego y o
In oduc ion 18
no med X-spec a NAlg(Sp⊗(X))wi h espec o a p o ini e g oupoid Xand p o ide
he undamen al p ope ies o his ca ego y, see De ini ion 3.203.
- We in oduce he Bachmann-Hoyois Z-g aded-no m unc o SH⊗(−)Zin s able mo-
i ic homo opy heo y, see Theo em 3.217. We hen p esen he undamen al p ope -
ies o SH⊗(−)Z. We hen in oduce he Z-g aded-no m sub unc o (SH⊗(−)Z)e and
he Z-g aded-no m sub unc o (SH⊗(−)Z) e and p esen also he ela ions o hese,
see Co olla y 3.230 and Rema k 3.239.
- We employ he Z-g aded-no m unc o SH⊗(−)Zin s able mo i ic homo opy heo y,
see Theo em 3.217, o he in oduce ∞-ca ego y o Z-g aded-no med mo i ic spec-
a NAlgC(SH⊗(−)Z)o e some ca ego y o schemes Cand p o ide he undamen al
p ope ies o his ca ego y, see De ini ion 3.223.
Chap e 4
- We ini ially de e mine he dualizable and compac gene a o s o he s able ∞-ca ego y
Sp⊗(
Π´e
1(S))wi h espec o a quasi-compac and quasi-sepa a ed base scheme S, see
P oposi ion 4.39 and P oposi ion 4.41. We hen also de e mine he dualizable and
compac gene a o s o he s able ∞-ca ego y Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z, see Co ol-
la y 4.49.
- We in oduce he s able ∞-ca ego y o A in mo i es DMA⊗(S), see he De ini-
ion 4.51, and also he s able ∞-ca ego y o A in-Ta e mo i es DMAT⊗(S), see
De ini ion 4.53, in he s able ∞-ca ego y o Spi zweck mo i es DM⊗(S)o e a quasi-
compac and quasi-sepa a ed base scheme S. We hen also p o ide he ela ions o he
s able ∞-ca ego ies DMA⊗(S)and DMT⊗(S)and DMAT⊗(S), see Rema k 4.54. We
hen p esen explici ly Spi zweck’s undamen al equi alence be ween he s able ∞-
ca ego y DMT⊗(S)and he s able ∞-ca ego y ModAS(D⊗(Ab)Z), see Theo em 4.67.
- We employ he na u al ans o ma ion c∶Sp⊗(−)○
Π´e
1(−)→SH⊗(−)o Bachmann-
Hoyois, see Theo em 3.168, and also a esul o H. Heine, see Theo em 4.71, wi h
espec o he dualizable and compac gene a o s o Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z, see
Co olla y 4.49, o es ablish a symme ic monoidal equi alence be ween he s able
∞-ca ego y DMAT⊗(S)o e a quasi-compac and quasi-sepa a ed base scheme S,
see De ini ion 4.53, and he s able ∞-ca ego y ModAS(Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z)wi h
espec o some Z-g aded E∞-algeb a ASassoccia ed o S, see Theo em 4.73.
In oduc ion 19
- We employ he Z-g aded-no m unc o SH⊗(−)Zin s able mo i ic homo opy heo y,
see Theo em 3.217, and he no m unc o Sp⊗(−)in s able equi a ian homo opy
heo y wi h espec o p o ini e g oupoids, see Theo em 3.99, o he cons uc ion o
aZ-g aded-no m unc o Sp⊗(−)Zin s able equi a ian homo opy heo y wi h espec
o p o ini e g oupoids, see P oposi ion 4.146.
- We employ he Z-g aded-no m unc o Sp⊗(−)Zin s able equi a ian homo opy he-
o y wi h espec o p o ini e g oupoids, see P oposi ion 4.146, along wi h he Z-
g aded-no m unc o SH⊗(−)Zin s able mo i ic homo opy heo y, see Theo em 3.217,
along wi h he p o ini e ´
e ale undamen al g oupoid unc o 
Π´e
1(−), see P oposi-
ion 3.162, and also he na u al ans o ma ion c∶Sp⊗(−)○
Π´e
1(−)→SH⊗(−)o
Bachmann-Hoyois, see Theo em 3.168, o he con uc ion o he ollowing na u al
ans o ma ion o unc o s cZ∶Sp⊗(−)Z○
Π´e
1(−)→SH⊗(−)Z, see P oposi ion 4.162.
- We p o e he mul iplica i e pe iodiza ion PMZSpi
So Spi zweck’s mo i ic Eilenbe g-
MacLane spec um MZSpi
S, see Co olla y 2.33, o e he spec um S=Spec(D)o a
Dedekind domain o mixed cha ac e is ic is unique, see Theo em 4.65.
- We employ he na u al ans o ma ion cZ, see P oposi ion 4.162, and he uniqueness
o he mul iplica i e pe iodiza ion PMZSpi
So Spi zweck’s mo i ic Eilenbe g-MacLane
spec um MZSpi
So e a base scheme S, see Theo em 4.65, o p o e ha he Z-g aded
E∞-algeb a ASis a Z-g aded-no med 
Π´e
1(S)-spec um, p o ided ha Sis a Dedekind
scheme, see Theo em 4.177. We expec his esul o ex end o e e y quasi-compac
and quasi sepa a ed base scheme S.
- We es ablish na u al s able ∞-subca ego ies o he s able ∞-ca ego y o A in-Ta e
mo i es DMAT⊗(S)o e a quasi-compac and quasi-sepa a ed base scheme Sand
show hem sepa a ely o be symme ically monoidally equi alen o s able ∞-ca ego ies
associa ed wi h he s able ∞-ca ego y ModAS(Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z), see Theo-
em 4.73 and ollowing.
- We in es iga e he s able ∞-ca ego y o A in-Ta e mo i es DMAT⊗(Spec(R)) o e
he spec um o he eal numbe s, see Theo em 4.73 and ollowing.
- We p esen a se ies o s aigh o wa d conclusions and examples o he main esul s
i he base scheme is he spec um S=Spec(k)o a ield ko cha ac e is ic ze o o
an algeb aically closed ield, see Theo em 4.73 and ollowing.
20
1 Voe odsky Mo i es
1.1 The Uns able A1-Mo i ic Homo opy Ca ego y
De ini ion 1.1 (Uni e sal P ope y):Le Sbe a scheme. Le (X, Z)be a closed pai
o schemes. The ∞-ca ego y H∗(S)is he ca ego y o unc o s F∶Smop
/S→Spc∗which
sa is ies he ollowing p ope ies:
(1) (Excision):Fo any pai o smoo h schemes Xand Yo e Sand any excisi e
mo phism p∶(Y, W )→(X, Z)o closed pai s is p∗∶F(Y, W)→F(X, Z)a weak
equi alence in he poin ed ∞-ca ego y o spaces Spc∗.
(2) (A1-In a iance):Fo any scheme X∈Ob(Sm/S) he mo phism πX∶X×SA1
S→X
induces an equi alence F(X)≃F(X×SA1
S)in he poin ed ∞-ca ego y Spc∗.
Re e ence. See [11, De ini ion 1.2].
Rema k 1.2 (Closed Pai o Schemes):A closed pai o schemes (X, Z)is a scheme X
wi h a closed subscheme Z. A mo phism o closed pai s o schemes p∶(Y, W )→(X, Z)
is a mo phism ∶X→Ysuch ha −1(Z)=W.
Rema k 1.3 (No a ion):Le Sbe a scheme. Le (X, Z)be a closed pai o schemes.
Le F∶Smop
/S→Spc∗be a unc o . Le F(X, Z)be he ibe o F(X)→F(X/Z)in he
∞-ca ego y o poin ed spaces Spc∗.
De ini ion 1.4: A mo phism o schemes X→Yis called a Nisne ich mo phism i i
is an ´e ale mo phism such ha o e e y x∈X he e exis s an y∈Yin he ibe −1(x)
o xsuch ha he induced mo phism o esidue ields k(x)→k(y)is an isomo phism.
De ini ion 1.5 (Nisne ich Si e):Le Sbe a scheme. A Nisne ich co e o X∈Ob(Sm/S)
is a ini e se o ´e ale mo phisms { α∶Uα→X}α∈A⊂Mo (Sm/S)such ha o any
x∈X he e is some α∈Aand a poin u∈Uαsuch ha he induced mo phism o esidue
ields k(x)→k(u)is an isomo phism. The ca ego y o smoo h schemes o e Swi h he
Nisne ich co e ings is called he Nisne ich si e (Sm/S)Nis o S.
Re e ence. See [39, De ini ion 1.2] and [1, Exe cise 3.40].
De ini ion 1.6 (Dis inguished Nisne ich Squa e):Le Sbe a Noe he ian scheme o
ini e K ull dimension. A ca esian squa e in (Sm/S)Nis is called an dis inguished Nis-
ne ich squa e i i∶U→Xis a Za iski open imme sion and i p∶V→Xis ´e ale and
1.1 The Uns able A1-Mo i ic Homo opy Ca ego y 21
i p−1(X/U)→(X/U)is an isomo phism o schemes whe e X/Uadmi s he educed
induced scheme s uc u e.
Re e ence. See [39, De ini ion 1.3].
De ini ion 1.7 (Nisne ich Excision):Le Sbe a Noe he ian scheme o ini e K ull
dimension. A simplicial p eshea F∈Ob(PSh((Sm/S)Nis,sSe Q)locinj)o e Ssa is ies
Nisne ich excision i i sends e e y dis inguished Nisne ich squa e o a homo opy pull-
back squa e in he model ca ego y o simplicial se s sSe Qand i F(∅)=∗.
De ini ion 1.8: Le Sbe a Noe he ian scheme o ini e K ull dimension and le he
homo opy ca ego y o simplicial p eshea es o e Sbe
Hs(S)∶=Ho(PSh((Sm/S)Nis ,sSe Q)locinj).
Re e ence. See he exposi ion ollowing [39, De ini ion 2.1].
De ini ion 1.9: Le Sbe a Noe he ian scheme o ini e K ull dimension.
(1) A simpliclal p eshea Fis called A1-local i o all simpliclal p eshea es G he
ollowing mo phism induced by he p ojec ion πG∶G×SA1
S→Gis a bijec ion
π∗
G∶HomHs(S)(G, F )→HomHs(S)(G×SA1
S, F ).
(2) A mo phism o simplicial p eshea es ∶F→Gis called an A1-weak equi alence i
o e e y A1-local simpliclal p eshea H he ollowing mo phism is a bijec ion
∗∶HomHs(S)(G, H)→HomHs(S)(F, H).
(3) A mo phism o simplicial p eshea es ∶F→Gis called an A1- ib a ion i i
admi s he igh li ing p ope y wi h espec o all co ib a ions in he model ca ego y
PSh((Sm/S)Nis ,sSe Q)locinj which a e also A1-weak equi alences.
Re e ence. See [39, De ini ion 2.1].
P oposi ion 1.10: Le Sbe a Noe he ian scheme o ini e K ull dimension. The e
exis s he ollowing commu a i e diag am o monoidal Quillen equi alences.
PSh((Sm/S)Nis ,sSe Q)locinj Sh((Sm/S)Nis ,sSe Q)locinj
PSh((Sm/S)Nis ,sSe Q)locp oj Sh((Sm/S)Nis,sSe Q)locp oj ⋅
aNis
i
aNis
i
Id Id Id Id

1.1 The Uns able A1-Mo i ic Homo opy Ca ego y 22
P oo . See [13] and [7] and [27] and [39].
P oposi ion 1.11: Le Sbe a Noe he ian scheme o ini e K ull dimension. The le
Bous ield localiza ion o he model ca ego ies abo e a he class o A1-weak equi alences
induces he ollowing commu a i e diag am o symme ic monoidal Quillen equi alences
o model ca ego ies.
PSh((Sm/S)Nis ,sSe Q)A1-locinj Sh((Sm/S)Nis ,sSe Q)A1-locinj
PSh((Sm/S)Nis ,sSe Q)A1-locp oj Sh((Sm/S)Nis,sSe Q)A1-locp oj ⋅
aNis
i
aNis
i
Id Id Id Id
P oo . See [32, A.3.7.3] and [39].
P oposi ion 1.12: Le Sbe a Noe he ian scheme o ini e K ull dimension. The e
exis s a canonical Quillen adjunc ion o model ca ego ies
Id ∶PSh((Sm/S)Nis ,sSe Q)locinj ⇆PSh((Sm/S)Nis ,sSe Q)A1-locinj ∶Id
ha sa is ies he ollowing p ope ies:
(1) The ib an objec s in he model ca ego y on he igh a e he A1-local and ib an
objec s in he model ca ego y on he le .
(2) The co ib a ions in he model ca ego y on he igh a e also he co ib a ions in he
model ca ego y on he le .
(3) The weak equi alances in he model ca ego y on he igh a e he A1-weak equi .
P oo . See [32, P oposi ion A.2.8.2].
De ini ion 1.13 (A1-In a iance):Le Sbe a Noe he ian scheme o ini e K ull dimen-
sion. A simplicial p eshea F∈Ob(PSh((Sm/S)Nis,sSe Q)A1-locinj)is called A1-in a ian
i o e e y X∈Ob((Sm/S)Nis) he canonical mo phism πX∶X×SA1
S→Xinduces a
weak equi alence F(X)≃F(X×SA1
S)in he model ca ego y sSe Q.
De ini ion 1.14 (Mo i ic Spaces):Le Sbe a Noe he ian scheme o ini e K ull di-
mension. The ull subca ego y PSh((Sm/S)Nis,sSe Q)
A1-locinj o he A1- ib an simplicial
p eshea es is called he ca ego y o mo i ic spaces o e S.
1.1 The Uns able A1-Mo i ic Homo opy Ca ego y 23
De ini ion 1.15 (Uns able A1-Homo opy Ca ego y):Le Sbe a Noe he ian scheme
o ini e K ull dimension. The (poin ed) uns able A1-homo opy ca ego y o e he base
scheme Sis he homo opy ca ego y
H(S)∶=Ho(PSh((Sm/S)Nis ,sSe Q)A1-locinj),
H∗(S)∶=Ho(PSh∗((Sm/S)Nis ,sSe Q)A1-locinj).
Re e ence. See he exposi ion ollowing [39, De ini ion 2.1 and P oposi ion 2.12].
De ini ion 1.16 (Simplicial Ci cle and Ta e Ci cle):Le Sbe a Noe he ian scheme o
ini e K ull dimension. Le he simplicial ci cle and Ta e ci cle o e Sbe:
(1) Le S1
sbe he canonical poin ed cons an simplicial p eshea ∆1/∂∆1.
(2) Le S1
be simplicial p eshea ep esen ed by Gm,S ∶=A1
S/{0}and poin ed by 1.
(3) Le Tbe he canonical poin ed simplicial p eshea ep esen ed by A1
S/Gm,S.
Re e ence. See he exposi ion ollowing [39, P oposi ion 2.14].
Rema k 1.17 (No a ion):
(1) Le Sn
s∶=(S1
s)∧⋯∧(S1
s)and Sn
∶=(S1
)∧⋯∧(S1
)and Tn∶=(T)∧⋯∧T.
(2) Le Sp,q ∶=(S1
s)p−q∧(S1
)q.
De ini ion 1.18 (Suspension):Le Sbe a Noe he ian scheme o ini e K ull dimension.
Le F∈Ob(PSh((Sm/S)Nis ,sSe Q)A1-locinj)be a simplicial p eshea o e S.
(1) Le Σs(F)∶=S1
s∧Fand le Σn
s(F)∶=Sn
s∧F.
(2) Le Σ (F)∶=S1
∧Fand le Σn
(F)∶=Sn
∧F.
(3) Le ΣT(F)∶=T∧Fand le Σn
T(F)∶=Tn∧F.
De ini ion 1.19 (Mo i ic Sphe e):Le Sbe a Noe he ian scheme o ini e K ull dimen-
sion. A simplicial p eshea F∈Ob(PSh∗((Sm/S)Nis,sSe Q)A1-locinj)is called a mo i ic
sphe e i he e exis na u al numbe s m, n ≥1and an A1-weak equi alence
Sn
s∧Sm
≃F.
1.2 The S able A1-Mo i ic Homo opy Ca ego y 24
Example 1.20: Le Sbe a Noe he ian scheme o ini e K ull dimension. The ollowing
a e mo i ic sphe es o e S.
(1) S1
s∧S1
≃T≃(P1
S,{∞}).
(2) S2n,n ≃Sn
s∧Sn
.
(3) T=S1
s∧Gm,S.
P oo . See [39, Lemma 2.15 and Co olla y 2.18] and [1, Chap e 4.6].
De ini ion 1.21 (Thom Space):Le Sbe a Noe he ian scheme o ini e K ull dimen-
sion. Le πX∶V→Xbe a ec o bundle o smoo h schemes o e Sand le i∶X→Vbe
he ze o sec ion. The Thom space o he ec o bundle is he ollowing homo opy co ibe
ThX(V)∶=V/(V−i(X)).
Re e ence. See [39, De ini ion 2.16].
Theo em 1.22 (Pu i y):Le Sbe a quasi-compac and quasi-sepa a ed scheme. Le
i∶Z→Xbe a closed imme sion in (Sm/S)Nis. Le also νZ∶NZ(X)→Zbe he no mal
bundle. The e exis s a canonical A1-weak equi alence
ThZ(NZ(X))≃X/(X−Z).
P oo . See [1, Theo em 7.6].
1.2 The S able A1-Mo i ic Homo opy Ca ego y
De ini ion 1.23 (Uni e sal P ope y):Le Sbe a scheme. Le C⊗∈Ob(CAlg(P L,⊗))
be a locally p esen able symme ic monoidal ∞-ca ego y. The s able A1-homo opy ca e-
go y o symme ic mo i ic spec a SH⊗(S)o e Sadmi s he uni e sal p ope y:
(1) The ollowing unc o o ∞-ca ego ies is ully ai h ul
Φ∶Fun⊗,L(SH⊗(S),C⊗)→Fun⊗(Sm/S,C⊗)
F↦F○Σ∞
P1(−)+
(2) The essen ial image o he unc o Φis he se o symme ic monoidal ∞- unc o s
F∶Sm/S→C⊗which a e cha ac e ized by he ollowing p ope ies:
1.2 The S able A1-Mo i ic Homo opy Ca ego y 25
●A1-in a iance.
●Nisne ich excision.
●(ΣP1-S abili y):The co ibe co ib(F(Gm,S)→F(A1
S)) is ⊗-in e ible.
Re e ence. See [48, Co olla y 5.11].
De ini ion 1.24: Le Sbe a Noe he ian scheme o ini e K ull dimension. The ca ego y
o sequen ial Σs-spec a SpN(PSh∗((Sm/S)Nis,sSe ),Σs)o e he base scheme Sis he
ollowing ca ego y:
(1) A sequen ial Σs-spec um F=(F●, σF●
●)is a sequence {Fn}n∈Nin he ollowing
poin ed ca ego y PSh∗((Sm/S)Nis,sSe )along wi h basepoin p ese ing mo phisms
σF●
n∶S1
s∧Fn→Fn+1.
(2) A mo phism o sequen ial Σs-spec a h●∶(F●, σF●
●)→(G●, σG●
●)is a sequence
{hn}n∈No basepoin p ese ing mo phisms hn∶Fn→Gnsuch ha he ollowing
diag ams commu e.
S1
s∧Fn→→
σF●
n
↓↓
S1
s∧Gn
σG●
n
↓↓
Fn+1hn+1
→→Gn+1
De ini ion 1.25: Le Sbe a Noe he ian scheme o ini e K ull dimension. Le also
F∈Ob(PSh∗((Sm/S)Nis ,sSe )) be a poin ed simplicial p eshea o e Sand le πn(F)
be he shea associa ed o he p eshea
(Sm/S)Nis →G p , U ↦πn(F(U)) , n ≥1.
Rema k 1.26: I n≥2 hen πn(F)is a Nisne ich shea o abelian g oups o e S.
De ini ion 1.27: Le Sbe a Noe he ian scheme o ini e K ull dimension. Le also
F=(F●, σF●
●)be a Σs-spec um in SpN(PSh∗((Sm/S)Nis,sSe ),Σs). The shea o s able
homo opy g oups o 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)→⋯).
Re e ence. See he exposi ion ollowing [14, Example 2.2].
1.2 The S able A1-Mo i ic Homo opy Ca ego y 32
●(Base Change):I αβ is sepa a ed o ini e ype hen any ca esian squa e
Xδ
δγ →→
gδα
↓↓
Xγ
gγβ
↓↓
Xα αβ
→→Xβ
implies he base change na u al isomo phisms
g∗
γβ ○ αβ!→ δγ!○g∗
δα , gδα∗○ !
δγ → !
αβ ○gγβ∗.
●(P ojec ion Fo mula):The p ojec ion o mula is sa is ied
!(G∧ ∗(F))≃ !(G)∧F.
P oo . See [11, Theo em 2.9].
Example 1.56 (Ta e Twis ):Le Sbe a Noe he ian scheme o ini e K ull dimension.
Le πS∶P1
S→Sbe he canonical p ojec ion and le iS∶S→P1
Sbe he inclusion and le
1S(1)∶=i∗
S(π!
S(1S))[−2],1S(n)∶=1S(1)∧⋯∧1S(1)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
n- imes
, n ≥1.
Rema k 1.57 (No a ion):Since he spec um 1S(1)is ⊗-in e ible in SH⊗(S)we se
1S(0)∶=1S,1S(1)∧1S(−1)=1S(0),1S(−n)∶=1S(−1)∧⋯∧1S(−1)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
n- imes
, n ≥1.
Rema k 1.58 (No a ion):Le he n- h Ta e wis o a symme ic ΣP1-spec um be
F(n)∶=F∧1S(n), F ∈Ob(SH⊗(S)) , n ∈Z.
De ini ion 1.59 (E ec i e Spec a):Le Sbe a Noe he ian scheme o ini e K ull di-
mension. The iangula ed ca ego y o e ec i e spec a in he A1-homo opy ca ego y
SH(S)is he ollowing localizing ull iangula ed subca ego y
SH(S)e ∶=LocSH(S)({Σ∞
P1(X)+∣X∈Ob((Sm/S)Nis)}).
Re e ence. See he in oduc ion o [4, Chap e 13].

1.2 The S able A1-Mo i ic Homo opy Ca ego y 33
De ini ion 1.60 (Homology):Le Sbe a Noe he ian scheme o ini e K ull dimension.
Le Ebe a Σs, -bispec um in he s able A1-homo opy ca ego y SH(S). The big aded
homology heo y o Eis he unc o
E●,●∶(Sm/S)Nis →AbZ×Z,
Ep,q(X)∶=HomSH(S)(Sp,q,Σ∞
P1(X)+∧E).
De ini ion 1.61 (Cohomology):Le Sbe a Noe he ian scheme o ini e K ull dimen-
sion. Le Ebe a Σs, -bispec um in he s able A1-homo opy ca ego y SH(S). The bi-
g aded cohomology heo y o Eis is he unc o
E●,●∶(Sm/S)op
Nis →AbZ×Z,
Ep,q(X)∶=HomSH(S)(Σ∞
P1(X)+, E ∧Sp,q).
Re e ence. See [14, Chap e 3.2] and [2, Chap e 4.2].
Example 1.62: Le Sbe a Noe he ian scheme o ini e K ull dimension and le also
X∈Ob((Sm/S)Nis). The e exis s canonical isomo phisms o abelian g oups
KGLp,q
S(X)≅K2q−p(X), p, q ∈Z.
P oo . See [10, Equa ion (13.2.1.2)].
Example 1.63 (Mo i ic Cohomology Theo y):Le kbe a ield o cha ac e is ic ze o
and le X∈Ob((Sm/k)Nis). The e exis s canonical isomo phisms o abelian g oups
MZp,q
k(X)∶=HomSH(Spec(k))(Σ∞
P1(X)+, Sp,q ∧MZk), p, q ∈Z.
P oo . See [14, Chap e 3.1].
Co olla y 1.64: Le kbe a ield o cha ac e is ic ze o and le X∈Ob((Sm/k)Nis ).
The e exis canonical isomo phisms o abelian g oups
MZp,q
k(X)≅Hp
Za (X, Zk(q))≅CHq(X, 2q−p), q ≥0,2q−p≥0.
P oo . See [14, Chap e 3.1] and [2, Theo em 4.16].
Theo em 1.65 (Naumann-Spi zweck):I Sis a Noe he ian scheme o ini e K ull di-
mension such ha S=⋃αSpec(Rα)and Rαa e coun able ings hen SH(S)sa is ies
B own ep esen abili y.
1.2 The S able A1-Mo i ic Homo opy Ca ego y 34
P oo . See [40].
Theo em 1.66 (Naumann-Øs ae -Spi zweck):Le Sbe a Noe he ian scheme o ini e
K ull dimension.
(1) Le M●be an Adams-g aded Landwebe exac MU●-module. The e is a Ta e spec-
um E∈Ob(SH(S)cell)and an isomo phism o homology heo ies on SH(S)whe e
E●,●(−)=MGL●,●(−)⊗MU●(∗)M●.
(2) I M●is a g aded MU●-algeb a hen Eadmi s a quasi-mul iplica ion which ep e-
sen s he ing s uc u e on he Landwebe exac heo y.
P oo . See [41].
De ini ion 1.67 (Complex Be i Realiza ion):The complex ealiza ion unc o on Sm/C
induces he ollowing unc o s on he uns able and he s able A1-homo opy ca ego y
(−)(C)∶Sm/C→SmM d , X ↦X(C),
H(Spec(C))→Spc ,ReC∶SH(Spec(C))→Sp.
Re e ence. See [64, Chap e 3.1].
De ini ion 1.68 (Real Be i Realiza ion):Le C2∶=Gal(C/R)be he absolu e Galois
g oup o R. Le e e y X(C)∈Ob(Sm/R)admi a g oup ac ion o C2by complex conju-
ga ion. Le ΦC2be he geome ic ixed poin unc o . The e exis s he ollowing unc o
ReR∶SH(Spec(R))ReC2
→Sp(C2)ΦC2
→Sp.
Re e ence. See [64, Chap e 3.2].
Rema k 1.69 (Slice Fil a ion):Le Sbe a ini e dimensional and Noe he ian scheme.
(1) The inclusion unc o inadmi s a igh adjoin nand es ablishes a il a ion
in∶Σn
P1(SH(S)e )→SH(S)∶ n,
n∶=in○ n∶SH(S)→SH(S),
⋯⊂Σn+1
P1(SH(S)e )⊂Σn
P1(SH(S)e )⊂⋯⊂SH(S).
1.2 The S able A1-Mo i ic Homo opy Ca ego y 35
(2) The slice il a ion and he n- h slices sn(E)in he dis inguished iangles o Ea e
⋯→ n+1(E)→ n(E)→⋯→E,
n+1(E)→ n(E)→sn(E)→ n+1(E)[1].
(3) The slice il a ion implies he so called slice spec al sequence
s−q(E)p+q,n(F)⇒Ep+q,n(F),
which is compa able o he A iyah-Hi zeb uch spec al sequence.
P oo . See [14, Chap e 4] and he in oduc ion o [4, Chap e 13.1].
Example 1.70 (E ec i e Algeb aic K-Theo y Spec um):Le Sbe a ini e dimensional
and Noe he ian scheme. The e ec i e algeb aic K- heo y spec um o e Sis
kglS∶= 0(KGLS).
P oo . See [2, De ini ion 3.19 and Exe cise 3.5].
Example 1.71: Le kbe a ield o cha ac e is ic ze o. Voe odsky’s mo i ic Eilenbe g-
MacLane spec um o e S=Spec(k)sa is ies
MZS=s0(KGLS).
P oo . See [2, De ini ion 3.19 and Exe cise 3.5].
Rema k 1.72 (Homo opy -S uc u e):Le Sbe ini e dimensional and Noe he ian.
(1) Le SH(S)≥dbe he iangula ed subca ego y o SH(S)gene a ed by he homo opy
colimi s and he ex ensions o
{Σp,q
s, (Σ∞
P1(X)+)∣X∈Ob((Sm/S)Nis), p −q≥d}.
(2) A spec um in SH(S)≥dis called d-connec i e o else connec ed i d=0.
(3) The iangula ed subca ego y SH(S)≥0is he nonnega i e pa o a unique -s uc u e
on SH(S)which is called he homo opy -s uc u e.
(4) I kis a p e ec ield hen Mo el’s A1-connec i i y heo em implies
SH(S)≥d={F∈Ob(SH(Spec(k)))∣πp,q(F)=0i p−q<d},
SH(S)≤d={F∈Ob(SH(Spec(k)))∣πp,q(F)=0i p−q>d}.
1.2 The S able A1-Mo i ic Homo opy Ca ego y 36
P oo . See [38, Chap e 5.2].
De ini ion 1.73 (Ve y E ec i e Spec a):Le Sbe a ini e dimensional and Noe he-
ian scheme. The iangula ed ca ego y o e y e ec i e spec a SH(S) e is he ull
iangula ed subca ego y o SH(S)gene a ed by he homo opy colimi s and ex ensions o
{Σ∞
P1(X)+∣X∈Ob((Sm/S)Nis)}.
Re e ence. See he exposi ion ollowing [4, Lemma 13.2].
Example 1.74: Le Sbe a ini e dimensional and Noe he ian scheme. The mo i ic
Thom spec um o e he base scheme Sis e y e ec i e
MGLS∈Ob(SH(S) e ).
P oo . See [38, Lemma 13.1].
De ini ion 1.75 (E ec i e Homo opy -S uc u e):Le Sbe a ini e dimensional and
Noe he ian scheme. The ca ego y o e y e ec i e spec a is he non nega i e pa o he
e ec i e homo opy -s uc u e
SH(S) e ∶=SH(S)e ∩SH(S)≥0.
Re e ence. See he in oduc ion o [4, Chap e 13.1].
Rema k 1.76 (Mo i ic S een od Algeb a):Le kbe a pe ec ield and le l≠cha (k).
Le MFlbe Voe odsky’s mo i ic Eilenbe g-MacLane spec um in SH(Spec(k)) ep esen -
ing mo i ic cohomology heo y wi h coe icien s in Fl.
(1) The mo i ic S een od algeb a and he dual mo i ic S een od algeb a a e espec i ely
A●,●∶=(MFl)●,●(MFl),A●,●∶=(MFl)●,●(MFl).
(2) Voe odsky cons uc ed educed powe ope a ions on he mo i ic S een od algeb a o
ields o cha ac e is ic ze o. These esul s we e ex ende by Hoyois-Kelly-Øs ae
and by F ankland and Spi zweck.
(3) The mo i ic S een od algeb a is c ucial o he p oo o he Bloch-Ka o conjec u e.
P oo . See [61].
1.3 Mixed Mo i es 37
1.3 Mixed Mo i es
De ini ion 1.77 (G oup o Algeb aic Cycles o X):Le Sbe a smoo h sepa a ed scheme
o ini e ype o e a pe ec ield.
(1) Le X∈Ob(Sm/S)and le z (X)be he ee abelian g oup gene a ed by he closed
in eg al subschemes o Xo dimension .
(2) The g oup o algeb aic cycles o he scheme Xis he g aded ee abelian g oup
z●(X)∶=⊕
∈Z
z (X).
De ini ion 1.78 (Fini e S-Co espondences):Le Sbe a smoo h sepa a ed scheme o
ini e ype o e a pe ec ield and le X, Y ∈Ob(Sm/S). The g aded abelian g oup o
ini e S-co espondences is he ollowing ee abelian subg oup
cS(X, Y )⊂z●(X×SY), X, Y ∈Ob(Sm/S)
which is gene a ed by he in eg al closed subschemes W⊂X×SYwi h he p ope ies:
(1) The canonical p ojec ion mo phism πX∶W→Xis ini e.
(2) The image πX(W)⊂Xis an i educible componen o X.
Re e ence. See [45, Chap e 2.1] and [29, Chap e 3.1].
De ini ion 1.79 (Smoo h Fini e S-Co espondences):Le Sbe a smoo h sepa a ed
scheme o ini e ype o e a pe ec ield.
(1) Ob(FinCo (Sm/S))∶=Ob(Sm/S).
(2) HomFinCo (Sm/S)([X],[Y])∶=cS(X, Y ),[X],[Y]∈Ob(FinCo (Sm/S)).
Re e ence. See [45, Chap e 2.1] and [29, Chap e 3.1].
Rema k 1.80: Le Sbe a smoo h sepa a ed scheme o ini e ype o e a pe ec ield.
Le X, Y ∈Ob(Sm/S)and le ( ∶X→Y)∈Mo (Sm/S)and le also Γ ⊂X×SYbe he
g aph o o e S. The e exis s a canonical inclusion unc o
iS∶Sm/S→FinCo (Sm/S), X ↦[X],( ∶X→Y)↦Γ .

1.3 Mixed Mo i es 38
De ini ion 1.81 (P eshea es wi h T ans e ):Le Sbe a smoo h sepa a ed scheme o
ini e ype o e a pe ec ield. The ca ego y o p eshea es wi h ans e o e Sis he
ollowing ca ego y o addi i e unc o s
PSh (Sm/S,Ab)∶=Fun⊕(FinCo (Sm/S)op,Ab).
De ini ion 1.82 (Nisne ich Shea wi h T ans e ):Le Sbe a smoo h sepa a ed scheme
o ini e ype o e a pe ec ield. Le iS∶Sm/S→FinCo (Sm/S)be he inclusion unc o .
A Nisne ich shea wi h ans e o e Sis a p eshea wi h ans e Fsuch ha F○iSis
a shea on (Sm/S)Nis. The ull subca ego y o Nisne ich shea es wi h ans e o e Sis
Sh ((Sm/S)Nis ,Ab)⊂PSh ((Sm/S)Nis ,Ab).
Re e ence. See [45, Chap e 2.3] and [29, Chap e 3.1].
Example 1.83: Le Sbe a smoo h sepa a ed scheme o ini e ype o e a pe ec ield.
Le X∈Ob(Sm/S)and le also he p eshea wi h ans e o e S ep esen ed by [X] ia
he ully ai h ul Yoneda embedding be
FinCo (Sm/S)op →PSh (Sm/S,Ab),[X]↦Z
S(X)(−)∶=cS(−, X).
Re e ence. See [45, Chap e 2.3] and [29, Chap e 3.1].
De ini ion 1.84 (E ec i e Mixed Mo i es):Le Sbe a smoo h sepa a ed scheme o
ini e ype o e a pe ec ield. The symme ic monoidal iagula ed ca ego y o e ec i e
mixed mo i es o e Sis he homo opy ca ego y o he ollowing s able model ca ego y
DMe (S)∶=D(Sh ((Sm/S)Nis,Ab))[{Z
S(X×SA1
S)π∗
X
→Z
S(X)}−1
X].
Re e ence. See [12, P oposi ion 3.6] and [29, Chap e 3.1].
De ini ion 1.85 (Big Mixed Mo i es):Le Sbe a smoo h sepa a ed scheme o ini e
ype o e a pe ec ield. The symme ic monoidal iagula ed ca ego y o big mixed
mo i es o e Sis he homo opy ca ego y o he ollowing s able model ca ego y
ZS(1)∶=coke (Z
S(S)→Z
S(P1
S))[−2],,ZS(n)∶=ZS(1)⊗S⋯⊗SZS(1)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
n- imes
, n ≥1,
DM(S)=Ho(SpΣ(PSh ((Sm/S)Nis,Ab)A1-s able,ΣZS(1))A1-s able),ΣZS(1)∶=ZS(1)⊗S(−).
Re e ence. See [29, De ini ion 3.1.4].
1.3 Mixed Mo i es 39
Rema k 1.86: Le Sbe a smoo h sepa a ed scheme o ini e ype o e a pe ec ield.
The Ta e mo i e ZS(1)becomes ⊗-in e ible in he iangula ed ca ego y DM(S)and
1S∶=ZS(0),ZS(1)⊗SZS(−1)=ZS(0),ZS(−n)∶=ZS(−1)⊗S⋯⊗SZS(−1)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
n- imes
, n ≥1.
De ini ion 1.87 (Geome ic Mixed Mo i es):Le Sbe a smoo h sepa a ed scheme o i-
ni e ype o e a pe ec ield. The symme ic monoidal iangula ed ca ego y o geome ic
mixed mo i es o e Sis he ollowing ull subca ego y o compac objec s
DMgm(S)∶=DM(S)c.
Theo em 1.88: Le Sbe a smoo h sepa a ed scheme o ini e ype o e a pe ec
ield. The e exis s he ollowing commu a i e diag am o symme ic monoidal locally
p esen able s able ∞-ca ego ies whe e he ho izon al unc o s a e ully ai h ul.
DMe
gm(S)→→
↓↓
DMe (S)
↓↓
DMgm(S)→→DM(S)
P oo . See [45, Theo em 2.18].
Theo em 1.89 (Röndigs-Øs ae ):Le kbe a ield o cha ac e is ic ze o. The e exis s
he ollowing equi alence o symme ic monoidal iangula ed ca ego ies
Ho(ModMZk(SH(Spec(k)))A1-s able)≃DM(Spec(k)).
P oo . See [49].
Theo em 1.90: Le Sbe a Noe he ian scheme o ini e K ull dimension. The e ex-
is s he ollowing commu a i e diag am whe e e e y unc o is symme ic monoidal and
admi s a igh adjoin .
Sm/SH⊗
∗(S)SH⊗(S)e SH⊗(S)
FinCo (Sm/S)DM⊗(S)e DM⊗(S)
iS
(−)+
Re e ence. See [11, Rema que 2.6 (2)].
1.3 Mixed Mo i es 40
Theo em 1.91 (Six-Func o Fo malism):Le Sbe a Noe he ian scheme o ini e K ull
dimension. Le also ∶X→Ybe a mo phism o e S.
(1) The unc o ∗is symme ic monoidal. I is sepe a ed o ini e ype hen he e
exis s and adjunc ion !⊣ !. The iangula ed ca ego y DM(X)admi s a closed
symme ic monoidal s uc u e. The six unc o s sa is y he ollowing adjunc ions
HomDM(X)( ∗(L), K)≅HomDM(Y)(L, ∗(K)),
HomDM(Y)( !(L), K)≅HomDM(X)(L, !(K)),
HomDM(X)(M⊗XK, L)≅HomDM(X)(M, HomDM(X)(K, L)).
(2) The e exis s a na u al ans o ma ion which is also an isomo phism i is p ope
α ∶ ∗→ !.
(3) Fo any smoo h and sepa a ed mo phism ∶X→So ela i e dimension d he e
exis s a canonical na u al isomo phism o unc o s
β ∶ ∗→ !○(−)(−d)[−2d].
(4) I αβ is a sepa a ed mo phism o ini e ype hen any ca esian squa e implies he
ollowing base change na u al isomo phisms
Xδ
δγ →→
gδα
↓↓
Xγ
gγβ
↓↓
Xα αβ
→→Xβ
g∗
γβ ○ αβ!→ δγ!○g∗
δα , gδα∗○ !
δγ → !
αβ ○gγβ∗.
(5) I ∶Y→Xis sepa a ed and o ini e ype hen he e exis na u al isomo phisms
!(K)⊗XL→ !(K⊗Y ∗(L)),
HomDM(X)( !(L), K)→ ∗(HomDM(Y)(L, !(K))),
!(HomDM(X)(L, M))→HomDM(Y)( ∗(L), !(M)).
1.3 Mixed Mo i es 41
(6) (Localiza ion):Fo any closed imme sion i∶Z→Swi h complemen a y open
imme sion j∶U→S he e exis s a dis inguished iangle o na u al ans o ma ions
j!○j!→Id →i∗○i∗→j!○j!○(−)[1].
P oo . See [10, In oduc ion A.5.1].
Theo em 1.92 (Duali y o he Six-Func o Fo malism):Le Sbe a Noe he ian scheme
o ini e K ull dimension and le KSbe ⊗-in e ible in DM(S). Le ∶Y→Xbe a
mo phism o sepa a ed S-schemes o ini e ype. Le ∶X→Sbe a sepa a ed mo phism
o ini e ype and le
KX∶= !(KS), DX(M)∶=HomDM(X)(M, KX).
(1) KXis a dualizing objec in DM(X)and he e exis s a canonical isomo phism
M≅DX(DX(M)).
(2) The e exis s he ollowing canonical isomo phism
DX(M⊗XDX(N))≅HomDM(X)(M, N).
(3) The e exis he ollowing canonical isomo phisms
DY( ∗(M))≅ !(DX(M)) , ∗(DX(M))≅DY( !(M)),
DX( !(N))≅ ∗(DY(N)) , !(DY(N))≅DX( ∗(N)).
P oo . See [10, In oduc ion A.5.2].
Example 1.93: Le kbe a ield and le S∈Ob((Sm/k)Nis )and le X∈Ob((Sm/S)Nis ).
The six- unc o o malism implies a na u al isomo phism o abelian g oups
HomDM(S)(MS(X),ZS(q)[p])≅HomDM(Spec(k))(Mk(X),Zk(q)[p]) , p, q ∈Z.
P oo . See [45, Theo em 2.17].
De ini ion 1.94 (Mo i ic Cohomology):Le kbe a ield and le X∈Ob((Sm/k)Za )
and le A∈Ob(Ab)be an abelian g oup. The mo i ic cohomology g oups o Xa e he
hype cohomology g oups o he cochain complexes o shea es Zk(q)on he Za iski si e
Hp,q(−, A)∶=Hp
Za (−, Ak(q))∶Smop
/k→AbZ×Z,
2.1 The S able ∞-Ca ego y o Spi zweck Mo i es 48
(3) (Localiza ion):Fo any closed imme sion i∶Z→Swi h complemen a y open
imme sion j∶U→S he e exis s a dis inguished iangle o na u al ans o ma ions
j!○j!→Id →i∗○i∗→j!○j!○(−)[1].
P oo . See [52, Cap e 9].
Rema k 2.11: Le ∶X→Ybe a mo phism o Noe he ian sepa a ed schemes o ini e
K ull dimension o e S=Spec(Z). The unc o ∗is symme ic monoidal and admi s
a le adjoin #i he mo phism is smoo h.
P oo . See [52, Cap e 9].
Rema k 2.12: The canonical o ge ul unc o U∶Ho(ModMZSpi
X(SH(X)))→SH(X)o
iangula ed ca ego ies induces a six- unc o o malism on he iangula ed ca ego y o
Spi zweck mo i es Ho(ModMZSpi
X(SH(X))) which includes he base change p ope y and
he localiza ion iangle and excep possibly absolu e pu i y.
P oo . See [52, Cap e 9].
Rema k 2.13: The six- unc o o malism o iangula ed ca ego ies o Spi zweck mo-
i es ex ends o a six- unc o o malism o s able ∞-ca ego ies o Spi zweck mo i es.
Co olla y 2.14: Le kbe a ield o cha ac e is ic ze o. Le MZkbe Voe odsky’s mo-
i ic Eilenbe g-MacLane spec um o e Spec(k). The iangula ed ca ego y o Spi zweck
mo i es o e he Spec(k)is symme ic monoidal equi alen o Voe odsky’s iangula ed
big ca ego y o mixed mo i es o e Spec(k)since he e exis he ollowing equi alences
DM(Spec(k))≃Ho(ModMZk(SH⊗(Spec(k)))A1-s able)
≃Ho(ModMZSpi
k(SH⊗(Spec(k)))A1-s able).
P oo . This ollows om Theo em 2.1 and Theo em 1.89.
Rema k 2.15: Le kbe a ield o cha ac e is ic ze o. The e exis s he equi alence
Ho(ModMZSpi
k∧HQ(SH⊗(Spec(k))Q)A1-s able)≃Ho(ModMZk∧HQ(SH(Spec(k))Q)A1-s able).
P oo . This ollows om Theo em 1.118 and Co olla y 2.14.

2.1 The S able ∞-Ca ego y o Spi zweck Mo i es 49
De ini ion 2.16 (Geome ic Spi zweck Mo i es):The symme ic monoidal iangula ed
ca ego y o geome ic Spi zweck mo i es o e Xis he ull iangula ed subca ego y o
compac objec s in he symme ic monoidal iangula ed ca ego y o Spi zweck mo i es
(DMgm(X),⊗)∶=(DM(X)c,⊗),DMgm(X)⊂DM(X).
Conjec u e 2.17 (Beilinson-Soul´e):Le Dbe a Dedekind domain o mixed cha ac e -
is ic and le S=Spec(D). I Xis a smoo h scheme o e S hen
HomSH(S)(Σ∞
P1(X)+,MZSpi
S(q)[p])=0i q≥0, p <0o else q>0, p =0.
Re e ence. See he exposi ion ollowing [54, Theo em 3.3].
Example 2.18: A1
Sand Gm,S o e a numbe ield S=Spec(k)and also Spec(Z)and
any localiza ion o a numbe ing sa is ies he Beilinson-Soul´econjec u e. Smoo h cu es
o e ini e ields sa is y he Beilinson-Soul´econjec u e. Moduli spaces o genus 0cu es
wi h nma ked poin s M0,n and i s Deligne-Mum o d compac i ica ions 
M0,n a e sa is y-
ing he Beilinson-Soul´econjec u e o n≥3. Also P1
S/Xo e a numbe ield S=Spec(k)
whe e Xis a ini e se S-poin s o P1
Ssa is ies he Beilinson-Soul´econjec u e.
P oo . See [54, Example 3.5].
Theo em 2.19 (Beilinson-Lich enbaum):Le Xbe a smoo h scheme o e S=Spec(D).
Le ε∶X´e →XZa be he canonical unc o o si es. Le Le ine’s cycle complexes be
ΓXZa (n)∈Ob(D(Sh(XZa ,Z))) whe e n∈Z. Le m>0be in e ible on X. The e exis s
a canonical isomo phism in he de i ed ca ego y D(Sh(XZa ,Z/mZ)) whe e
ΓXZa (n)⊗L
ZZ/mZ≅τ≤n(Rε∗(µ⊗n
m,X´e )).
P oo . See [54, Theo em 3.3].
De ini ion 2.20 (Mo i ic Cohomology):Le Dbe a Dedekind domain o mixed cha -
ac e is ic and le S=Spec(D). Le A∈Ob(Ab)and le 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.
Re e ence. See he exposi ion ollowing [54, De ini ion 3.1].
Example 2.21: Le Xbe a smoo h scheme o e S=Spec(D)and le A∈Ob(Ab).
2.2 Ta e Mo i es 50
(1) Hp,q(X, A)=0i p∈Zand q<0.
(2) Hp,q(X, A)=0i p>qand i Xis he spec um o a local ing.
(3) Hp,q(X, A)=0i p>q+dim(X).
P oo . See [54, Theo em 3.2].
Theo em 2.22 (M. Spi zweck):Le Dbe a Dedekind domain o mixed cha ac e is ic
and le S=Spec(D). Le also X∈Ob((Sm/S)Nis). The e exis canonical isomo phisms
HomSH(S)(Σ∞
P1(X)+,MZSpi
S(q)[p])≅HomDM(X)(ZX(0),ZX(q)[p])≅Hp(X, q), p, q ∈Z.
P oo . See [52, Theo em 7.18 and Co olla y 7.19 and Co olla y 7.20].
2.2 Ta e Mo i es
De ini ion 2.23 (Ta e Mo i es):Le Xbe a Noe he ian sepa a ed scheme o ini e
K ull dimension. The symme ic monoidal iangula ed ca ego y o Ta e mo i es o e
Xis he ollowing ull localizing iangula ed subca ego y
(DMT(X),⊗)∶=Loc(DM(X),⊗)({ZX(n)}n∈Z),DMT(X)⊂DM(X).
De ini ion 2.24 (Geome ic Ta e Mo i es):Le Xbe a Noe he ian sepa a ed scheme
o ini e K ull dimension. The symme ic monoidal iangula ed ca ego y o geome ic
Ta e mo i es o e Xis he ollowing ull iangula ed subca ego y o compac objec s
(DMTgm(X),⊗)∶=(DMT(X)c,⊗).
Example 2.25: Le S=Spec(Z). Le M0,n be he moduli space o genus 0algeb aic
cu es wi h nma ked poin s. Le also 
M0,n be he Deligne-Mum o d compac i ica ion o
M0,n. The ollowing a e Ta e mo i es
MS(M0,n),MS(
M0,n)∈Ob(DMT(S)) , n ≥3.
P oo . See [51, Chap e 3].
Example 2.26: Le S=Spec(C)and le Con n(R2)be he opological space o un-
o de ed con igu a ions o npoin s in he plane. Fo e e y posi i e na u al numbe n
he e exis s a smoo h scheme Cno e Ssuch ha he se o complex poin s Cn(C)wi h
he canonical complex analy ic opology is Con n(R2). The ollowing a e Ta e mo i es
MS(Cn(C))∈Ob(DMT⊗(S)) , n ≥1.
2.2 Ta e Mo i es 51
P oo . See he in oduc ion o [25, Chap e 1].
Rema k 2.27: I kis a ield o cha ac e is ic ze o hen he e exis he ollowing equi -
alences o iangula ed ca ego ies
DMT(Spec(k))Q≃DMT(Spec(k),Q),DMAT(Spec(k))Q≃DMAT(Spec(k),Q).
P oo . This ollows om Rema k 2.15. See also De ini ion 4.53 and De ini ion 4.51.
De ini ion 2.28 (Mul iplica i e Pe iodiza ion):Le Xbe any base scheme. A mul i-
plica i e (n, m)-pe iodiza ion o a mo i ic E∞-algeb a Eis a g aded mo i ic E∞-algeb a
PE whose mul iplica ion mo phism sa is ies he ollowing canonical equi alence
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.
Re e ence. See he exposi ion ollowing [23, Theo em 1.1].
Example 2.29: Le Xbe a Noe he ian sepa a ed scheme o ini e K ull dimension. The
ollowing mo i ic E∞-algeb as a e mul iplica i ely (2,1)-pe iodizable
KGLX,MGLX,MZX∈Ob(CAlg(SH⊗(X))).
P oo . See [23, Co olla y 4.11].
Example 2.30: Le Xbe a Noe he ian sepa a ed scheme o ini e K ull dimension. The
E∞-algeb a MZSpi
Xadmi s a unique E∞-o ien a ion. The E∞-algeb a MGLXadmi s a
mul iplica i e pe iodiza ion. The E∞-o ien a ion φ∶MGLX→MZSpi
Ximplies he e o e
MZSpi
Xalso o admi a mul iplica i e (2,1)-pe iodiza ion.
P oo . See [52, Rema k 10.2].
Rema k 2.31: Le Dbe a Dedekind domain o mixed cha ac e is ic and le S=Spec(D).
An explici pe iodiza ion o he E∞-algeb a MZSpi
Swas cons uc ed by M. Spi zweck.
Re e ence. See [52, Appendix C].
Example 2.32: Le Xbe a Noe he ian sepa a ed scheme o ini e K ull dimension.
Any E∈Ob(CAlg(SH⊗(X))) wi h an E∞-o ien a ion is mul iplica i ely pe iodizable.
2.2 Ta e Mo i es 52
P oo . See he exposi ion ollowing [23, Theo em 1.1].
Co olla y 2.33: Le Xbe a Noe he ian sepa a ed scheme o ini e K ull dimension.
Spi zweck’s mo i ic E∞-algeb a o e Xis mul iplica i ely (2,1)-pe iodizable wi h
MZSpi
X∈Ob(CAlg(DM⊗(X))) ,PMZSpi
X∈Ob(CAlg(DM⊗(X)Z)),
PMZSpi
X∶=⊕
k∈Z
MZSpi
X∧S2k,k.
P oo . This ollows om Example 2.30 and De ini ion 2.28.
De ini ion 2.34: Le Xbe Noe he ian sepa a ed scheme o ini e K ull dimension and
G Z
0(−)∶DM⊗(X)Z→DM⊗(X),⊕
n∈Z
En↦E0.
Rema k 2.35: Le Xbe a Noe he ian sepa a ed scheme o ini e K ull dimension.
The e exis s an equi alence o mo i ic E∞-algeb as
MZSpi
X
≃
→G Z
0(PMZSpi
X).
P oo . See he exposi ion ollowing [56, De ini ion 4.1].
De ini ion 2.36: Le Xbe a Noe he ian sepa a ed scheme o ini e K ull dimension.
Le E∈Ob(CAlg(SH⊗(X))). The s able ∞-ca ego y o cellula E-module spec a in
SH⊗(X)is he smalles ull s able subca ego y o he s able ∞-ca ego y SH⊗(X)which
includes {E∧Sn,1}n∈Zand is closed unde small colimi s
ModE(SH⊗(X))cell ⊂SH⊗(X).
Rema k 2.37: Le Xbe a Noe he ian sepa a ed scheme o ini e K ull dimension.
Le E∈Ob(CAlg(SH⊗(X))). Since he mo i ic sphe es {Sn,1}n∈Za e ⊗-in e ible in
SH⊗(X), he E-module spec a {E∧Sn,1}n∈Za e ⊗-in e ible in ModE(SH⊗(X)).
P oo . See [23, Example 2.8].
Example 2.38: Le Xbe a Noe he ian sepa a ed scheme o ini e K ull dimension.
The e exis s a symme ic monoidal equi alence o s able ∞-ca ego ies
DMT⊗(X)≃DM⊗(X)cell ≃ModMZSpi
X(SH⊗(X))cell.
2.2 Ta e Mo i es 53
De ini ion 2.39: Le he symme ic monoidal s able ∞-ca ego y o Z-g aded chain com-
plexes in he ca ego y o abelian g oups Ab wi h he Day con olu ion ⊗-p oduc be
D⊗(Ab)Z∶=Fun(N(Z)⊗,D⊗(Ab))⊗Day .
De ini ion 2.40 (∞-Ca ego y o Z-G aded Spec a):Le he symme ic monoidal s able
∞-ca ego y o Z-g aded spec a wi h he Day con olu ion ⊗-p oduc be
(Sp⊗)Z∶=Fun(N(Z)⊗,Sp⊗)⊗Day .
Theo em 2.41 (M. Spi zweck and H. Heine):Le Xbe a Noe he ian sepa a ed scheme
o ini e K ull dimension. Le Ebe mul iplica i ely (n, 1)-pe iodizable. The e exis s a
Z-g aded E∞-algeb a AXand a symme ic monoidal equi alence o s able ∞-ca ego ies
E∈Ob(CAlg(SH⊗(X))) , AX∈Ob(CAlg((Sp⊗)Z)),
ModE(SH⊗(X))cell ≃ModAX((Sp⊗)Z).
P oo . See [23, Theo em 4.4].
Example 2.42: Le Xbe a Noe he ian sepa a ed scheme o ini e K ull dimension.
Since he E∞-algeb a MZSpi
Xis mul iplica i ely (2,1)-pe iodizable he e exis s a Z-g aded
E∞-algeb a AXand a symme ic monoidal equi alence o s able ∞-ca ego ies
AX∈Ob(CAlg((Sp⊗)Z)),
DMT⊗(X)≃ModAX((Sp⊗)Z).
P oo . This ollows om Theo em 2.41 and Co olla y 2.33 and Example 2.38.
Theo em 2.43 (M. Spi zweck):Le Xbe a Noe he ian sepa a ed scheme o ini e K ull
dimension. The e exis s a Z-g aded E∞-algeb a AXand a symme ic monoidal equi a-
lence o s able ∞-ca ego ies
AX∈Ob(CAlg(D⊗(Ab)Z)),
DMT⊗(X)≃ModAX(D⊗(Ab)Z).
P oo . See [52, Co olla y C.3].

54
3 No m Func o s in Homo opy Theo y
3.1 No m Func o s in Mo i ic Homo opy Theo y
Theo em 3.1: The e exis s he ollowing Quillen equi alence o model ca ego ies
C[−]∶sSe J⇆sSe -Ca B∶N∆,
LC[−]∶Ho(sSe J)⇆Ho(sSe -Ca B)∶RN∆.
P oo . See [19, Theo em 1.38].
Co olla y 3.2: I C∈Ob(sSe -Ca B)is a ca ego y en iched in sSe such ha all o i s
mapping spaces a e Kan complexes hen he simplicial se N∆(C)is an ∞-ca ego y.
P oo . See [19, Co olla y 1.39].
Example 3.3: Le CMbe a simplicially en iched model ca ego y. Le X, Y ∈Ob(CM). I
Xis co ib an and Y ib an hen MapCM(X, Y )is a Kan complex. In pa icula a e he
mapping spaces o he ull subca ego y o ib an and co ib an objec s C c
MKan complexes.
The e o e is he simplicial se N∆(C c
M)an ∞-ca ego y.
P oo . See [19, Example 1.40 (1)].
Example 3.4 (∞-Ca ego y o ∞-Ca ego ies):The ca ego y o small ∞-ca ego ies qCa
is a simplicially en iched ca ego y such ha all o i s mapping spaces a e Kan complexes.
The la ge ∞-ca ego y o small ∞-ca ego ies is he e o e
Ca ∞∶=N∆(qCa ).
P oo . See [17, Example 2.51].
De ini ion 3.5 (Localiza ion o ∞-Ca ego y):Le Cbe an ∞-ca ego y and le Wbe a
class o mo phisms in C. A unc o F∶C→Dexhibi s Das he ∞-ca ego y ob ained om
Cby localiza ion a he class Wi o e e y ∞-ca ego y E he unc o o ∞-ca ego ies
(−)○F∶Fun(D,E)→Fun(C,E)is a ully ai h ul embedding, whose essen ial image
is he se o unc o s which maps each mo phism in W o an equi alence in E. The
∞-ca ego y Dis unique up o equi alence and we w i e
D≃C[W−1].
Re e ence. See [31, Theo em 1.3.4.1].
3.1 No m Func o s in Mo i ic Homo opy Theo y 55
De ini ion 3.6 (Unde lying ∞-Ca ego y o a Model Ca ego y):Le CMbe a model
ca ego y and le Cc
Mbe he ull subca ego y o co ib an objec s in CM. Le Wbe he
class o weak equi alences in Cc
Mand le Dbe an ∞-ca ego y. A unc o F∶N(Cc
M)→D
exhibi s Das he unde lying ∞-ca ego y o CMi i induces an equi alence o ∞-ca ego ies
D≃N(Cc
M)[W−1].
Re e ence. See [31, Theo em 1.3.4.15].
Theo em 3.7 (Dwye -Kan):Le CMbe a simplicial model ca ego y and le C c
Mbe he ull
subca ego y o ib an and co ib an objec s in CM. Le Wbe he class o weak equi alences
in Cc
M. The e exis s a unc o Θ∶N∆(Cc
M)→N∆(C c
M)which exhibi s N∆(C c
M)as he
unde lying ∞-ca ego y o CMi.e.
N∆(Cc
M)[W−1]≃N∆(C c
M).
P oo . See [31, Theo em 1.3.4.20].
Example 3.8 (∞-Ca ego y o Spaces):The model ca ego y o simplicial se s sSe Qis
a simplicial model ca ego y. The ull subca ego y sSe c
Qo ib an and co ib an objec s
in sSe Qis he ca ego y o Kan complexes Kan. The unde lying ∞-ca ego y o he model
ca ego y o simplicial se s is he locally p esen able ∞-ca ego y o spaces
Spc ∶=N∆(sSe c
Q)=N∆(Kan).
P oo . See [19, Example 1.40 (2)].
Example 3.9: Le CMand DNbe simplicial model ca ego ies. The ollowing (simplicial)
Quillen adjunc ion o model ca ego ies ex ends o an adjunc ion o ∞-ca ego ies
F∶CM⇆DN∶G,
N∆○F∶N∆(C c
M)⇆N∆(D c
N)∶N∆○G,
LF∶Ho(N∆(C c
M))⇆Ho(N∆(D c
N))∶RG.
P oo . See [32, P oposi ion 5.2.4.6 and Rema k 5.2.4.7].
De ini ion 3.10 (Localiza ion Func o ):A unc o F∶C→Dbe ween ∞-ca ego ies is
called a localiza ion unc o i i admi s a ully ai h ul igh adjoin unc o G∶D→C.
3.1 No m Func o s in Mo i ic Homo opy Theo y 56
Example 3.11: Le Cbe an ∞-ca ego y and le Dbe a ull subca ego y o C. Le also he
inclusion unc o i∶D↪Cadmi a le adjoin unc o L. I Wis a class o mo phisms
in Csuch ha o e e y mo phism ∈W he mo phism L( )is an equi alence in D
hen he e exis s an equi alence o ∞-ca ego ies
D≃C[W−1].
P oo . See [31, Example 1.3.4.3].
Rema k 3.12: Le Cbe some ∞-ca ego y wi h a e minal objec ∗and ini e colimi s.
Le C∗/be he coslice ca ego y. The ollowing o ge ul unc o Uadmi s a le adjoin
x∈Ob(C), x+∶=x∐∗,C∗∶=C∗/,(−)+∶C⇆C∗∶U.
De ini ion 3.13: Le Sbe a qcqs scheme. Le also PShA1(Sm/S,Spc)be he ull sub-
ca ego y o A1-in a ian p eshea es on he si e o smoo h schemes o e Sand le
PShA1(Sm/S,Spc)⊂PSh(Sm/S,Spc),
H(S)∶=Sh((Sm/S)Nis ,Spc)∩PShA1(Sm/S,Spc).
Re e ence. See he in oduc ion o [4, Chap e 2.2].
Rema k 3.14: Le Sbe a qcqs scheme. The ∞-ca ego y H(S)is he pullback in Ca ∞
along he ollowing inclusion unc o s o ull subca ego ies
H(S)→→
↓↓
PShA1(Sm/S,Spc)
↙↖
↓↓
Sh((Sm/S)Nis ,Spc)↘↙→→PSh(Sm/S,Spc)
De ini ion 3.15: Le Sbe a qcqs scheme. Le PShA1(Sm/S,Spc)be he ull subca ego y
o A1-in a ian p eshea es o e S. The canonical inclusion unc o s admi a le adjoin
LA1∶PSh(Sm/S,Spc)⇆PShA1(Sm/S,Spc)∶i,
LNis ∶PSh(Sm/S,Spc)⇆Sh((Sm/S)Nis,Spc)∶i.
Rema k 3.16: Le Sbe a quasi-compac and quasi-sepa a ed scheme and le also w i e
H(S)∶=LNis(PSh(Sm/S,Spc))∩LA1(PSh(Sm/S,Spc)).
Re e ence. [2, De ini ion 2.18].
3.1 No m Func o s in Mo i ic Homo opy Theo y 57
Rema k 3.17: Le F∈Ob(PSh(Sm/S,Spc)). I Fis A1-in a ian hen LNis (F)has
no o be and i Fis a Nisne ich shea on he si e (Sm/S)Nis hen LA1(F)has no o be.
Rema k 3.18: Le F∈Ob(PSh(Sm/S,Spc)). Since he ∞-ca ego ies PShA1(Sm/S,Spc)
and Sh((Sm/S)Nis ,Spc)a e s able unde il e ed colimi s i ollows
LM(F)∶=colim
n∈N(LA1○LNis)○n(F), LM(F)∈Ob(H(S)).
De ini ion 3.19 (Mo i ic Equi alence):We call a mo phism ∈Mo (PSh(Sm/S,Spc))
a Nisne ich equi alence and A1-equi alence and mo i ic equi alence i he mo phism
LNis( )and LA1( )and LM( )is an equi alence, espec i ely.
Re e ence. See he in oduc ion o [4, Chap e 2.2].
De ini ion 3.20 (Ex ensi e ∞-Ca ego y):An ∞-ca ego y Cis called ex ensi e i i
admi s ini e cop oduc s and i o all x, y ∈Ob(C) he cop oduc x×x∐yy∈Ob(C)exis s
and is an ini ial objec and i ini e cop oduc decomposi ions a e s able unde pullbacks.
Re e ence. See [4, De ini ion 2.3].
De ini ion 3.21 (Ex ensi e Co e age):Le Cbe some ex ensi e ∞-ca ego y. Le also
Sh∐(C,Spc)be he ∞-ca ego y o shea es on Cwi h espec o he G o hendieck opology
gene a ed by he co e s { ∶∐α∈Axα→x}such ha is an equi alence and Aa ini e
se .
Re e ence. See he exposi ion ollowing [4, Lemma 2.2].
De ini ion 3.22 (Non-Abelian De i ed ∞-Ca ego y):Le Cbe a small ∞-ca ego y wi h
all ini e cop oduc s. Le he ull subca ego y o ini e p oduc s p ese ing p eshea es be
PShΣ(C,Spc)⊂PSh(C,Spc).
Re e ence. See he in oduc ion o [4, Chap e 2.1].
P oposi ion 3.23: Le Cbe a small ∞-ca ego y which admi s ini e cop oduc s. The
∞-ca ego y PShΣ(C,Spc)is compac ly gene a ed and s able unde si ed colimi s. Fu -
he mo e p ese es he le adjoin unc o LΣ o he inclusion unc o isi ed colimi s
LΣ∶PSh(C,Spc)⇆PShΣ(C,Spc)∶i.
P oo . See he in oduc ion o [4, Chap e 2.1].
3.1 No m Func o s in Mo i ic Homo opy Theo y 64
De ini ion 3.51: Le Co (Sch,all, ´e )be he 2-ca ego y o co espondences in Sch wi h:
(1) Ob(Co (Sch,all, ´e ))=Ob(Sch).
(2) The 1-mo phisms a e co espondences o schemes Y
←Xp
→Swhe e pis a ini e
´e ale mo phism. The 2-mo phisms a e he isomo phisms o 1-mo phisms.
Rema k 3.52: Le SmQP/Sbe he ca ego y o smoo h quasi-p ojec i e schemes o e S.
Since e e y smoo h scheme o e Sadmi s an open co e by quasi-p ojec i e schemes o e
S he inclusion o ca ego ies SmQP/S⊂Sm/Sinduces an equi alence o ∞-ca ego ies
Sh((SmQP/S)Nis,Spc)≃Sh((Sm/S)Nis,Spc).
P oo . See he in oduc ion o [4, Chap e 6.1].
Rema k 3.53: I p∶X→Sis a ini e locally ee mo phism o schemes and Ya smoo h
scheme o e X hen Rp(Y)is smoo h and quasi-p ojec i e o e S. I also ollows
p∗∶PSh(SmQP/X,Spc)→PSh(SmQP/S,Spc),
p∗(SmQP/X)⊂SmQP/S, p⊗(SmQP/X+)⊂SmQP/S+.
P oo . See he in oduc ion o [4, Chap e 6.1].
Rema k 3.54: Le SmQP/Sbe he ca ego y o smoo h quasi-p ojec i e schemes o e S.
(1) Le l be he class o locally ee mo phisms o schemes. I p∶X→Sis a ini e
locally ee mo phism o schemes and Rp he Weil es ic ion unc o along he
mo phism p hen he e exis s he ollowing pai o adjoin unc o s
p∗∶SmQP/S→SmQP/X∶Rp.
Th ough Ba wick’s un u ling cons uc ion we ob ain he ollowing induced unc o
Co (Sch,all, l )→Ca 1, S ↦SmQP/S,(Y
←Xp
→S)↦Rp○ ∗.
(2) Le clopen be he class o clopen imme sions in SmQP/S. The unc o s Rpand ∗
p ese e clopen imme sions and he e exis s a canonical equi alence o ca ego ies
SmQP/S+≃Co (SmQP/S,clopen,all).

3.1 No m Func o s in Mo i ic Homo opy Theo y 65
(3) Th ough he unc o iali y o 2-ca ego ies o co espondences we ob ain he unc o
SmQP⊗
+(−)∶Co (Sch,all, l )→Ca 1, S ↦SmQP/S+,(Y
←Xp
→S)↦p⊗○ ∗.
The o ge ul unc o below induces a unique li o he unc o abo e o he nex
U∶CAlg(Ca 1)→Ca 1,
SmQP⊗
+(−)∶Co (Sch,all, l )→CAlg(Ca 1),
S↦SmQP/S+,(Y
←Xp
→S)↦p⊗○ ∗.
P oo . See he in oduc ion o [4, Chap e 6.1].
Rema k 3.55:
(1) Le Ube he canonical o ge ul unc o . The e exis s an adjunc ion o ∞-ca ego ies
such ha he alue o he unc o PShΣon a symme ic monoidal ∞-ca ego y is
PShΣ(−)∶CAlg(Ca ∞)→CAlg(Ca si
∞)∶U , C⊗↦PShΣ(C,Spc)⊗Day .
(2) Le ibe he inclusion unc o and le he ollowing unc o be he composi ion
i∶CAlg(Ca 1)↪CAlg(Ca ∞),
(PShΣ○i○SmQP⊗
+)(−)∶Co (Sch,all, l )→CAlg(Ca si
∞),
S↦PShΣ∗(SmQP/S,Spc).
(3) Le OCa ∞be he ∞-ca ego y o he ∞-ca ego ies ha a e endowed wi h a ce ain
se o equi alence classes o objec s. Le MCa ∞be he ∞-ca ego y o he ∞-
ca ego ies ha a e endowed wi h a ce ain se o equi alence classes o mo phisms.
Le Mbe he class o mo i ic equi alences in he ∞-ca ego y PShΣ∗(SmQP/S,Spc).
Since he unc o s ∗and p⊗p ese e mo i ic equi alences and he unc o p⊗is
s able unde he smash p oduc he unc o abo e ex ends o he ollowing unc o
Co (Sch,all, l )→CAlg(MCa si
∞), S ↦(PShΣ∗(SmQP/S,Spc),M).
P oo . See he exposi ion ollowing [4, Rema k 6.1].
Rema k 3.56:
3.1 No m Func o s in Mo i ic Homo opy Theo y 66
(1) Le Ebe he class o equi alences in a p esc ibed ∞-ca ego y C. By he uni e sal
p ope y o localiza ion o symme ic monoidal ∞-ca ego ies he ∞-ca ego y H⊗
∗(S)
is he image o (PShΣ∗(SmQP/S,Spc),M)by a pa ial le adjoin o he unc o
CAlg(Ca si
∞)→CAlg(MCa si
∞),C→(C,E).
(2) We he e o e ob ain a unc o
H⊗
∗(−)∶Co (Sch,all, l )→CAlg(Ca si
∞), S ↦H⊗
∗(S),(Y
←Xp
→S)↦p⊗○ ∗.
(3) The e is ∗(SV)≃S ∗(V)and i pis ini e ´e ale hen p⊗(SV)≃SRp(V)and also
Co (Sch,all, ´e )→CAlg(OCa si
∞), S ↦(H⊗
∗(S),{SV}V/S).
P oo . See he exposi ion ollowing [4, Equa ion (6.2)].
Rema k 3.57:
(1) SH⊗(S)is he image o (H⊗
∗(S),{SV}V/S)by a pa ial le adjoin o he unc o
CAlg(Ca si
∞)→CAlg(OCa si
∞),C⊗↦(C⊗, π0(Pic(C⊗))).
(2) We he e o e ob ain a unc o
SH⊗(−)∶Co (Sch,all, ´e )→CAlg(Ca si
∞), S ↦SH⊗(S),(Y
←Xp
→S)↦p⊗○ ∗.
(3) By s a ing wi h he ca ego y SmQP ins ead o SmQP+we ob ain also he unc o
H⊗(−)∶Co (Sch,all, l )→CAlg(Ca si
∞), S ↦H⊗(S),(Y
←Xp
→S)↦p∗○ ∗.
P oo . See he exposi ion ollowing [4, Equa ion (6.2)].
Co olla y 3.58: The e exis he ollowing canonical na u al ans o ma ions o unc o s
H⊗(−)(−)+
→H⊗
∗(−)Σ∞
→SH⊗(−).
P oo . See [4, Rema k 6.3].
3.2 No m Func o s in S able Equi a ian Homo opy Theo y 67
Rema k 3.59: The na u al ans o ma ions abo e a e cons uc ed analogously o he
unc o s hemsel s. In pa icula is he na u al ans o ma ion Σ∞cons uc ed ia
Co (Sch,all, ´e )×∆1→CAlg(OCa si
∞),
(S, 0→1)↦((H⊗
∗(S),{1S})→(H⊗
∗(S),{SV}V/S)).
P oo . See [4, Rema k 6.3].
Rema k 3.60: Le Sbe a scheme. The unc o SH⊗(−) eco e s he symme ic monoidal
s uc u e o he s able ∞-ca ego y SH⊗(S) ia he ollowing composi ion o unc o s
FinSe ∗≃
→Co (FinSe ,inj,all)↪Co (FinSe )→Co (FE /S)→Co (Sch,all, ´e )SH⊗(−)
→Ca ∞.
P oo . See [4, Rema k 6.4].
Theo em 3.61 (Bachmann-Hoyois):The e exis s he ollowing unc o
SH⊗(−)∶Co (Sch,all, ´e )→CAlg(Ca si
∞),
S↦SH⊗(S),(Y
←Xp
→S)↦p⊗○ ∗.
P oo . See he in oduc ion o [4, Chap e 6.1].
3.2 No m Func o s in S able Equi a ian Homo opy Theo y
De ini ion 3.62 (Delooping g oupoid):Le Gbe a g oup. The delooping g oupoid BG
o he g oup Gis he ollowing ca ego y:
(1) Ob(BG)={G}.
(2) Mo (BG)is he se o elemen s o Gand he composi ion is he g oup ope a ion.
De ini ion 3.63 (G oupoid):Le Cbe a small ca ego y.
(1) The ca ego y Cis called a g oupoid i all mo phisms o Ca e isomo phisms.
(2) The ca ego y Cis called a ini e g oupoid i i is a g oupoid and i i admi s a ini e
se o objec s and a ini e se o mo phisms.
Example 3.64: The delooping g oupoid BGo a g oup Gis a g oupoid.
3.2 No m Func o s in S able Equi a ian Homo opy Theo y 68
Example 3.65: The delooping g oupoid BGo a ini e g oup Gis a ini e g oupoid.
Example 3.66: The ne e N(C)o a small ca ego y Cis a Kan complex i and only i
he ca ego y Cis a g oupoid.
De ini ion 3.67 (2-Ca ego y o G oupoids):Le he 2-ca ego y o g oupoids G pd be:
(1) Ob(G pd)is he se o g oupoids.
(2) A mo phism o g oupoids is a unc o . A 2-mo phism o g oupoids is a na u al
ans o ma ion o unc o s and necessa ily a na u al isomo phism.
Rema k 3.68: The ca ego y o g oupoids i s also in he ollowing diag am o ca ego ies
Ca ∞Ho
→Ca Co e
→G pd ↪Ca N
→sSe π0
→Se .
Rema k 3.69: The homo opy cohe en ne e induces an adjunc ion o ∞-ca ego ies
Co e ∶qCa ⇆Kan ∶i,
N∆○Co e ∶Ca ∞⇆Spc ∶N∆○i.
P oo . See [17, Example 3.53].
De ini ion 3.70 (Fundamen al G oupoid o X):The undamen al g upoid Π1(X)o
X∈Ob(TopQ)is he ollowing ca ego y:
(1) Ob(Π1(X))=X.
(2) HomΠ1(X)(x, y)is he se o homo opy classes o con inuous pa hs om x o y.
Co olla y 3.71: Le X∈Ob(TopQ)and le x∈Xbe a poin . The undamen al g oup
o Xwi h he basepoin xis isomo phic o he ollowing au omo phism g oup
π1(X, x)≅Au Π1(X)(x).
P oo . See [19, Example 1.1].
Rema k 3.72 (Homo opy Hypo hesis):
(1) The unde lying ∞-ca ego y o TopQis he ∞-ca ego y G pd∞o ∞-g oupoids.
3.2 No m Func o s in S able Equi a ian Homo opy Theo y 69
(2) The ollowing Quillen equi alence o model ca ego ies induces an equi alence be-
ween he ∞-ca ego y o spaces and he ∞-ca ego y o ∞-g oupoids
∣−∣∶sSe Q⇆TopQ∶Sing,
Spc ⇆G pd∞.
P oo . See [17, Commen a y 1.40 and Commen a y 1.43].
Rema k 3.73: Le FinG pd be he 2-ca ego y o ini e g oupoids. Le G pd∞be he ∞-
ca ego y o ∞-g oupoids. The e exis s he ollowing diag am o inclusions o ca ego ies
FinG pd↘↙→→
↙↖
↓↓
G pd↘↙→→G pd∞≃→→Spc
↙↖
↓↓
P o(FinG pd)↘↙→→P o(Spc)
Example 3.74: I X∈Ob(TopQ) hen Sing(X)is a Kan complex and he e o e also
an ∞-ca ego y. In pa icula is he undamen al g oupoid o Xis equi alen o
Π1(X)≃Ho(Sing(X)).
P oo . See [17, Exe cise 1.36].
Example 3.75 (Pica d ∞-G oupoid):The Pica d ∞-g oupoid o a symme ic monoidal
∞-ca ego y C⊗is he maximal ∞-g oupoid in C⊗spanned by he ⊗-in e ible objec s
Pic(C⊗)∈Ob(G p(Spc)).
De ini ion 3.76 (Fini ely P esen ed Mo phism):A mo phism h∈Mo (P o(Spc)) is
called ini ely p esen ed i i is a pullback h= ∗(g)o a mo phism go spaces
X×ZY
h= ∗(g)
↓↓
→→Y
g∈Mo (Spc)
↓↓
X →→Z
.
Re e ence. See he in oduc ion o [4, Chap e 9.1].
Example 3.77: The class o ini ely p esen ed mo phisms in he ∞-ca ego y P o(Spc)
is closed unde composi ion and base change and sa is ies he 2-ou -o -3p ope y.

3.2 No m Func o s in S able Equi a ian Homo opy Theo y 70
P oo . See [4, Lemma 9.2].
De ini ion 3.78 (Fini e Co e ing Mo phism):A mo phism h∈Mo (P o(Spc))is called
a ini e co e ing i i is ini ely p esen ed and i i s ibe s a e (possibly emp y) ini e se s.
Re e ence. See he exposi ion ollowing [4, Lemma 9.3].
Example 3.79: Le Gbe a ini e g oup and le i∶H↪Gbe a subg oup. The induced
mo phism o ini e g oupoids B(i)∶BH→BGis a ini e co e ing mo phism.
P oo . See [4, Chap e 1.4].
Rema k 3.80: The e exis he ollowing inclusions o se s o mo phisms
co ⊂ p ⊂Mo (P o(FinG pd)).
De ini ion 3.81 (Ca ego y o Fini e Co e s o X):Le Xbe a p o ini e g oupoid. Le
he ca ego y o ini e co e s o Xbe he ull subca ego y FC/X⊂P o(FinG pd)/Xwi h:
(1) The se o objec s in he ca ego y o ini e co e s o he p o ini e g oupoid Xis
Ob(FC/X)∶={( ∶Y→X)∣Y∈Ob(P o(FinG pd)) , is a ini e co e ing}.
(2) A mo phism om ( ∶Y→X) o ( ′∶Y′→X)is a commu a i e diag am
Y
g
↙↙
↘↘
Y′
′→→X
Example 3.82: Le Gbe a ini e g oup. The ollowing ull subca ego y o ini e co e s
o he delooping g oupoid BGis equi alen o he ca ego y o ini e G-se s
G-FinSe ≃FC/BG,FC/BG⊂FinG pd/BG⊂P o(FinG pd)/BG.
P oo . See [4, Chap e 1.4].
Rema k 3.83: Le Cbe some ∞-ca ego y. The unc o o ∞-ca ego ies Fun(−,C)ex-
ends uniquely o a colimi p ese ing unc o on he opposi e ∞-ca ego y o p o-spaces
Spcop Fun(−,C)→→
↓↓
Ca ∞
P o(Spc)op
↗↗
3.2 No m Func o s in S able Equi a ian Homo opy Theo y 71
P oo . See he exposi ion ollowing [4, Lemma 9.2].
Rema k 3.84: Le X∈Ob(P o(Spc)) be a p o ini e space. The e is he ollowing ully
ai h ul unc o whose image consis s o he ini ely p esen ed mo phims and which is
∫∶Fun(X, Spc)→P o(Spc)/X.
P oo . See [4, Lemma 9.3].
Rema k 3.85: The ollowing a e symme ic monoidal embeddings na u al in Xwi h
Fun(X, Spc)(−)+
→Fun(X, Spc)+↪Co (Fun(X, Spc)) , X ∈Ob(P o(Spc)op).
P oo . See he exposi ion ollowing [4, Equa ion 9.4].
Rema k 3.86: Le p∶Y→Xbe a ini ely p esen ed map o p o ini e ∞-g oupoids.
(1) The unc o p∗has a igh adjoin p∗ unc o and a le adjoin p# unc o and is
p∗∶Fun(X, Spc)⇆Fun(Y, Spc)∶p∗, p#∶Fun(Y, Spc)⇆Fun(X, Spc)∶p∗.
(2) The i s symme ic monoidal unc o es ic s o he ollowing one
p⊗∶=Co (p∗)∶Co (Fun(Y, Spc))→Co (Fun(X, Spc)),
p⊗∶Fun(Y, Spc)+→Fun(X, Spc)+.
P oo . See [4, Lemma 9.5].
De ini ion 3.87: Le Xbe a p o ini e g oupoid. Le he 1-ca ego y o ini e X-se s be
X∈Ob(P o(FinG pd)) , X-FinSe ∶=Fun(X, FinSe ),Fun(X, FinSe )⊂Fun(X, Spc).
P oposi ion 3.88: Le X, Y ∈Ob(P o(FinG pd)) be p o ini e g oupoids.
(1) The e exis s he ollowing equi alence o ca ego ies
X-FinSe ≃FC/X.
(2) I p∶Y→Xis a ini ely p esen ed mo phism o p o ini e g oupoids hen he ibe s
o he mo phism pha e ini ely many connec ed componen s.
3.2 No m Func o s in S able Equi a ian Homo opy Theo y 72
P oo . See he exposi ion ollowing [4, Lemma 9.5].
P oposi ion 3.89: Le X, Y ∈Ob(P o(FinG pd)) be p o ini e g oupoids.
(1) I p∶Y→Xis a ini ely p esen ed mo phism hen he e exis s an adjunc ion
p∗∶X-FinSe ⇆Y-FinSe ∶p∗.
(2) I in addi ion pis a ini e co e ing hen he e is also he ollowing adjunc ion
p#∶Y-FinSe ⇆X-FinSe ∶p∗.
P oo . See he exposi ion ollowing [4, Lemma 9.5].
De ini ion 3.90 (∞-Ca go y o Le Exac ∞-Ca ego ies):Le Ca lex
∞be he ∞-ca ego y:
(1) Ob(Ca lex
∞)is he class o he ∞-ca ego ies wi h ini e limi s.
(2) MapCa lex
∞(C,D)is he space o le exac unc o s be ween Cand D.
Rema k 3.91:
(1) The e exis s he ollowing unc o
FinSe ⊗(−)∶Co (P o(FinG pd),all, p)→Ca lex
1,
X↦X-FinSe ,(Y
←Xp
→Z)↦p∗○ ∗.
(2) Since he unc o s ∗and p∗a e le exac he unc o abo e ex ends o he unc o
Co (−)∶Ca lex
∞→CAlg(Ca ∞),
(Co ○FinSe ⊗)(−)∶Co (P o(FinG pd),all, p)→CAlg(Ca 2),
X↦Co (X-FinSe ),(Y
←Xp
→Z)↦p⊗○ ∗.
(3) The e exis s he ollowing symme ic monoidal sub unc o and we also se
FinSe ⊗
+(−)⊂(Co ○FinSe ⊗)(−)∶Co (P o(FinG pd),all, p)→CAlg(Ca 1),
X↦X-FinSe +,(Y
←Xp
→Z)↦p⊗○ ∗.
H⊗(−),H⊗
∗(−)∶Co (P o(FinG pd),all, p)→CAlg(Ca si
∞),
H⊗(−)∶=(PShΣ○FinSe ⊗)(−),H⊗(X)∶=PShΣ(X-FinSe ,Spc),
H⊗
∗(−)∶=(PShΣ○FinSe ⊗
+)(−),H⊗
∗(X)∶=PShΣ(X-FinSe +,Spc).
3.2 No m Func o s in S able Equi a ian Homo opy Theo y 73
P oo . See he in oduc ion o [4, Chap e 9.2].
Theo em 3.92 (Bachmann-Hoyois):The e exis s he ollowing unc o
H⊗
∗(−)∶Co (P o(FinG pd),all, p)→CAlg(Ca si
∞),
X↦H⊗
∗(X),(Y
←Xp
→Z)↦p⊗○ ∗.
P oo . See he in oduc ion o [4, Chap e 9.2].
Example 3.93 (O bi Ca ego y):Le Gbe a compac Lie g oup. The o bi ca ego y
OGo Gis he ull subca ego y o he ∞-ca ego y o G-equi a ian spaces G-Spc whe e
Ob(OG)∶={G/H∣H⊂Gis a closed subg oup , G/His a cose space}.
P oo . See [8, De ini ion 1.3.1].
De ini ion 3.94 (G-Spaces):Le Gbe a ini e g oup wi h he disc e e opology o else
a compac Lie g oup. The symme ic monoidal ∞-ca ego y o G-equi a ian spaces is
G-Spc ∶=Fun(Oop
G,Spc).
Re e ence. See he in oduc ion o [8, Chap e 1.1].
Example 3.95 (Bachmann-Hoyois):
(1) I Gis a ini e g oup hen he e exis he ollowing symme ic monoidal equi alences
H⊗(BG)≃G-Spc⊗,H⊗
∗(BG)≃G-Spc⊗
∗.
(2) I B( )∶BG→BHis induced by a g oup homomo phism ∶G→Hwi h he
ke nel N hen B( )⊗∶H⊗
∗(BG)→H⊗
∗(BH)is he unc o NH
G/N○ΦN.
(3) I p∶∗→BG hen p⊗(S1)∈Ob(H⊗
∗(BG)) is he egula ep esen a ion sphe e.
P oo . See he in oduc ion o [4, Chap e 9.2].
De ini ion 3.96: Le X∈Ob(P o(FinG pd)). The compac ly gene a ed symme ic
monoidal ∞-ca ego y Sp⊗(X)is ob ained om he symme ic monoidal ∞-ca ego y
H⊗
∗(X)by in e ing he compac objec p⊗(S1) o each ini e co e ing p∶Y→Xi.e.
Sp⊗(X)∶=H⊗
∗(X)[{p⊗(S1)−1}p].
Re e ence. See he in oduc ion o [4, Chap e 9.2].
3.3 The É ale Fundamen al G oup 80
P oo . See [58, Tag 0BN8].
De ini ion 3.124 (Fini e Galois Co e ):A ini e ´e ale mo phism o schemes ∶Y→X
is called a Galois co e i he scheme Yis connec ed and i
∣G∣=deg( ), G ∶=Au FE /X(Y/X).
Re e ence. See [58, Tag 03SF].
De ini ion 3.125 (Fibe Func o ):Le Xbe a connec ed scheme and le kbe an alge-
b aically closed ield. Le x∶Spec(k)→Xbe a mo phism o schemes and le ∶Y→X
be a ini e ´e ale mo phism. Le also he ollowing unc o be
Fx∶FE /X→Se , Y ↦HomSch/X(Spec(k), Y ).
Re e ence. See he exposi ion ollowing [58, Tag 0BL7].
P oposi ion 3.126: Le Xbe a scheme and le kbe an algeb aically closed ield and
le x∶Spec(k)→Xbe a mo phism o schemes.
(1) The se Fx(Y)is ini e o e e y Y∈Ob(FE /X).
(2) (FE /X, Fx)is a Galois ca ego y.
P oo . See [58, Tag 0BNB].
De ini ion 3.127 (´
E ale Fundamen al G oup):Le Xbe a connec ed scheme and le k
be an algeb aically closed ield and le x∶Spec(k)→Xbe a mo phism o schemes. The
´e ale undamen al g oup o Xwi h he basepoin xis he p o ini e g oup

π´e
1(X, x)∶=Au Fun(FE /X,Se )(Fx).
Re e ence. See [58, Tag 0BNC].
Co olla y 3.128: Le Xbe a connec ed scheme and le kbe an algeb aically closed ield
and le x∶Spec(k)→Xbe a mo phism. The e is a canonical equi alence o ca ego ies
FE /X
≃
→
π´e
1(X, x)-FinSe , Y ↦(Fx(Y), φFx).
P oo . See [58, Tag 0BND].

3.4 The P o ini e É ale Fundamen al G oupoid 81
Example 3.129: Le kbe a ield and le x∶Spec(k)→Xbe a mo phism o schemes.
(1) The e exis s a canonical isomo phism o p o ini e g oups

π´e
1(Spec(k), x)≅Gal(ksep/k).
(2) The e exis s a canonical equi alence o ca ego ies
FE /Spec(k)≃
→Gal(ksep/k)-FinSe .
(3) I kis an algeb aically closed ield hen i s ´e ale undamen al g oup is i ial
{e}≅Gal(ksep/k)≅
π´e
1(Spec(k), x).
P oo . See [58, Tag 0BNE].
Example 3.130: Le Xbe a connec ed a ie y o e Spec(C). The G o hendieck-
Riemann exis ence heo em implies an isomo phism o p o ini e g oups

π´e
1(X, x)≅
π1(X(C), x).
P oo . See [58, Tag 03VD].
P oposi ion 3.131: Le Xbe a connec ed scheme and le kbe an algeb aically closed
ield and le x∶Spec(k)→Xbe a mo phism. The e exis s an equi alence o ca ego ies
Sh(X´e ,Se ) lc ≃
π´e
1(X, x)-FinSe .
P oo . See [58, Tag 0DV5].
3.4 The P o ini e É ale Fundamen al G oupoid
De ini ion 3.132 (∞-Topos):An ∞-ca ego y Xis called an ∞- opos i he e exis s a
small ∞-ca ego y Cand an accessible le exac localiza ion unc o
L∶PSh(C,Spc)⇆X∶i.
Re e ence. See [32, De ini ion 6.1.0.4].
Theo em 3.133 (Lu ie):An ∞-ca ego y Xis an ∞- opos i and only i i sa is ies he
∞-ca ego ical Gi aud’s axioms.
Re e ence. See [32, Theo em 6.4.1.5].
3.4 The P o ini e É ale Fundamen al G oupoid 82
Example 3.134: The ∞-ca ego y o p eshea es on a small ∞-ca ego y Cis an ∞- opos
X∶=PSh(C,Spc).
Example 3.135 (Te minal ∞-Topos):The ∞-ca ego y o spaces Spc is a e minal
objec in he ∞-ca ego y o ∞- opoi T∞.
P oo . See [46, Example 6.3].
Example 3.136 (Slice ∞-Topos):Le Xbe an ∞- opos and le U∈Ob(X). The slice
∞-ca ego y X/Uis an ∞- opos.
P oo . See [46, P oposi ion 4.2].
Example 3.137 (´
E ale ∞-Topos):The small ´e ale ∞- opos and he big ´e ale ∞- opos
o a scheme Xis he ∞-ca ego y o shea es on he small ´e ale si e X´e and he big ´e ale
si e (Sch/X)´e , espec i ely.
P oo . See [46, Chap e 2.3].
Example 3.138 (G o hendieck Topos):Le Xbe an ∞- opos. The ull subca ego y o
0- unca ed objec s τ≤0(X)⊂Xis a G o hendieck opos.
P oo . See [46, Chap e 2.8].
Rema k 3.139: E e y ∞- opos Xis powe ed o e he e minal ∞- opos o spaces
MapSpc(K, MapX(Y, X))≃MapX(Y, XK).
De ini ion 3.140 (Homo opy G oups in ∞-Topos):Le Xbe an ∞- opos and le also
F∈Ob(X). Le Sn=∂∆n+1be he simplicial n-sphe e wi h he basepoin x∶ ∗ →Sn.
Le e x∶FSn→Xbe he canonical mo phism in he slice ∞- opos X/Fand le
πn(F, x)∶=τ≤0(FSn), τ≤0(FSn)∈Ob(τ≤0(X/F)) , n ≥0.
Re e ence. See [32, De ini ion 6.5.1.1].
Example 3.141: Le Spc be he ∞- opos o spaces. Le Xbe a space. The n- h homo-
opy g oup πn(X)o he space Xis p ecisely i s n- h simplicial homo opy g oup.
P oo . This ollows om Example 3.135 and De ini ion 3.140.
3.4 The P o ini e É ale Fundamen al G oupoid 83
Example 3.142: Le Cbe a small ∞-ca ego y. Le PSh(C)be he ∞- opos o p eshea es
and le and le F∈Ob(PSh(C)) be a p eshea wi h some basepoin . The powe ing FSn
is he limi o he cons an diag am cons F∶Sn→PSh(C). Limi s in he ca ego y
o p eshea es a e compu ed poin wise. The n- h homo opy g oup o Fin he ∞- opos
PSh(C)is he e o e he o dina y n- h homo opy g oup o he p eshea Fon Cand is
πn(F(x))≅τ≤0(FSn(x))≅τ≤0(F(x)Sn), n ≥0.
P oo . This ollows om Example 3.134 and De ini ion 3.140.
De ini ion 3.143 (Geome ic Mo phism):A geome ic mo phism ∶X→Yo ∞- opoi
is a pai o adjoin unc o s
∗∶X→Y∶ ∗.
De ini ion 3.144: Le Cbe an accessible ∞-ca ego y which admi s ini e limi s. Le he
ollowing be he ull subca ego y spanned by he accessible le exac unc o s
P o(C)⊂Fun(C,Spc)op.
Example 3.145 (∞-Ca ego y o Shapes):The ∞-ca ego y o shapes is he ollowing
ull subca ego y spanned by he accessible le exac unc o s
P o(Spc)⊂Fun(Spc,Spc)op.
P oo . See [32, De ini ion 7.1.6.1].
De ini ion 3.146 (Shape o ∞-Topos):Le Xbe an ∞- opos and le q∶X→Spc be
he unique up o homo opy geome ic mo phism o he e minal ∞- opos o spaces. The
shape Sh(X)o he ∞- opos Xis he accessible le exac unc o
(q∗○q∗∶Spc →Spc)∈Ob(P o(Spc)).
Re e ence. See [32, De ini ion 6.5.1.1].
Example 3.147 (T i ial Shape):We say an ∞- opos Xis o i ial shape i Sh(X)is
equi alen o he iden i y unc o on he ∞-ca ego y o spaces Spc.
P oo . See [32, De ini ion 7.1.6.11].
De ini ion 3.148 (π-Fini e Space):A space X∈Ob(Spc)is called π- ini e i i sa is ies:
3.4 The P o ini e É ale Fundamen al G oupoid 84
(1) The space Xis n- unca ed o some in ege n.
(2) The se o connec ed componen s π0(X)is ini e.
(3) Fo each e ex x∈Xand each in ege m≥1is he g oup πm(X, x) ini e.
Re e ence. See [33, De ini ion E.0.7.8].
De ini ion 3.149 (∞-Ca ego y o π-Fini e Spaces):The ∞-ca ego y o π- ini e spaces
is he ull subca ego y o he ∞-ca ego y o spaces spanned by he π- ini e spaces
Spcπ⊂Spc.
De ini ion 3.150 (∞-Ca ego y o P o ini e Spaces):The ∞-ca ego y o p o ini e spaces
is he P o-comple ion o he ∞-ca ego y o π- ini e spaces

Spc ∶=P o(Spcπ),P o(Spcπ)⊂Fun(Spcπ,Spc)op.
Re e ence. See [33, De ini ion E.0.7.11].
De ini ion 3.151 (P o ini e Comple ion):The p o ini e comple ion o a space Xis
X∈Ob(Spc),
Xπ∈Ob(
Spc),
Xπ∶Spcπ→Spc , T ↦
Xπ(T)∶=MapSpc(X, T ).
Re e ence. See [33, Example E.0.7.12].
De ini ion 3.152 (P o ini e Shape o ∞-Topos):Le T∞be he ∞-ca ego y o ∞- opoi
and le Ube he o ge ul unc o . The p o ini e shape o an ∞- opos Xis

Shπ(X)∶=U(Sh(X)) ,T∞Sh
→P o(Spc)U
→P o(Spcπ).
Re e ence. See [33, Va ian E.2.2.2].
De ini ion 3.153: Le Xbe an ∞- opos. We call F∈Ob(X) ini e locally cons an i
he e exis s an e ec i e epimo phism ∐α∈AUα→∗along wi h {Lα}α∈A⊂Ob(Spcπ)such
ha o e e y α∈A he e is an equi alence Uα×F≃Uα×Lαin he slice ∞- opos X/Uα.
I he se Ais ini e hen Fis called locally cons an cons uc ible.
Re e ence. See he in oduc ion o [4, Chap e 10.1].
3.4 The P o ini e É ale Fundamen al G oupoid 85
De ini ion 3.154: Le Xbe an ∞- opos. Le he ull subca ego ies o Xspanned by he
ini e locally cons an objec s and he locally cons an cons uc ible objec s be
Xlcc ⊂X lc ⊂X.
Re e ence. See he in oduc ion o [4, Chap e 10.1].
P oposi ion 3.155: Le Xbe an ∞- opos. The e exis equi alences o ∞-ca ego ies
X lc ≃Fun(Sh(X),Spcπ),Xlcc ≃Fun(
Shπ(X),Spcπ).
P oo . See [4, P oposi ion 10.1].
De ini ion 3.156 (´
E ale Fundamen al G oupoid):The ´e ale undamen al g oupid o a
scheme Xis he 1- unca ion o he shape o he small ´e ale ∞- opos o Xand is
Π´e
1(X)∶=τ≤1(Sh(Sh(X´e ,Spc))) ,Π´e
1(X)∈Ob(P o(G pd)).
Re e ence. See he exposi ion ollowing [4, P oposi ion 10.1].
De ini ion 3.157 (P o ini e ´
E ale Fundamen al G oupoid):The p o ini e ´e ale unda-
men al g oupid o a scheme Xis he 1- unca ion o he p o ini e shape o he small
´e ale ∞- opos o Xand is

Π´e
1(X)∶=τ≤1(
Shπ(Sh(X´e ,Spc))) ,
Π´e
1(X)∈Ob(P o(FinG pd)).
Re e ence. See he exposi ion ollowing [4, P oposi ion 10.1].
Co olla y 3.158:
(1) I Xis an a bi a y scheme hen he e exis s a na u al equi alence o ca ego ies
FE /X≃Fun(Π´e
1(X),FinSe ).
(2) I Xis quasi-compac in he clopen opology hen he e exis s a na u al equi alence
FE /X≃Fun(
Π´e
1(X),FinSe ).
P oo . See [4, Co olla y 10.2].

3.4 The P o ini e É ale Fundamen al G oupoid 86
P oposi ion 3.159: Le Xbe a scheme.
(1) The ollowing squa e is commu a i e
FE /X
≃→→
↙↖
↓↓
Fun(
Π´e
1(X),FinSe )
∫
↓↓
Sch 
Π´e
1(−)→→P o(FinG pd)
(2) The unc o 
Π´e
1(−)sends ini e ´e ale mo phisms o schemes o ini e co e ing
mo phisms o p o ini e g oupoids.
(3) The unc o 
Π´e
1(−)p ese es pullbacks along ini e ´e ale mo phisms o schemes.
P oo . See [4, Lemma 10.4].
Example 3.160: Le Xbe a connec ed scheme and le kbe an algeb aically closed ield
and le x∶Spec(k)→Xbe a mo phism. Le ˆπ´e
1(X, x)be he ´e ale undamen al g oup
o Xwi h basepoin x. The e exis s a canonical equi alence o p o ini e g oupoids

Π´e
1(X)≃Bˆπ´e
1(X, x).
P oo . See [4, Rema k 10.3].
Example 3.161: Le Sbe a scheme. The ollowing inclusion unc o induces a sym-
me ic monoidal equi alence o ∞-ca ego ies

Π´e
1(S)-FinSe +↪Co (
Π´e
1(S)-FinSe ),
Sp⊗(
Π´e
1(S))≃Sp⊗(PShΣ(Co (
Π´e
1(S)-FinSe ),Spc)).
P oo . This ollows om P oposi ion 3.107.
P oposi ion 3.162: The p o ini e ´e ale undamen al g oupoid unc o ex ends o

Π´e
1(−)∶Co (Sch,all, ´e )→Co (P o(FinG pd),all, co ).
P oo . See [4, Lemma 10.4].
3.5 Galois-Equi a ian Spec a and Mo i ic Spec a 87
3.5 Galois-Equi a ian Spec a and Mo i ic Spec a
Rema k 3.163: In he ollowing we p esen he cons uc ion o he na u al ans o -
ma ion c, see Theo em 3.168. We i s collec he ele an unc o s o he cons uc ion
o he unc o c, see in pa icula Rema k 3.164. We hen employ he ollowing s a egy.
(1) In Rema k 3.165 we i s cons uc he ollowing na u al ans o ma ion o unc o s
FinSe ⊗
+(−)○
Π´e
1(−)↪SmQP⊗
+(−).
(2) In Rema k 3.166 we hen p omo e he na u al ans o ma ion o unc o s in (1) o
c∶H⊗
∗(−)○
Π´e
1(−)→H⊗
∗(−)∶Co (Sch,all, ´e )→CAlg(Ca si
∞).
(3) In Rema k 3.167 we hen p omo e he na u al ans o ma ion o unc o s in (2) o
c∶Sp⊗(−)○
Π´e
1(−)→SH⊗(−)∶Co (Sch,all, ´e )→CAlg(Ca si
∞).
Rema k 3.164:
(1) The e exis s he ollowing unc o
SmQP⊗
+(−)∶Co (Sch,all, l )→CAlg(Ca 1),
S↦SmQP/S+,(Y
←Xp
→S)↦p⊗○ ∗.
(2) The e exis s he ollowing unc o
FinSe ⊗(−)∶Co (P o(FinG pd),all, p)→Ca lex
1,
X↦X-FinSe ,(Y
←Xp
→Z)↦p∗○ ∗.
(3) The e exis s he ollowing symme ic monoidal sub unc o
FinSe ⊗
+(−)⊂(Co ○FinSe ⊗)(−)∶Co (P o(FinG pd),all, p)→CAlg(Ca 1),
X↦X-FinSe +,(Y
←Xp
→Z)↦p⊗○ ∗
P oo . See he exposi ion ollowing [4, Equa ion (10.5)].
Rema k 3.165:
3.5 Galois-Equi a ian Spec a and Mo i ic Spec a 88
(1) The equi alence o ca ego ies below is na u al in S∈Ob(Schop)and admi s a
canonical li o he ollowing na u al equi alence o unc o s
FE /S
≃
→Fun(
Π´e
1(S),FinSe )
FinSe ⊗(−)○
Π´e
1(−)≃
→FE ⊗
/(−)∶Co (Sch,all, ´e )→Ca lex
1.
(2) Applying he unc o o co espondences and es ic ing o poin ed objec s we ob ain
Co (−)∶Ca lex
1→CAlg(Ca 2),
FinSe ⊗
+(−)○
Π´e
1(−)≃
→FE ⊗
/(−)+∶Co (Sch,all, ´e )→CAlg(Ca 1).
(3) The inclusion o ca ego ies below is na u al in S∈Ob(Schop)and li s uniquely o
a na u al ans o ma ion o unc o s which also p omo es o he ollowing one
FE ⊗
/S+⊂SmQP/S+,
FE ⊗
+(−)↪SmQP⊗
+(−)∶Co (Sch,all, ´e )→CAlg(Ca 1),
FinSe ⊗
+(−)○
Π´e
1(−)↪SmQP⊗
+(−).
P oo . See he exposi ion ollowing [4, Equa ion (10.5)].
Rema k 3.166:
(1) The na u al ans o ma ion abo e is a unc o
Co (Sch,all, ´e )×∆1→CAlg(Ca ∞).
(2) By composi ion wi h he unc o PShΣ he unc o abo e li s o he ollowing one
PShΣ∶CAlg(Ca ∞)→CAlg(Ca si
∞),
Co (Sch,all, ´e )×∆1→CAlg(MCa si
∞),
(S, 0→1)↦((H⊗
∗(
Π´e
1(S)),E)→(PShΣ(SmQP/S+,Spc),M)).
(3) Th ough composi ion wi h a pa ial le adjoin unc o o he unc o below we
ob ain he ollowing na u al ans o ma ion o unc o s
CAlg(Ca si
∞)→CAlg(MCa si
∞),C→(C,E),
c∶H⊗
∗(−)○
Π´e
1(−)→H⊗
∗(−)∶Co (Sch,all, ´e )→CAlg(Ca si
∞).
3.5 Galois-Equi a ian Spec a and Mo i ic Spec a 89
P oo . See he exposi ion ollowing [4, Equa ion (10.5)].
Rema k 3.167:
(1) I p∶Y→
Π´e
1(S)is a ini e co e ing mo pism hen he ollowing is ⊗-in e ible
Σ∞(cS(p⊗(S1)))∈Ob(SH⊗(S)).
(2) The unc o Σ∞○cadmi s he e o e a li o he ollowing unc o o ca ego ies
Co (Sch,all, ´e )×∆1→CAlg(OCa si
∞),
(S, 0→1)↦((H⊗
∗(
Π´e
1(S)),{p⊗(S1)}p)→(SH⊗(S), π0(Pic(SH⊗(S)))).
(3) Th ough composi ion wi h a pa ial le adjoin unc o o he unc o below we
ob ain he ollowing na u al ans o ma ion o unc o s
CAlg(Ca si
∞)→CAlg(OCa si
∞),C⊗↦(C⊗, π0(Pic(C⊗))),
c∶Sp⊗(−)○
Π´e
1(−)→SH⊗(−)∶Co (Sch,all, ´e )→CAlg(Ca si
∞).
P oo . See he exposi ion ollowing [4, Equa ion (10.5)].
Theo em 3.168 (Bachmann-Hoyois):The e exis s a na u al ans o ma ion o unc o s
c∶Sp⊗(−)○
Π´e
1(−)→SH⊗(−)∶Co (Sch,all, ´e )→CAlg(Ca si
∞).
P oo . See he exposi ion ollowing [4, Equa ion (10.5)].
Theo em 3.169 (Bachmann-Hoyois):
(1) The e exis s he ollowing adjunc ion o symme ic monoidal ∞-ca ego ies whe e
cSis a le adjoin symme ic monoidal unc o na u al in Swi h
cS∶Sp⊗(
Π´e
1(S))⇆SH⊗(S)∶uS, S ∈Ob(Co (Sch,all, ´e )).
(2) I Sis a connec ed scheme wi h a ini e Galois co e S′/Sand wi h he au omo -
phism g oup G hen he e exis s a le adjoin symme ic monoidal unc o
G∶=Au FE /S(S′/S),
cS′/S∶Sp⊗(BG)⇆SH⊗(S)∶uS′/S.
3.7 No med Mo i ic Spec a 96
(1) A no med mo i ic spec um o e Cis a sec ion o SH⊗(−)o e he ca ego y
Co (C,all, ´e ) ha is coca esian o e Cop.
(2) Le he ∞-ca ego y o no med mo i ic spec a o e Cbe he ull subca ego y
NAlgC(SH⊗(−))⊂Sec (SH⊗(−)∣Co (C,all, ´e )).
Re e ence. See [4, De ini ion 7.1].
Rema k 3.189 (No a ion):We also w i e
NAlgFE (SH⊗(S))∶=NAlgFE /S(SH⊗(−)).
P oposi ion 3.190: Le Sbe a scheme. Le C⊂ ´e Sch/Sbe a ull subca ego y ha
con ains Sand is closed unde ini e sums and ini e ´e ale ex ensions.
(1) The ∞-ca ego y NAlgC(SH⊗(−)) admi s all colimi s and ini e limi s. I Cis small
hen i is locally p esen able and he e o e admi s all limi s.
(2) The canonical o ge ul unc o U∶NAlgC(SH⊗(−))→SH⊗(S)is conse a i e and
p ese es si ed colimi s and ini e limi s. I C⊂Sm/S hen i p ese es limi s and
is he e o e monadic.
(3) The canonical o ge ul unc o U∶NAlgC(SH⊗(−))→CAlg(SH⊗(S)) is conse a-
i e and p ese es colimi s and ini e limi s. I C⊂Sm/S hen i p ese es limi s
and is he e o e monadic and comonadic.
(4) The e exis he ollowing unc o along wi h he equi alence o ∞-ca ego ies
ModE(SH⊗(−))∶Co (C,all, ´e )→CAlg(Ca si
∞), E ∈Ob(NAlgC(SH⊗(−))),
NAlgC(SH⊗(−))E/≃NAlgC(ModE(SH⊗(−))).
P oo . See [4, P oposi ion 7.6].
Example 3.191: Le Sbe any scheme. The monoidal uni 1Sis canonically a no med
mo i ic spec um o e he ca ego y Sch/Si.e.
1S∈Ob(NAlgSch/S(SH⊗(−))).
P oo . See [4, Example 7.11].

3.7 No med Mo i ic Spec a 97
Example 3.192: Voe odsky’s mo i ic Eilenbe g-MacLane spec um MZSo e a Noe he-
ian scheme Sis a no med mo i ic spec um o e he ca ego y Sm/Si.e.
MZS∈Ob(NAlgSm/S(SH⊗(−))).
P oo . See [4, Example 7.12].
Rema k 3.193: Le Sbe a Noe he ian scheme. Le p∶X→Sbe ini e ´e ale. Since
MZSis no med and o ien ed he e exis s some E∞-mul iplica i e ans e mo phism
νp∶⊕
n∈Z
MapSH(X)(1X,Σn
P1(MZX))→⊕
m∈Z
MapSH(S)(1S,Σm
P1(MZS)).
I Sis a smoo h scheme o e a ield hen νpis a mo phism o Bloch’s cycle complexes
νp∶z●(X, ●)→z●(S, ●).
P oo . See he exposi ion ollowing [4, Rema k 14.9].
Rema k 3.194: Le kbe a ield and le Sbe a smoo h quasi-p ojec i e scheme o e
Spec(k). Le p∶X→Sbe a ini e ´e ale mo phism. Le z●(S, ●)be Bloch’s cycle complex
o S. Since MZSis no med and o ien ed he e exis s some E∞-mul iplica i e ans e
mo phism νp ha induces he Ful on-MacPhe son no m νFM
pon ollowing Chow g oups
MapSH⊗(S)(1S,Σn
P1(MZS))≃zn(S, ●), n ∈Z,
νp∶z●(X, ●)→z●(S, ●),
CH●(−)∶Co (SmQP/k,all, ´e )→Se , S ↦CH●(S),(Y
←Xp
→S)↦νFM
p○ ∗.
P oo . See [4, P oposi ion 14.12].
Example 3.195: The algeb aic K- heo y spec um KGLSo e any scheme Sis a no med
mo i ic spec um o e he ca ego y Sch/Si.e.
KGLS∈Ob(NAlgSch/S(SH⊗(−))).
P oo . See [4, Example 7.13].
Example 3.196: The algeb aic cobo dism spec um MGLSo e any scheme Sis a
no med mo i ic spec um o e he ca ego y Sch/Si.e.
MGLS∈Ob(NAlgSch/S(SH⊗(−))).
3.7 No med Mo i ic Spec a 98
P oo . See [4, Example 7.14].
Example 3.197: Le Dbe a Dedekind domain o mixed cha ac e is ic and S=Spec(D).
The ollowing a e no med mo i ic spec a o e he ca ego y Sm/Si.e.
MZSpi
S/m∈Ob(NAlgSm/S(SH⊗(−))) , m ≥0.
P oo . See [4, Theo em 13.9].
Rema k 3.198 (Powe Ope a ions):Le Sbe a scheme. Le Tbe a ini e ´e ale scheme
o e S. Le Gbe a ini e ´e ale g oup scheme o e S. Le B´e Gbe he ´e ale classi ying
space o G. Le Tbe G-equi a ian . Le Ebe a no med mo i ic spec um o e Sm/S.
The e exis s he ollowing o al T-powe ope a ion PTin he ∞-ca ego y H(S)whe e
E∈Ob(NAlgSm/S(SH⊗(−))) , PT∶Ω∞
P1(E)→MapH(S)(B´e G, Ω∞
P1(E)).
P oo . See [4, Example 7.25].
Example 3.199: Le kbe a ield. Le Sbe a smoo h scheme o e Spec(k). Le Tbe a
ini e ´e ale scheme o e S. Le Gbe a ini e ´e ale g oup scheme o e S. Le Tbe also G-
equi a ian . The ollowing o al T-powe ope a ion PTna u al in X∈Ob(Sm/S) eco e s
on homogeneous elemen s Voe edosky’s o al powe ope a ion in mo i ic cohomology
⊕
n∈Z
Σn
P1(MZS)∈Ob(NAlgSm/S(SH⊗(−))) , PT∶z●(X, ●)→z●(X×B´e G, ●).
P oo . See [4, Example 7.25].
Theo em 3.200: Le ∶S′→Sbe a ini e ´e ale mo phism o schemes. Le C⊂ ´e Sch/S
and le C′=C/S′. The no m unc o ⊗li s o he ollowing adjunc ion o ∞-ca ego ies
⊗∶SH⊗(S′)→SH⊗(S),
⊗∶NAlgC′(SH⊗(−))⇆NAlgC(SH⊗(−))∶ ∗.
P oo . See [4, Theo em 8.1].
Theo em 3.201: Le ∶S′→Sbe a ini e ´e ale mo phism o schemes. Le C⊂ ´e Sch/S
and le C′⊂ ´e Sch/S′be subca ego ies such ha #(C′)⊂Cand ∗(C)⊂C′. Le also
∗∶SH⊗(S′)→SH⊗(S)be compa ible wi h any base change (g∶T→S)∈Mo (C).
The e exis s he ollowing adjunc ion o ∞-ca ego ies
∗∶NAlgC(SH⊗(−))⇆NAlgC′(SH⊗(−))∶ ∗.
3.7 No med Mo i ic Spec a 99
P oo . See [4, Theo em 8.2].
Example 3.202: I ∶S′→Sis a p o-smoo h mo phism o schemes hen he e exis s
∗∶NAlgSm/S(SH⊗(−))⇆NAlgSm/S′(SH⊗(−))∶ ∗.
P oo . See [4, Example 8.4].
De ini ion 3.203 (No med X-Spec um):Le Xbe a p o ini e g oupoid.
(1) Le a no med X-spec um be a sec ion o he unc o Sp⊗(−)o e he ca ego y
Co (X-FinSe ) ha is coca esian o e backwa d mo phisms.
(2) Le he ∞-ca ego y o no med X-spec a be NAlg(Sp⊗(X)).
Re e ence. See [4, De ini ion 9.14].
Example 3.204: I Xis a p o ini e se hen he e is an equi alence o ∞-ca ego ies
NAlg(Sp⊗(X))≃CAlg(Sp⊗(X)) , X ∈Ob(P o(FinSe )).
P oo . See [4, Example 9.15].
P oposi ion 3.205 (Bachmann-Hoyois):The i s adjunc ion o symme ic monoidal
∞-ca ego ies whe e cSis a symme ic monoidal unc o na u al in Sli s o he ollowing
S∈Ob(Co (Sch,all, ´e )),
cS∶Sp⊗(
Π´e
1(S))⇆SH⊗(S)∶uS,
NAlg(Sp⊗(
Π´e
1(S)))⇆NAlgFE (SH⊗(S)).
P oo . See [4, P oposi ion 10.8].
Example 3.206: The C2-equi a ian Be i ealiza ion unc o li s o he adjunc ion
ReC2∶SH⊗(Spec(R))⇆Sp⊗(BC2)∶SingC2,
NAlgFE (SH⊗(R))⇆NAlg(Sp⊗(BC2)).
P oo . See he exposi ion ollowing [4, Lemma 11. 4].
3.8 G aded-No med Mo i ic Spec a 100
Example 3.207: Le Sbe an essen ially smoo h scheme o e a ield. Le MZSbe
Voe odsky’s mo i ic Eilenbe g-MacLane spec um. Since ⊕n∈ZΣn
P1(MZS)is a no med
mo i ic spec um o e he ca ego y Sm/S, i ollows ha Bloch’s cycle complex z●(S, ●)
o Sp omo es o a no med 
Π´e
1(S)-spec um.
P oo . See [4, Example 10.9].
Example 3.208: Since he algeb aic K- heo y spec um KGLSo e any scheme Sis
a no med mo i ic spec um o e Sch/S, i ollows ha Weibel’s homo opy in a ian K-
heo y spec um KHSo e any scheme Sp omo es o a no med 
Π´e
1(S)-spec um.
P oo . See [4, Example 10.10].
Example 3.209: Le Sbe a smoo h scheme o e a ield o cha ac e is ic ze o. The
algeb aic cobo dism cohomology heo y o Le ine-Mo el sa is ies Ωn(S)≃MGL2n,n
S(S).
Since ⊕n∈ZΣn
P1(MGLS)is a no med mo i ic spec um o e he ca ego y Sch/S, he e
exis s a no med 
Π´e
1(S)-spec um whose 0- h homo opy g oup is ⊕n∈ZΩn(S).
P oo . See [4, Example 10.11].
3.8 G aded-No med Mo i ic Spec a
De ini ion 3.210: Le Sbe a scheme. Le Γbe a disc e e commu a i e monoid. Le
ΓSmQP/S+be he ollowing ca ego y:
(1) Le (X, γ)∈Ob(ΓSmQP/S+)be X∈Ob(SmQP/S)along wi h a locally cons an
unc ion γ∶X→Γ.
(2) Le ( ∶(X, γ)→(Y, δ)) ∈Mo (ΓSmQP/S+)be ( ∶X+→Y+)∈Mo (SmQP/S+)
such ha he locally cons an unc ions γand δcoincide on −1(Y)⊂X.
Re e ence. See he exposi ion ollowing [4, Equa ion (13.12)].
Rema k 3.211: Le Sbe a scheme. Le Γbe a disc e e commu a i e monoid.
(1) The ca ego y ΓSmQP/S+admi s all ini e cop oduc s. The ollowing canonical
unc o is he ini ial unc o ha p ese es ini e cop oduc s in i s second a gumen
and he e exis s an equi alence o ∞-ca ego ies
Γ×SmQP/S+→ΓSmQP/S+,
PShΣ(ΓSmQP/S+,Spc)≃PShΣ∗(SmQP/S,Spc)Γ.
3.8 G aded-No med Mo i ic Spec a 101
(2) Fo e e y mo phism o disc e e commu a i e monoids φ he unc o φ!is he le
Kan ex ension o he ollowing 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 commu a i e monoid s uc u e o Γinduces a symme ic monoidal s uc u e
on he ca ego y ΓSmQP/S+. The Day con olu ion induces a symme ic monoidal
s uc u e on he ∞-ca ego y PShΣ∗(SmQP/S,Spc)Γ.
P oo . See he exposi ion ollowing [4, Equa ion (13.12)].
De ini ion 3.212: Le Sbe a scheme. Le Γbe a disc e e commu a i e monoid.
(1) Le (ΓS,+)be he commu a i e monoid o locally cons an unc ions and le
γ∶S→Γ,( ∶X→Y)∈Mo (SmQP/S), ∗∶ΓY→ΓX, γ ↦γ○ .
(2) Le p∶X→Sbe ini e locally ee and γ∶X→Γa locally cons an unc ion and
p∗(γ)∶S→Γ, p∗(γ)(s)∶=∑
x∈p−1(s)
degx(p)⋅γ(x),degx(p)=dimκ(s)(OXs,x).
(3) I pis ´e ale a x∈X hen OXs,x =κ(x)and he e o e degx(p)=[κ(x)∶κ(s)].
Re e ence. See he exposi ion ollowing [4, Equa ion (13.12)].
Rema k 3.213: Le Sbe a scheme. Le Γbe a disc e e commu a i e monoid.
(1) Le l be he class o ini e locally ee mo phisms in Sch. The e exis s a unc o
Co (Sch,all, l )→CAlg(Se ), S ↦ΓS,(Y
←Xp
→S)↦p∗○ ∗.
(2) The unc o abo e ex ends o he ollowing unc o
ΓSmQP⊗
+(−)∶Co (Sch,all, l )→CAlg(Ca 1),
S↦ΓSmQP/S+,(Y
←Xp
→S)↦p⊗○ ∗,
∗(U, γ)∶=(U×YX, ∗
U(γ)) , p⊗(U, γ)∶=(Rp(U), q∗(e∗(γ))),
Ue
←Rp(U)×SXq
→Rp(U).

3.8 G aded-No med Mo i ic Spec a 102
(3) The alues o he unc o s ∗and p⊗on mo phisms as well as hei symme ic
monoidal s uc u es a e uniquely de e mined by he equi emen ha he ai h ul
unc o ΓSmQP/S+→SmQP/S+is na u al in S∈Ob(Co (Sch,all, l )).
P oo . See he exposi ion ollowing [4, Lemma 13.13].
Rema k 3.214: Le Sbe a scheme. Le Γbe a disc e e commu a i e monoid.
(1) Fo e e y mo phism φ he ollowing unc o is na u al in Swi h
φ∶Γ→Γ′,
ΓSmQP/S+→Γ′SmQP/S+.
(2) We ob ain he e o e he ollwing unc o
(−)SmQP⊗
+(−)∶CAlg(Se )×Co (Sch,all, l )→CAlg(Ca 1).
(3) By epea ing he cons uc ion o he unc o SH⊗(−)we ob ain also he unc o
SH⊗(−)(−)∶CAlg(Se )×Co (Sch,all, ´e )→CAlg(Ca si
∞),(Γ, S)↦SH⊗(S)Γ.
P oo . See he exposi ion ollowing [4, Lemma 13.13].
Rema k 3.215: Le Sbe any scheme. Le Γbe a disc e e commu a i e monoid. The
symme ic monoidal ∞-ca ego y H⊗
∗(S)Γis ob ained om he symme ic monoidal ∞-
ca ego y PShΣ∗(SmQP/S,Spc)Γby in e ing he amilies o mo i ic equi alences which
a e gene a ed by he Nisne ich sie es and he A1-homo opy equi alences in ΓSmQP/S+.
P oo . See he exposi ion ollowing [4, Lemma 13.13].
De ini ion 3.216: Le Sbe scheme. Le Γbe a disc e e commu a i e monoid. The
compac ly gene a ed symme ic monoidal ∞-ca ego y SH⊗(S)Γis ob ained om he sym-
me ic monoidal ∞-ca ego y H⊗
∗(S)Γby in e ing he mo i ic sphe es in deg ee 0∈Γi.e.
SH⊗(S)Γ∶=H⊗
∗(S)Γ[{G Γ
0(Sp,q)−1}p,q∈Z].
Theo em 3.217 (Bachmann-Hoyois):The e exis s he ollowing unc o
SH⊗(−)(−)∶CAlg(Se )×Co (Sch,all, ´e )→CAlg(Ca si
∞),(Γ, S)↦SH⊗(S)Γ.
3.8 G aded-No med Mo i ic Spec a 103
P oo . See he exposi ion ollowing [4, Lemma 13.13].
Lemma 3.218: Le Γbe a disc e e commu a i e monoid. The ollowing canonical
mo phism o commu a i e monoids yields a na u al ans o ma ion
Γ→0,
⊕
Γ∶SH⊗(−)Γ→SH⊗(−).
P oo . See [4, Equa ion (13.12)].
Lemma 3.219: Le Γbe a disc e e commu a i e monoid and le δ∈Γ. Le ∶X→Y
be a mo phism o schemes. The g aded pullback unc o ∗is compu ed deg eewise. I
he mo phism is smoo h ∗admi s a le adjoin #which is also compu ed deg eewise
#∶SH⊗(X)Γ⇆SH⊗(Y)Γ∶ ∗,{Eγ}γ∈Γ∈Ob(SH⊗(Y)Γ), ∗({Eγ}γ∈Γ)δ= ∗(Eδ).
P oo . See [4, Rema k 13.15].
Lemma 3.220: Le Γbe a disc e e commu a i e monoid and le δ∈Γ. Le Sbe a
scheme. The unc o δ∗admi s a le adjoin δ!wi h
δ!∶SH⊗(S)⇆SH⊗(S)Γ∶δ∗, δ∗({Eγ}γ∈Γ)∶=Eδ, δ!(F)ξ∶=⎧
⎪
⎪
⎨
⎪
⎪
⎩
Fi ξ=δ
0else
P oo . See he exposi ion ollowing [4, Rema k 13.16].
Lemma 3.221: Le Γbe a disc e e commu a i e monoid and le γ∈Γ. I p∶X→Sis
ini e ´e ale o cons an deg ee d hen he e is an equi alence o SH⊗(X)-module unc o s
p⊗○γ!
≃
→(d⋅γ)!○p⊗∶SH⊗(X)→SH⊗(S)Γ.
P oo . See [4, Lemma 13.17].
Co olla y 3.222: Le Γbe a disc e e commu a i e monoid. The ollowing is also a
p esen ably no med ∞-ca ego y o e he ca ego y o schemes
SH⊗(−)Γ∶Co (Sch,all, ´e )→CAlg(Ca si
∞), S ↦SH⊗(S)Γ,(Y
←Xp
→S)↦p⊗○ ∗.
P oo . See he exposi ion ollowing [4, Rema k 13.16].
3.8 G aded-No med Mo i ic Spec a 104
De ini ion 3.223 (Γ-G aded-No med Mo i ic Spec um):Le Sbe a scheme. Le
C⊂ ´e Sch/Sbe a ull subca ego y ha con ains Sand is closed unde ini e sums and
ini e ´e ale ex ensions.
(1) A Γ-g aded-no med mo i ic spec um o e Cis a sec ion o SH⊗(−)Γo e he
ca ego y Co (C,all, ´e ) ha is coca esian o e Cop.
(2) The ∞-ca ego y o Γ-g aded-no med mo i ic spec a o e Cbe he ull subca ego y
NAlgC(SH⊗(−)Γ)⊂Sec (SH⊗(−)Γ∣Co (C,all, ´e )).
Re e ence. See he exposi ion ollowing [4, Lemma 13.17].
Rema k 3.224: Le Sbe a scheme. Le C⊂ ´e Sch/Sbe a ull subca ego y ha con ains
Sand is closed unde ini e sums and ini e ´e ale ex ensions.
(1) Le φbe a mo phism o disc e e commu a i e monoids. The e exis s an adjunc ion
φ∶Γ→Γ′,
φ!∶Sec (SH⊗(−)Γ∣Co (C,all, ´e ))⇆Sec (SH⊗(−)Γ′∣Co (C,all, ´e ))∶φ∗,
(2) Since φ!and φ∗p ese e Γ-g aded-no med spec a we ob ain he adjunc ion
φ!∶NAlgC(SH⊗(−)Γ)⇆NAlgC(SH⊗(−)Γ′)∶φ∗.
(3) The ollowing i ial mo phism o commu a i e g oupoids yields he adjunc ion
Γ→0,
⊕
Γ∶NAlgC(SH⊗(−)Γ)⇆NAlgC(SH⊗(−))∶cons Γ,
⊕
γ∈Γ({Eγ}γ∈Γ)∶=⊕
γ∈Γ
Eγ,
cons Γ(E)γ∶=E , ∀γ∈Γ.
P oo . See he exposi ion ollowing [4, Lemma 13.17].
3.8 G aded-No med Mo i ic Spec a 105
Example 3.225: Le Sbe a scheme. Le C⊂ ´e Sch/Sbe a ull subca ego y ha con ains
Sand is closed unde ini e sums and ini e ´e ale ex ensions. Le NSymC(−)be he ee
no med spec um unc o o e C. Le Ube he o ge ul unc o . The ee no med mo i ic
spec a o e Ca e canonically N-g aded and he e exis s an adjunc ion o ∞-ca ego ies
NSymC(−)∶SH⊗(S)⇆NAlgC(SH⊗(−))∶U.
P oo . See [4, Example 13.19].
De ini ion 3.226: Le Sbe a scheme. Le C⊂ ´e Sch/Sbe a ull subca ego y ha
con ains Sand is closed unde ini e sums and ini e ´e ale ex ensions. Le he ull
subca ego y o Z-g aded e ec i e spec a o e Sbe
(SH⊗(S)Z)e ⊂SH⊗(S)Z,{En}n∈Z∈Ob((SH⊗(S)Z)e ), En∈Ob(Σn
P1(SH⊗(S)e )).
Rema k 3.227: Le Sbe a scheme. The ∞-ca ego y o Z-g aded e ec i e spec a o e
Sis a symme ic monoidal subca ego y since
Σn
P1(SH⊗(S)e )∧Σm
P1(SH⊗(S)e )⊂Σn+m
P1(SH⊗(S)e ).
P oo . See he in oduc ion o [4, Chap e 13.4].
Lemma 3.228: Le Sbe a scheme. The ∞-ca ego y o Z-g aded e ec i e spec a o e
Sis gene a ed unde colimi s by he se
{n!(Σn
P1(E))∈Ob((SH⊗(S)Z)e )∣E∈Ob(SH⊗(S)e )and n∈Z}.
P oo . See he in oduc ion o [4, Chap e 13.4].
Lemma 3.229: I p∶X→Sis a ini e ´e ale mo phism o cons an deg ee d hen
p⊗(n!(Σn
P1(E)))≃(n⋅d)!(p⊗(S2n,n)∧p⊗(E)) , E ∈Ob(SH⊗(X)e ), n ∈Z.
P oo . See he in oduc ion o [4, Chap e 13.4].
Co olla y 3.230: The e exis s a sub unc o
(SH⊗(−)Z)e ⊂SH⊗(−)Z∶Co (Sch,all, ´e )→CAlg(Ca si
∞).
P oo . See he in oduc ion o [4, Chap e 13.4].
4.1 Z-G aded Galois-Equi a ian Spec a 112
P oo . See [16, Lemma 9.1.5].
Lemma 4.25: Le C⊗be a symme ic monoidal locally p esen able s able ∞-ca ego y.
Le C⊗be gene a ed unde a bi a y shi s and small colimi s by a se o compac ob-
jec s {Gα}α∈A⊂Ob(Cc)and le A∈Ob(CAlg(C⊗)). The symme ic monoidal locally
p esen able s able ∞-ca ego y ModA(C⊗)is gene a ed unde a bi a y shi s and small
colimi s by he se o compac objec s {A⊗Gα}α∈A⊂Ob(ModA(C⊗)c).
P oo . See [31, Chap e 4.6.2].
Lemma 4.26: Le C⊗be a symme ic monoidal locally p esen able s able ∞-ca ego y.
Le A∈Ob(CAlg(C⊗)) be some E∞-algeb a. I X∈Ob(C⊗)is dualizable and admi s
he dual X∗ hen A⊗X∈Ob(ModA(C⊗)) is dualizable and admi s he dual A⊗X∗. In
pa icula is Adualizable and sel -dual as a module o e i sel .
P oo . See [31, Chap e 4.6.2].
De ini ion 4.27: The ∞-ca ego y P Lis he subca ego y o he ∞-ca ego y o no nec-
essa ily small ∞-ca ego ies 
Ca ∞wi h he ollowing p ope ies:
(1) Ob(P L)is he class o locally p esen able ∞-ca ego ies in 
Ca ∞.
(2) MapP L(C,D)is he space o colimi p ese ing unc o s o ∞-ca ego ies FunL(C,D).
Theo em 4.28: The e exis s a closed symme ic monoidal ∞-ca ego y P L,⊗such ha :
(1) Le FunL,L(C×D,E)be he ∞-ca ego y o unc o s which p ese e small colimi s
independen ly in each a gumen . The e exis s an equi alence o ∞-ca ego ies
FunL,L(C×D,E)≃FunL(C⊗D,E).
(2) The ∞-ca ego y Spc is he ⊗-uni o he symme ic monoidal ∞-ca ego y P L,⊗.
(3) The e exis s he ollowing equi alence o ∞-ca ego ies
MapP L(C,FunL(D,E))≃MapP L(C⊗D,E).
P oo . See [19, Theo em 5.31].
Theo em 4.29: Le P L
S be he ull subca ego y o P Lspanned by he locally p esen able
s able ∞-ca ego ies. The e is a closed symme ic monoidal ∞-ca ego y P L,⊗such ha :

4.1 Z-G aded Galois-Equi a ian Spec a 113
(1) Le FunL,L(C×D,E)be he ∞-ca ego y o unc o s which p ese e small colimi s
independen ly in each a gumen . The e exis s an equi alence o ∞-ca ego ies
FunL,L(C×D,E)≃FunL(C⊗D,E).
(2) The s able ∞-ca ego y o spec a Sp is also he ⊗-uni o he ∞-ca ego y P L,⊗
S .
(3) The e exis s he ollowing equi alence o ∞-ca ego ies
MapP L
S (C,FunL(D,E))≃MapP L
S (C⊗D,E).
P oo . See [19, Theo em 5.34].
Rema k 4.30: The ollowing P oposi ion 4.31 is used in Co olla y 4.49 o iden i y he
dualizable and compac objec s o he s able ∞-ca ego y Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z. The
Co olla y 4.49 is hen employed in he p oo o Theo em 4.73, see also Rema k 4.72.
P oposi ion 4.31: Le C,D∈Ob(CAlg(P L,⊗
S )) be locally p esen able and s able ∞-
ca ego ies. Le Cand Dbe compac ly gene a ed. Le Cc⊂Cand Dc⊂Dbe he ull
subca ego ies o compac objec s. Le c0∈Ob(Cc)and d0∈Ob(Dc). The locally p e-
sen able s able ∞-ca ego y C⊗Dis compac ly gene a ed by he ollowing se
{c⊗d∈Ob(C⊗D)∣c∈Ob(Cc), d ∈Ob(Dc)}.
P oo . See [16, P oposi ion 7.4.2].
Rema k 4.32: In he ollowing Rema k 4.33 and Rema k 4.34 and De ini ion 4.35
and De ini ion 4.36 we p esen he undamen al componen s o a boo s apping a gu-
men in P oposi ion 4.39, applied o he dualizable and sel -dual and compac gene a o s
{Σ∞
G(G/H)+}H≤G, see P oposi ion 4.38 and P oposi ion 4.40, o he symme ic monoidal
s able ∞-ca ego y o genuine G-equi a ian spec a Sp⊗(G)wi h espec o a ini e g oup
G, see De ini ion 3.103.
Rema k 4.33: Le G∈Ob(P o(FinG pd)) be a p o ini e g oupoid wi h he co il e ed
limi G=lim
α∈A(Gα). Fo e e y α∈A he ini e g oupoid Gα∈Ob(FinG pd)sa is ies
Gα=∐β∈BαBGα,β such ha Gα,β ∈Ob(FinG p)is a ini e g oup and BGα,β is he
delooping g oupoid o Gα,β and Bαis a ini e se .
4.1 Z-G aded Galois-Equi a ian Spec a 114
Rema k 4.34: Le G∈Ob(P o(FinG pd)) be a p o ini e g oupoid wi h he co il e ed
limi G=lim
α∈A(Gα). Le Gα=∐β∈BαBGα,β. Le H⊂Gα,β be a ini e subg oup and le
Fun(Gα,Se )≃Fun(∐
β∈Bα
BGα,β,Se )≃∏
β∈Bα
Fun(BGα,β,Se ),
Gα,β/H∈Ob(Fun(BGα,β,Se )),
Sα,β,H ∶=(∅,⋯,∅, Gα,β/H, ∅,⋯,∅), Sα,β,H ∈Ob(∏
β∈Bα
Fun(BGα,β,Se )).
De ini ion 4.35: Le G∈Ob(P o(FinG pd)) be a p o ini e g oupoid wi h he co il e ed
limi G=lim
α∈A(Gα). Le he ca ego y G-Se be he il e ed colimi in he ca ego y P Lwi h
G-Se ∶=colim
α∈A(Fun(Gα,Se )).
De ini ion 4.36: Le G∈Ob(P o(FinG pd)) be a p o ini e g oupoid wi h he co il e ed
limi G=lim
α∈A(Gα). Le α∈Awi h Gα=∐β∈BαBGα,β such ha Gα,β ∈Ob(FinG p)is
a ini e g oup and BGα,β is he delooping g oupoid o Gα,β and Bαis a ini e se . Le
also H⊂Gα,β be a ini e subg oup. Le Φα∶Fun(Gα,Se )→colim
α∈A(Fun(Gα,Se )) be he
canonical mo phism in he o dina y ca ego y P L. Le Sα,β,H be as abo e and le also
Xα,β,H ∶=Φα(Sα,β,H), Sα,β,H ∈Ob(Fun(Gα,Se )),
{Xα,β,H ∣α∈A , β ∈Bα, H ⊂Gα,β}⊂Ob(G-Se ).
P oposi ion 4.37: Le Gis a compac Lie g oup and le H⊂Gbe a closed subg oup.
The suspension spec a {Σ∞
G(G/H)+}H≤Ga e dualizable in he symme ic monoidal s a-
ble ∞-ca ego y o genuine G-equi a ian spec a Sp⊗(G).
P oo . See [34, Chap e 5.3].
P oposi ion 4.38: Le Gbe a g oup o ini e o de and le H⊂Gbe a subg oup.
The suspension spec a {Σ∞
G(G/H)+}H≤Ga e dualizable and sel -dual in he symme ic
monoidal s able ∞-ca ego y o genuine G-equi a ian spec a Sp⊗(G).
P oo . See [8, Example 3.2.6 (1)].
P oposi ion 4.39: Le G∈Ob(P o(FinG pd)). Le he Bachmann-Hoyois unc o o
spec a o e p o ini e g oupoids be Sp⊗(−)∶Co (P o(FinG pd),all, p)→CAlg(Ca si
∞).
4.1 Z-G aded Galois-Equi a ian Spec a 115
The ollowing G-spec a in he symme ic monoidal ∞-ca ego y Sp⊗(G)a e dualizable
and sel -dual
{Σ∞
Gα,β (Xα,β,H )∣α∈A , β ∈Bα, H ⊂Gα,β}⊂Ob(Sp⊗(G)).
P oo . This ollows om P oposi ion 4.38 and De ini ion 4.36.
P oposi ion 4.40: Le Gbe a g oup o ini e o de . The symme ic monoidal s able
∞-ca ego y o genuine G-equi a ian spec a Sp⊗(G)is compac ly gene a ed by he se
o dualizable and sel -dual spec a {Σ∞
G(G/H)+}H≤G.
P oo . See [22, P oposi ion 2.1].
P oposi ion 4.41: Le G∈Ob(P o(FinG pd)). Le he Bachmann-Hoyois unc o o
spec a o e p o ini e g oupoids be Sp⊗(−)∶Co (P o(FinG pd),all, p)→CAlg(Ca si
∞).
The symme ic monoidal ∞-ca ego y Sp⊗(G)is compac ly gene a ed by he se o dual-
izable and sel -dual G-spec a.
P oo . This ollows om P oposi ion 4.40 and De ini ion 4.36.
Example 4.42: Le Sbe a quasi-compac and quasi-sepa a ed scheme. Le 
Π´e
1(S)
be he p o ini e ´e ale undamen al g oupid o S. The symme ic monoidal locally p e-
sen able s able ∞-ca ego y Sp⊗(
Π´e
1(S)) is compac ly gene a ed by he se o o dualizable
and sel -dual 
Π´e
1(S)-spec a.
P oo . This ollows om P oposi ion 4.39 and P oposi ion 4.41.
Co olla y 4.43: Le Sbe a quasi-compac and quasi-sepa a ed scheme and le 
Π´e
1(S)
be he p o ini e ´e ale undamen al g oupid o S. The ull subca ego y o he compac

Π´e
1(S)-spec a Sp⊗(
Π´e
1(S))cis gene a ed unde a bi a y shi s and ini e colimi s and
e ac s by he se o dualizable and sel -dual 
Π´e
1(S)-spec a.
P oo . This ollows om Example 4.42.
De ini ion 4.44 (Unbounded ∞-Ca ego y o Chain Complexes):Le Abe a G o hendieck
abelian ca ego y. The unbounded ∞-ca ego y o chain complexes is he ollowing sym-
me ic monoidal locally p esen able s able ∞-ca ego y
D(A)∶=Ndg(Ch●(A) c
inj).
Re e ence. See [31, De ini ion 1.3.5.8].
4.1 Z-G aded Galois-Equi a ian Spec a 116
Example 4.45: Le Rbe a commu a i e ing. The e exis s he ollowing equi alence o
symme ic monoidal iangula ed ca ego ies
Ho(D(ModR))≃D(ModR).
Example 4.46 (∞-Ca ego y o Connec ed Chain Complexes):Th ough he use o
N∆(−)and he Dold-Kan co espondence N∶sAbp oj ⇆Ch≥0(Ab)p oj ∶Γ, we eco e
he ∞-ca ego y D≥0(Ab), o connec ed chain complexes.
P oo . See [19, Example 1.40 (3)].
Lemma 4.47: Le D⊗(Ab)Zbe he symme ic monoidal s able ∞-ca ego y o Z-g aded
chain complexes in Ab. The ull subca ego y (D⊗(Ab)Z)co compac unbounded Z-g aded
chain complexes is
D⊗(Ab)c=Pe ⊗(Ab),
A●,●∈Ob(D⊗(Ab)Z), A●,●∶=(⋯, A●,m, A●,m+1, A●,m+2,⋯),
Ob((D⊗(Ab)Z)c)={A●,●∈Ob(Pe ⊗(Ab)Z)∣A●,m =0almos all m∈Z}.
P oo . This ollows om Example 4.8.
Co olla y 4.48: Le D⊗(Ab)Zbe he symme ic monoidal ∞-ca ego y o unbounded
Z-g aded chain complexes in Ab. The ull subca ego y o dualizable unbounded Z-g aded
chain complexes in Ab is equi alen o (D⊗(Ab)Z)c.
P oo . This ollows om Example 4.8 and Lemma 4.47.
Co olla y 4.49: Le Sbe a quasi-compac and quasi-sepa a ed scheme. Le 
Π´e
1(S)be
he p o ini e ´e ale undamen al g oupid o S. Le D⊗(Ab)Zbe he locally p esen able
s able ∞-ca ego y o unbounded Z-g aded chain complexes in he ca ego y Ab. Le
Sp⊗(−)∶Co (P o(FinG pd),all, p)→CAlg(Ca si
∞)be he Bachmann-Hoyois unc o
o spec a o e p o ini e g oupoids. Le Sp⊗(
Π´e
1(S))⊗D(Ab)Zbe he ⊗-p oduc o sym-
me ic monoidal locally p esen able s able ∞-ca ego ies in he symme ic monoidal ∞-
ca ego y CAlg(P L,⊗
S ). Le AS∈Ob(CAlg(Sp⊗(
Π´e
1(S))⊗D(Ab)Z))be some E∞-algeb a.
The ollowing symme ic monoidal locally p esen able s able ∞-ca ego y
ModAS(Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z)∈Ob(CAlg(P L,⊗
S ))
is gene a ed unde shi s and small colimi s by he se o dualizable compac AS-modules
{AS⊗(P●⊗Q●,●)∣P●∈Ob(Sp⊗(
Π´e
1(S))c), Q●,●∈Ob(D⊗(Ab)Z)c)}.
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 117
P oo . This ollows om P oposi ion 4.31 and Co olla y 4.43 and Co olla y 4.48.
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a
De ini ion 4.50 (A in Spec a):Le Sbe a quasi-compac and quasi-sepa a ed scheme.
The ca ego y o A in spec a o e Sin he iangula ed ca ego y SH(S)is he ull
localizing iangula ed subca ego y
A(SH(S))∶=LocSH(S)({Σ∞
P1(X)+∣X∈Ob((Sm/S)Nis), ∶X→Sis ini e ´e ale}).
De ini ion 4.51 (A in Mo i es):Le Sbe a quasi-compac and quasi-sepa a ed scheme.
The iangula ed ca ego y o A in mo i es DMA(S)o e Sin he iangula ed ca ego y
o Spi zweck mo i es DM(S)is he ull localizing iangula ed subca ego y
DMA(S)∶=LocDM(S)({MS(X)∣X∈Ob((Sm/S)Nis), ∶X→Sis ini e ´e ale}).
Rema k 4.52: Le Sbe a quasi-compac and quasi-sepa a ed scheme and le also be
X∈Ob((Sm/S)Nis)and ∶X→S ini e ´e ale. The ollowing A in spec a o e Sand
A in mo i es o e Sa e dualizable and sel -dual and compac
Σ∞
P1(X)+∈Ob(A(SH(S))c),MS(X)∈Ob(DMA(S)c).
P oo . This ollows om Lemma 4.14, by applying he symme ic monoidal unc o
c, see Theo em 3.168, o he dualizable and sel -dual gene a o s o Sp⊗(
Π´e
1(S)), see
Example 4.42. The Co olla y 3.158, implies e e y ini e é ale co e ing o S o be hi by
he unc o c. A a ian o he p oo is possible ia A iyah duali y.
De ini ion 4.53 (A in-Ta e Mo i es):Le Sbe a quasi-compac and quasi-sepa a ed
scheme. The iangula ed ca ego y o A in-Ta e mo i es DMA(S)o e Sin he ian-
gula ed ca ego y o Spi zweck mo i es DM(S)is he ollowing ull localizing iangula ed
subca ego y closed unde ⊗-p oduc s
DMAT(S)∶=Loc⊗
DM(S)(Ob(DMA(S))∪Ob(DMT(S))).
Rema k 4.54: Le Sbe a quasi-compac and quasi-sepa a ed scheme. The e exis
canonical inclusions o symme ic monoidal locally p esen able s able ∞-ca ego ies
Loc⊗
DM⊗(S)({MZSpi
S})↘↙→→
↙↖
↓↓
DMA⊗(S)
↙↖
↓↓
DMT⊗(S)↘↙→→DMAT⊗(S)

4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 118
Rema k 4.55: Since Sp⊗∈Ob(CAlg(P L,⊗
S )) is ini ial he e exis s a unique up o
homo opy mo phism (α∶Sp⊗→SH⊗(S)) ∈Mo (CAlg(P L,⊗
S )). Le HZ∈Ob(Sp⊗)be
he Eilenbe g-MacLane spec um. The e exis s a canonical mo phism φo E∞-algeb as
in SH⊗(S)along wi h he ollowing unc o s o symme ic monoidal s able ∞-ca ego ies
(φ∶α(HZ)→MZSpi
S)∈Mo (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)).
Rema k 4.56: The ollowing na u al diag am o s able ∞-ca ego ies is commu a i e
LocModα(HZ)(SH⊗(S))({α(HZ)}) →→
↓↓
Modα(HZ)(SH⊗(S))
↓↓
LocModMZSpi
S
(SH⊗(S))({MZSpi
S})↘↙→→DMA⊗(S)↘↙→→DM⊗(S)
De ini ion 4.57 (Day Con olu ion):Le C⊗be a small syme ic monoidal ∞-ca ego y.
Le D⊗be a syme ic monoidal ∞-ca ego y which admi s small colimi s and in which
he ⊗-p oduc bi unc o p ese es small colimi s in each a iable. The Day con olu ion
⊗-p oduc F⊗Day Go he unc o s F, G ∈Ob(Fun(C⊗,D⊗)) is he le Kan ex ension o
he unc o C⊗×C⊗F×G
→D⊗×D⊗⊗D
→D⊗along he unc o C⊗×C⊗⊗C
→C⊗.
Re e ence. See he in oduc ion o [18, Chap e 1].
P oposi ion 4.58: Le C⊗be a syme ic monoidal ∞-ca ego y. Le D⊗be a syme ic
monoidal ∞-ca ego y which admi s small colimi s and in which he ⊗-p oduc bi unc o
p ese es small colimi s in each a iable. The e exis s an equi alence o ∞-ca ego ies
CAlg(Fun(C⊗,D⊗)⊗Day )≃Fun⊗,lax(C⊗,D⊗).
P oo . See [18, P oposi ion 2.12].
Example 4.59: Le Sbe a quasi-compac and quasi-sepa a ed scheme. Le DM⊗(S)be
he symme ic monoidal locally p esen able s able ∞-ca ego y o Spi zweck mo i es o e
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 119
S. Le D⊗(Ab)be he symme ic monoidal s able ∞-ca ego y o chain complexes in he
ca ego y o abelian g oups. The e exis s a canonical equi alence o ∞-ca ego ies
DM⊗(S)Z∶=Fun(N(Z)⊗,DM⊗(S))⊗Day ,
CAlg(Fun(N(Z)⊗,DM⊗(S))⊗Day )≃Fun⊗,lax(N(Z)⊗,DM⊗(S)).
P oo . This ollows om P oposi ion 4.58.
Rema k 4.60: Le Sbe a quasi-compac and quasi-sepa a ed scheme. The ollowing
symme ic monoidal le adjoin unc o is induced by he unc o o ∞-ca ego ies 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).
Rema k 4.61: The le adjoin unc o o he adjunc ion abo e admi s he ac o iza ion
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)).
Re e ence. See also Theo em 4.73.
Rema k 4.62: Le Sbe a quasi-compac and quasi-sepa a ed scheme. Le DM⊗(S)be
he symme ic monoidal locally p esen able s able ∞-ca ego y o Spi zweck mo i es o e
S. The e exis he ollowing adjunc ions whe e each le adjoin is symme ic 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 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 120
P oo . See [56, Theo em 4.3].
Rema k 4.63: The e exis s he ollowing diag am o symme ic monoidal unc o s.
Loc⊗
DM⊗(S)({MZSpi
S}) DMA⊗(S)Sp⊗(
Π´e
1(S))
DMT⊗(S)DMAT⊗(S)
D(Ab)ZSp⊗(
Π´e
1(S))⊗D(Ab)Z
MS(−)○Θ
L
P oo . See Rema k 4.54 and Theo em 4.73. See also Rema k 4.83.
Rema k 4.64: The ollowing Theo em 4.65 is used in Theo em 4.177. An explici
pe iodiza ion PMZSpi
So he E∞-algeb a MZSpi
Swi h espec o he spec um S=Spec(D)
o a Dedekind domain Do mixed cha ac e is ic was cons uc ed by M. Spi zweck, see
also Rema k 2.31.
Theo em 4.65 (M. Spi zweck):Le Dbe a Dedekind domain o mixed cha ac e is ic
and S=Spec(D). The mul iplica i e pe iodiza ion PMZSpi
So he E∞-algeb a MZSpi
Sis
unique up o equi alence.
P oo .
(1) The e exis s he ollowing symme ic monoidal ∞-ca ego y wi h he equi alences
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)).
Le φbe he ollowing lax symme ic monoidal unc o o Pica d ∞-g oupoids wi h
essen ial image Im(φ). The essen ial image Im(φ)is a ull Pica d ∞-subg oupoid
φ∶N(Z)⊗→Pic(DM⊗(S)) , n ↦ZS(n)[2n],
Im(φ)={ZS(n)[2n]∣n∈Z},Im(φ)⊂Pic(DM⊗(S)),
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 121
Im(φ)≃N(Z×BC2),Im(φ)∈Ob(G p(Spc)) , π0(Im(φ))∈Ob(G p(Spc)).
Le Sp≥0be he ∞-ca ego y o connec i e spec a. The e exis s also he equi alence
G p(Spc)≃Sp≥0,G p(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 unc o φinduces also he ollowing sequence o spec a whe e Im(φ)spli s
N(Z)→Im(φ)→π0(Im(φ)) ≃
→N(Z).
Mo i ic cohomology in weigh ze o implies he ollowing isomo phisms o g oups
Au DM(S)(ZS(0))=C2,EndDM(S)(ZS(0))=Z,Au DM(S)(ZS(0))∈Ob(G p(Spc)).
The essen ial image o he unc o φsa is ies he e o e also he ollowing p ope y
π0(Au DM(S)(ZS(0)))=C2, πn(Au DM(S)(ZS(0)))=0, n ≥1.
Im(φ)=π0(Im(φ))⊕(π1(Im(φ)))[1],
(3) Le Sp⊗be he s able ∞-ca ego y o spec a. Le HZ∈Ob(Sp⊗)be he Eilenbe g-
MacLane spec um. The i s sequence whe e Im(φ)spli s implies he ollowing
N(Z)→Im(φ)→π0(Im(φ)) ≃
→N(Z),
HZ→Im(φ)≃
→HZ×HF2[1].
Le Sp⊗be admi he canonical -s uc u e. The e is he ollowing dis inguished
iangle in he iangula ed ca ego y o spec a placed in he ollowing diag am
τ≥1(S)→→S→→
↘↘
HZ→→
↓↓
τ≥1(S)[1]
↙↙
HF2[1]
I ollows by p ope ies o he -s uc u e ha he e exis s only he ze o mo phism
HomHo(Sp)(S,HF2[1])=π0(HF2[1])=0,HomHo(Sp)(τ≥1(S)[1],HF2[1])=0.
I ollows he e o e ha he e exis s only he ze o mo phism o ollowing spec a
HomHo(Sp)(HZ,HF2[1])=0.
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 128
The uni e sal p ope y o he cop oduc implies he exis ence o he dashed unc o
DMA⊗(S)
Sp⊗(
Π´e
1(S))
MS(−)○Θ→→
→→Sp⊗(
Π´e
1(S))⊗D⊗(Ab)
Ψ
↑↑
D⊗(Ab)
Λ
←←
←←
I ollows he e is he ollowing symme ic monoidal unc o o s able ∞-ca ego ies
Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Ψ
→DMA⊗(S)↪DMAT⊗(S).
(3) Acco ding o Rema k 4.54, he e exis s he ollowing diag am o canonical inclusion
unc o s o symme ic monoidal s able ∞-ca ego ies
Loc⊗
DM⊗(S)({MZSpi
S})↘↙→→
↙↖
↓↓
DMA⊗(S)
↙↖
↓↓
DMT⊗(S)↘↙→→DMAT⊗(S)
Acco ding o Rema k 4.60, he e exis s he ollowing adjunc ion o s able ∞-
ca ego ies whe e he le adjoin is symme ic monoidal
L∶D⊗(Ab)Z→DM⊗(S)∶R.
The le adjoin unc o o he adjunc ion abo e admi s he ollowing ac o iza ion
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 symme ic monoidal unc o s Ψand Λinduce he ollowing adjunc ion o
s able ∞-ca ego ies whe e he le adjoin unc o Lis also symme ic monoidal
L∶(Sp⊗(
Π´e
1(S))⊗D⊗(Ab))⊗D⊗(Ab)D⊗(Ab)Z⇆DMAT⊗(S)∶R.
P ope ies o he ela i e ⊗-p oduc imply he adjunc ion o be equi alen o
L∶Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z⇆DMAT⊗(S)∶R

4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 129
The adjunc ion o s able ∞-ca ego ies abo e implies also he ollowing ela ions
ZS(0)∶=MZSpi
S,
ZS(0)∈Ob(CAlg(DMAT⊗(S))),
R(ZS(0))∈Ob(CAlg(Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z)),
M∈Ob(ModR(ZS(0))(Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z)),
L(M)∈Ob(ModL(R(ZS(0)))(DMAT⊗(S))).
The couni mo phism below induces he ollowing symme ic monoidal unc o
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) Le C⊗and D⊗be he ollowing cocomple e s able symme ic monoidal ∞-ca ego ies
whe e he ⊗-p oduc s p ese e small colimi s. Le L∶C⊗⇆D⊗∶Rbe he adjunc-
ion o ∞-ca ego ies abo e whe e he le adjoin unc o Lis symme ic monoidal
C⊗∶=Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z,D⊗∶=DMAT⊗(S).
Le Aand Bbe he ollowing E∞-algeb as in he ∞-ca ego ies C⊗and D⊗ esp.
A∶=R(ZS(0)) , B ∶=ZS(0), A ∈Ob(CAlg(C⊗)) , B ∈Ob(CAlg(D⊗)).
Le also be he ollowing mo phism whose adjoin mo phism gis an equi alence
( ∶=εZS(0)∶L(A)→B)∈Mo (CAlg(D⊗)),
(g∶=idZS(0)∶A→R(B))∈Mo (CAlg(C⊗)),
Acco ding o Co olla y 4.49, he locally p esen able symme ic monoidal ∞-ca ego y
Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z∈Ob(CAlg(P L,⊗
S ))
gene a ed unde shi s and small colimi s by he se o dualizable compac objec s
{P●⊗Q●,●∣P●∈Ob(Sp⊗(
Π´e
1(S))c), Q●,●∈Ob(D⊗(Ab)Z)c)}.
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 130
Acco ding o De ini ion 4.36 and also P oposi ion 4.39 and P oposi ion 4.41 and
also Example 4.42, he dualizable and sel -dual and compac 
Π´e
1(S)-spec a a e
{Xα,β,H ∣α∈A , β ∈Bα, H ⊂Gα,β}⊂Ob(
Π´e
1(S)-Se ),
Ob(Sp⊗(
Π´e
1(S))c)={P●∈Ob(Sp⊗(
Π´e
1(S))∣P●=Σ∞
Gα,β (Xα,β,H )}.
Acco ding o Lemma 4.47, he dualizable and compac Z-g aded chain complexes
in he ca ego y o abelian g oups a e
D⊗(Ab)c=Pe ⊗(Ab),
Q●,●∈Ob(D⊗(Ab)Z), Q●,●∶=(⋯, Q●,m, Q●,m+1, Q●,m+2,⋯),
Ob((D⊗(Ab)Z)c)={Q●,●∈Ob(Pe ⊗(Ab)Z)∣Q●,m =0almos all m∈Z}.
Since he monoidal uni ZS(0)is compac in he ∞-ca ego y DMAT⊗(S), i ollows
om P oposi ion 4.24, ha e e y dualizable A in-Ta e mo i e o e Sis also
compac . Acco ding o Lemma 4.14 and also o Rema k 4.52, he le adjoin
unc o Lsends e e y dualizable and compac Z-g aded 
Π´e
1(S)-spec um P●⊗Q●,●
o a compac A in-Ta e mo i e o e S, ha is o say
L(P●⊗Q●,●)∈Ob(DMAT⊗(S)c).
Since he condi ions o Theo em 4.71 a e sa is ied, we ob ain he implica ions:
●The ollowing symme ic monoidal unc o o ∞-ca ego ies is ully ai h ul
ModR(ZS(0))(Sp⊗(
Π´e
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 essen ial image o he unc o abo e is compac ly gene a ed by he se
{ZS(0)∧L(R(ZS(0))) L(P●⊗Q●,●)} ⊂Ob(DMAT⊗(S)).
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 131
I ollows he e exis s some E∞-algeb a ASdepending on he base scheme Sand
also a symme ic monoidal equi alence o locally p esen able s able ∞-ca ego ies
AS∶=R(ZS(0)),
AS∈Ob(CAlg(Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z)),
ModAS(Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z)≃DMAT⊗(S).
Rema k 4.74: Th oughou he emainde o Sec ion 4.2 we p esen s aigh o wa d
co olla ies and examples o Theo em 4.73, as men ioned in he In oduc ion, while also
employing Theo em 4.177, i possible. The p oo s o he co olla ies and examples o
Theo em 4.73, which a e e e ing o Theo em 4.73 and Theo em 4.177 a e hemsel
analogous o he p oo s o Theo em 4.73 and Theo em 4.177 and a e hus supp essed.
Rema k 4.75: In he ollowing emainde o Sec ion 4.2 we es ablish na u al s able ∞-
subca ego ies o DMAT⊗(S)wi h espec o a quasi-compac and quasi-sepa a ed scheme
S. We hen show hem sepa a ely, o be symme ically monoidally equi alen o s able
∞-ca ego ies associa ed wi h he s able ∞-ca ego y ModAS(Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z),
see Theo em 4.73. We hen exploi hese equi alences o examine he s able ∞-ca ego y
o A in-Ta e mo i es DMAT⊗(Spec(R)) o e he spec um o he eal numbe s. We
hen also es ablish some s aigh o wa d conclusion and examples i he base scheme S
is he spec um S=Spec(k)o a ield ko cha ac e is ic ze o o an algeb aically closed
ield, see in pa icula Co olla y 4.110 and Co olla y 4.113 and also Co olla y 4.122.
Example 4.76: Le C2∶=Gal(C/R)be he absolu e Galois g oup o he eal numbe s.
(1) The ollowing A in-Ta e E∞-algeb a ARis also a Z-g aded no med BC2-spec um
L∶Sp⊗(BC2)⊗D⊗(Ab)Z⇆DMAT⊗(Spec(R))∶R,
AR∈Ob(NAlg(Sp⊗(BC2)⊗D⊗(Ab)Z)),
AR∶=R(MZSpi
R).
(2) The e exis s he ollowing symme ic monoidal equi alence o s able ∞-ca ego ies
ModAR(Sp⊗(BC2)⊗D⊗(Ab)Z)≃DMAT⊗(Spec(R)).
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 132
P oo . This ollows om Theo em 4.73 and Theo em 4.177 and Example 3.160.
Co olla y 4.77: Le Sbe a quasi-compac and quasi-sepa a ed scheme.
(1) The ollowing Ta e E∞-algeb a ASis a Z-g aded chain complex in abelian g oups
L∶D⊗(Ab)Z⇆DMT⊗(S)∶R,
AS∈Ob(CAlg(D⊗(Ab)Z)),
AS∶=R(MZSpi
S).
(2) The e exis s he ollowing symme ic monoidal equi alence o s able ∞-ca ego ies
ModAS(D⊗(Ab)Z)≃DMT⊗(S).
P oo . This ollows om Theo em 4.73.
Co olla y 4.78: Le Sbe a quasi-compac and quasi-sepa a ed scheme.
(1) The ollowing A in E∞-algeb a ASis in pa icula a no med 
Π´e
1(S)-spec um
L∶Sp⊗(
Π´e
1(S))⇆DMA⊗(S)∶R,
AS∈Ob(NAlg(Sp⊗(
Π´e
1(S)))),
AS∶=R(MZSpi
S).
(2) The e exis s he ollowing symme ic monoidal equi alence o s able ∞-ca ego ies
ModAS(Sp⊗(
Π´e
1(S)))≃DMA⊗(S).
P oo . This ollows om Theo em 4.73 and Theo em 4.177.
Example 4.79: Le C2∶=Gal(C/R)be he absolu e Galois g oup o he eal numbe s.
(1) The ollowing A in E∞-algeb a ARis in pa icula also a no med BC2-spec um
L∶Sp⊗(BC2)⇆DMA⊗(Spec(R))∶R,
AR∈Ob(NAlg(Sp⊗(BC2))),
AR∶=R(MZSpi
R).
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 133
(2) The e exis s he ollowing symme ic monoidal equi alence o s able ∞-ca ego ies
ModAR(Sp⊗(BC2))≃DMA⊗(Spec(R)).
P oo . This ollows om Co olla y 4.78 and Example 3.160 and Example 3.129.
Co olla y 4.80: Le Sbe a quasi-compac and quasi-sepa a ed scheme.
(1) The ollowing A in E∞-algeb a ASis in pa icula a no med 
Π´e
1(S)-spec um
L∶Sp⊗(
Π´e
1(S))⊗D⊗(Ab)⇆DMA⊗(S)∶R,
AS∈Ob(NAlg(Sp⊗(
Π´e
1(S))⊗D⊗(Ab))),
AS∶=R(MZSpi
S).
(2) The e exis s he ollowing symme ic monoidal equi alence o s able ∞-ca ego ies
ModAS(Sp⊗(
Π´e
1(S))⊗D⊗(Ab))≃DMA⊗(S).
P oo . This ollows om Theo em 4.73 and Theo em 4.177 and Co olla y 4.78.
Co olla y 4.81: Le Sbe a connec ed scheme wi h a ini e Galois co e S′/Sand i ial
au omo phism g oup {e}. Le also cS′/Sbe he le adjoin symme ic monoidal unc o
o he ollowing adjunc ion o locally p esen able s able ∞-ca ego ies.
(1) The ollowing A in-Ta e E∞-algeb a AS′/Sis a Z-g aded-no med B({e})-spec um
{e}∶=Au FE /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 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 134
(2) The e exis he ollowing symme ic monoidal equi alences o s able ∞-ca ego ies
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 Au FE /S(S′/S)is he i ial g oup {e} he scheme S′is isomo phic o S. We
ob ain he e o e he symme ic monoidal equi alence es ablished by M. Spi zweck
AS∶=R(MZSpi
S),
AS∈Ob(CAlg(D⊗(Ab)Z)),
ModAS(D⊗(Ab)Z)≃DMT⊗(S).
P oo . This ollows om Theo em 4.73 and Theo em 4.177 and Theo em 3.169.
Example 4.82: Le kbe an algeb aically closed ield and le S=Spec(k). Since he
unique up o isomo phism ini e Galois co e S′/Sis he iden i y mo phism he au o-
mo phism g oup o he ini e Galois co e is he i ial g oup {e}. The e o e is e e y
A in-Ta e E∞-algeb a AS′/SaZ-g aded-no med B({e})-spec um.
P oo . This ollows om Co olla y 4.81 and Example 3.160 and Example 3.129.
Rema k 4.83: The e exis s he ollowing diag am o symme ic monoidal unc o s,
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 135
Loc⊗
DM⊗(S)({MZSpi
S}) DMA⊗(S)ModAβ
S(Sp⊗(
Π´e
1(S))⊗D(Ab))
DMT⊗(S)DMAT⊗(S)
ModAα
S(D(Ab)Z)ModAγ
S(Sp⊗(
Π´e
1(S))⊗D(Ab)Z)
≃
≃≃
along wi h he ollowing E∞-algeb as
Aα
S∈Ob(CAlg(Sp⊗(D⊗(Ab)Z)),
Aβ
S∈Ob(NAlg(Sp⊗(
Π´e
1(S))⊗D⊗(Ab))),
Aγ
S∈Ob(CAlg(Sp⊗(
Π´e
1(S))⊗D⊗(Ab)Z)).
P oo . See Theo em 4.73 and Co olla y 4.77 and Co olla y 4.80 and C . Rema k 4.63.
Co olla y 4.84: Le Sis a connec ed scheme wi h a ini e Galois co e S′/Sand
au omo phism g oup G. Le H⊂Gbe a subg oup and le T=S′/Hbe he quo ien in
he ca ego y o schemes. Le also cS′/Sbe he le adjoin symme ic monoidal unc o
o he ollowing adjunc ion o o s able ∞-ca ego ies.
(1) The ollowing A in E∞-algeb a AS′/Sis in pa icula a no med BG-spec um
G∶=Au FE /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 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 136
(2) The e exis he ollowing symme ic monoidal equi alences o s able ∞-ca ego ies
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).
P oo . This ollows om Theo em 4.73 and Theo em 4.177 and Theo em 3.169.
Example 4.85: Le kbe an algeb aically closed ield and le S=Spec(k). Since he
unique up o isomo phism ini e Galois co e S′/Sis he iden i y mo phism he au o-
mo phism g oup o he ini e Galois co e is he i ial g oup {e}. The e o e he e exis s
he ollowing symme ic monoidal equi alence o s able ∞-ca ego ies
ModAS′/S(Sp⊗(B{e}))≃LocDM(S)({MS(S)}).
P oo . This ollows om Co olla y 4.84 and Example 3.160 and Example 3.129.
Example 4.86: Le Spec(C)/Spec(R)be he unique up o isomo phism non i ial
ini e Galois co e o he eal numbe s. The au omo phism g oup o he ini e Galois
co e is C2. Le H⊂C2be a subg oup and le T=Spec(C)/Hbe he quo ien in he
ca ego y o schemes. Le also cC/Rbe he le adjoin symme ic monoidal unc o o he
ollowing adjunc ion o o s able ∞-ca ego ies.
(1) The ollowing A in E∞-algeb a AC/Ris in pa icula a no med BC2-spec um
C2≅Au FE /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 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 137
(2) The e exis he ollowing symme ic monoidal equi alences o s able ∞-ca ego ies
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) The e exis s he ollowing symme ic monoidal equi alence o s able ∞-ca ego ies
Spec(C)/C2≅Spec(R),
ModAC/R(Sp⊗(BC2))≃LocDM(Spec(R))({MR(Spec(R)),MR(Spec(C))}).
P oo . This ollows om Co olla y 4.84.
Example 4.87: Le Spec(R)/Spec(R)be he i ial ini e Galois co e o he eal num-
be s. The au omo phism g oup o he ini e Galois co e is {e}. Le also cR/Rbe he le
adjoin symme ic monoidal unc o o he ollowing adjunc ion o o s able ∞-ca ego ies.
(1) The ollowing A in E∞-algeb a AR/Ris in pa icula a no med B({e})-spec um
{e}≅Au FE /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) The e exis he ollowing symme ic monoidal equi alence o s able ∞-ca ego ies
ModAR/R(Sp⊗(B({e})))≃LocDM(Spec(R))({MR(Spec(R))}).
P oo . This ollows om Co olla y 4.84.
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 144
(2) The e exis he ollowing symme ic monoidal equi alences o s able ∞-ca ego ies
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))).
P oo . This ollows om Co olla y 4.88.
De ini ion 4.101 (Pe mu a ion Modules):Le Rbe a commu a i e and uni al ing.
Le Gbe a p o ini e g oup. Le Mod(G;R)be he ca ego y o ee modules o e Ron
e e y basis X∈Ob(G-Se )wi h he linea ly ex ended G-ac ion. The ca egoy o (G;R)-
pe mu a ion modules is he essen ial image o he ollowing symme ic monoidal unc o
R(−)∶G-Se →Mod(G;R), R(X)∶=⊕
x∈X
R,
Pe m(G;R)∶=Im(R(−)).
Re e ence. See [5, No a ion 2.4].
Rema k 4.102: Le Rbe a commu a i e and uni al ing. Le Gbe a p o ini e g oup.
The ca ego y Pe m(G;R)is G o hendieck abelian wi h he ini ely p esen ed gene a o s
{R(G/H)∣H⊂Gis open}⊂Ob(Pe m(G;R)).
P oo . See [5, P oposi ion 2.6].
De ini ion 4.103: Le Rbe a commu a i e and uni al ing. Le Gbe a p o ini e g oup.
A(G;R)-chain complex A●∈Ob(K(Pe m(G;R))) is called G-acyclic i o e e y open
subg oup H⊂G he (G;R)-chain complex AH
●o H- ixed poin s is acyclic.
Re e ence. See [5, De ini ion 3.1].
De ini ion 4.104: Le Rbe a commu a i e uni al ing. Le Gbe a p o ini e g oup. A
mo phism ∈Ob(K(Pe m(G;R))) is called a G-quasi-isomo phism i he cone Cone( )
is G-acyclic. Le he ull subca ego y o he G-acyclic (G;R)-chain complexes be deno ed
Ac(Pe m(G;R)). Le he associa ed class o G-quasi-isomo phisms be WG.

4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 145
De ini ion 4.105: Le Rbe a commu a i e and uni al ing. Le Gbe a p o ini e g oup.
The de i ed ca ego y o (G;R)-pe mu a ion modules is
D(Pe m(G;R))∶=K(Pe m(G;R))[W−1
G]
=K(Pe m(G;R))/Ac(Pe m(G;R)).
Re e ence. See [5, De ini ion 3.6].
Co olla y 4.106: Le Rbe a commu a i e and uni al ing. Le Gbe a p o ini e g oup.
The de i ed ca ego y o (G;R)-pe mu a ion modules D(Pe m(G;R)) is a symme ic
monoidal and compac ly gene a ed iangula ed ca ego y.
Re e ence. See [5, Co olla y 3.10].
De ini ion 4.107 (Cohomological (G;R)-Mackey Func o ):Le Rbe a commu a i e
and uni al ing. Le Gbe a p o ini e g oup. An R-linea G-Mackey unc o is an addi i e
unc o F∶Bop
G→ModR. The R-linea G-Mackey unc o is called cohomological i o
all closed subg oups K⊂H⊂Gwi h he index [H∶K]and wi h he canonical p ojec ion
mo phism π∶G/K→G/H he alue o Fon (G/Hπ
←G/Kπ
→G/H)is he mul iplica ion
mo phism [H∶K]⋅(−).
Re e ence. See [5, De ini ion 4.5].
De ini ion 4.108 (Mixed A in Mo i es):Le Rbe a commu a i e and uni al ing. Le
kbe a ield. Le he inagula ed ca ego y o mixed A in mo i es o e he base scheme
Spec(k)wi h R-coe icien s be he ollowing ull localizing iangula ed subca go y in he
iangula ed ca ego y o Voe odsky’s mixed mo i es
DMA(Spec(k), R)⊂DM(Spec(k), R),
LocDM(Spec(k),R)({MS(X)∣X∈Ob((Sm/k)Nis), ∶X→Spec(k)is ini e ´e ale}).
Theo em 4.109 (G o hendieck-Voe odsky-Yoshida):Le Rbe a commu a i e and uni-
al ing. Le kbe a ield and le Gal(ksep/k)be he absolu e Galois g oup o k. Le
DMA(Spec(k), R)be he iangula ed ca ego y o Voe odsky’s mixed A in mo i es o e
he base scheme Spec(k)wi h R-coe icien s. Le Fun⊕
coh(Bop
Gal(ksep/k),ModR)be he
abelian ca ego y o cohomological (Gal(ksep/k);R)-Mackey unc o s. The e exis he
ollowing symme ic monoidal equi alences o iangula ed ca ego ies
D(Pe m(Gal(ksep/k);R))≃DMA(Spec(k), R)≃D(Fun⊕
coh(Bop
Gal(ksep/k),ModR)).
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 146
P oo . See [5, Equa ion (1.2)].
Co olla y 4.110: Le kbe a ield o cha ac e is ic ze o and le Gal(ksep/k)be he abso-
lu e Galois g oup o k. Le DMA⊗(Spec(k),Z)be he s able ∞-ca ego y o Voe odsky’s
mixed A in mo i es o e Spec(k)wi h Z-coe icien s. Le Fun⊕
coh(Bop
Gal(ksep/k),Ab)be
he abelian ca ego y o he cohomological (Gal(ksep/k);Z)-Mackey unc o s. Le also
Ak∈Ob(CAlg(Sp⊗(BGal(ksep/k)))) be he A in E∞-algeb a o Spec(k). The e exis
he ollowing symme ic monoidal equi alences o s able ∞-ca ego ies
ModAk(Sp⊗(BGal(ksep/k))) ≃→→
≃
↓↓
D⊗(Pe m(Gal(ksep/k);Z))
≃
↓↓
D⊗(Fun⊕
coh(Bop
Gal(ksep/k),Ab)) ≃→→DMA⊗(Spec(k),Z).
P oo . This ollows om Theo em 4.109 and Co olla y 4.78 and Co olla y 2.14.
Example 4.111: Le kbe an algeb aically closed ield.
(1) The e exis s he ollowing symme ic monoidal equi alences o abelian ca ego ies
{e}≅Gal(ksep/k)≅
π´e
1(Spec(k), x),
Ab ≃Fun⊕
coh(Bop
Gal(ksep/k),Ab)≃Pe m(Gal(ksep/k);Z).
(2) The e exis s he ollowing symme ic monoidal equi alence o s able ∞-ca ego ies
ModAk(Sp⊗)≃LocDM(Spec(k))({Mk(Spec(k))}).
(3) I kis an algeb aically closed ield o cha ac e is ic ze o hen he e exis also he
ollowing symme ic monoidal equi alences o s able ∞-ca ego ies
ModAk(Sp⊗)Ak↦Z→→
Ak↦HZ
↓↓
D⊗(Ab)
Z↦Mk(Spec(k))
↓↓
ModHZ(Sp⊗)HZ↦Mk(Spec(k)) →→DMA⊗(Spec(k),Z).
P oo . This ollows om Co olla y 4.110 and Example 4.85 and Example 3.129.
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 147
Theo em 4.112: Le kbe a pe ec ield. Le also DMA(Spec(k),Q)cbe he ull i-
angula ed subca ego y o compac a ional mixed A in mo i es o e he base scheme
Spec(k). Le Vecc
Qbe he ca ego y o ini e dimensional Q- ec o spaces. The e exis s
he ollowing symme ic monoidal equi alence o iangula ed ca ego ies
DMA(Spec(k),Q)c≃Db(Fun(BGal(ksep/k),Vecc
Q)).
P oo . See [65, P oposi ion 1.4].
Co olla y 4.113: Le kbe a ield o cha ac e is ic ze o. Le also DMA(Spec(k),Q)cbe
he ull iangula ed subca ego y o compac a ional mixed A in mo i es o e Spec(k).
Le Vecc
Qbe he ca ego y o ini e dimensional Q- ec o spaces. The e exis s he ollowing
symme ic monoidal equi alences o iangula ed ca ego ies
DMA(Spec(k),Q)c≃Ho(ModAk(Sp⊗(BGal(ksep/k)))c
Q)≃Db(Fun(BGal(ksep/k),Vecc
Q)).
P oo . This ollows om Theo em 4.112 and Co olla y 4.78 and Co olla y 2.14.
Example 4.114: I kis an algeb aically closed ield o cha ac e is ic ze o hen he e
exis also he ollowing symme ic monoidal equi alences o iangula ed ca ego ies
DMA(Spec(k),Q)c≃Ho(ModAk(Sp⊗)c
Q)≃Db(Vecc
Q).
P oo . This ollows om Co olla y 4.113 and Example 4.111.
De ini ion 4.115 (Fibe Func o ):Le kbe a ield. Le (A,⊗,1A)be a igid symme ic
monoidal abelian ca ego y en iched o e he ca ego y Veck. Le also Vecc
kbe he ca ego y
o ini e dimensional k- ec o spaces. A ibe unc o ωis a ai h ul exac symme ic
monoidal k-linea unc o
ω∶A→Vecc
k.
Re e ence. See [45, De ini ion 1.10].
De ini ion 4.116 (Neu al Tannakian Ca ego y):Le kbe a ield. A neu al Tan-
nakian ca ego y (A, ω)o e kis a igid symme ic monoidal abelian ca ego y (A,⊗,1A)
en iched o e he ca ego y Veckalong wi h a ibe unc o ω∶A→Vecc
k.
Re e ence. See [45, De ini ion 1.10].
Theo em 4.117 (M. Le ine):I Sis a base scheme sa is ying he a ional Beilinson-
Soule conjec u e hen he iangula ed ca ego y o geome ic a ional mixed Ta e mo i es
4.2 A in-Ta e Mo i es and Z-G aded Galois-Equi a ian Spec a 148
DMTgm(S, Q)o e Sadmi s a na u al -s uc u e whose hea includes {Q(n)}n∈Z. I
Sis also connec ed hen his hea MTgm(S, Q)is a neu al Tannakian ca ego y o e Q
wi h a canonical ibe unc o ωSand Tannakian undamen al g oup scheme GS. I S
sa is ies he K(π, 1)-conjec u e hen he e exis s also he ollowing symme ic monoidal
equi alence o iangula ed ca go ies
ωS∶MTgm(S, Q)→Vecc
Q,MTgm(S, Q)≃RepQ(GS),DMTgm(S, Q)≃Db(RepQ(GS)).
P oo . See [54, Theo em 5.1].
Co olla y 4.118: I kis a numbe ield hen he e exis he ollowing symme ic monoidal
equi alences o iangula ed ca ego ies
DMTgm(Spec(k),Q)≃Ho(ModAk(D⊗(Ab)Z)c
Q)≃Db(RepQ(Gk)).
P oo . This ollows om Theo em 4.117 and Co olla y 4.77 and Co olla y 2.14.
Theo em 4.119 (M. Spi zweck):Le kbe a pe ec ield. Le Nkbe he Adams-g aded
commu a i e di e en ial g aded algeb a o algeb aic cycles wi h espec o k. The de i ed
ca ego y D
Nko ini e Adams-g aded di e en ial g aded modules o e Nkis symme i-
cally monoidally equi alen o he iangula ed ca ego y o a ional geome ic mixed Ta e
mo i es o e Spec(k). I he base scheme Spec(k)sa is ies he Beilinson-Soule conjec-
u e hen he e exis canonical -s uc u es on bo h sides o he equi alence and an induced
equi alence o Tannakian ca ego ies
D
Nk≃DMTgm(Spec(k),Q),H
Nk≃MTgm(Spec(k),Q).
P oo . See [45, Theo em 3.31].
Co olla y 4.120: I kis a numbe ield hen he e exis he ollowing symme ic monoidal
equi alences o iangula ed ca ego ies along wi h canonical -s uc u es on he iangu-
la ed ca ego ies o he equi alences and an induced equi alence o Tannakian ca ego ies
DMTgm(Spec(k),Q)≃Ho(ModAk(D⊗(Ab)Z)c
Q)≃D
Nk.
P oo . This ollows om Theo em 4.119 and Co olla y 4.77 and Co olla y 2.14.
Theo em 4.121 (M. P eisinge ):Le kbe a numbe ield. The iangula ed homo opy
ca ego y D
Gal(ksep/k)o Gal(ksep/k)-equi a ian ini e cell modules wi h espec o kis
symme ically monoidally equi alen o he iangula ed ca ego y o compac a ional
4.3 G aded No m Func o s in S able Equi a ian Homo opy Theo y 149
mixed A in-Ta e mo i es o e Spec(k). Mo eo e he e exis canonical -s uc u es on
bo h sides o he equi alence along wi h an induced equi alence o Tannakian ca ego ies
D
Gal(ksep/k)≃DMAT(Spec(k),Q)c,H
Gal(ksep/k)≃(DMAT(Spec(k),Q)c)♡.
P oo . See [45, Theo em 4.41].
Co olla y 4.122: I kis a numbe ield hen he e exis he ollowing symme ic monoidal
equi alences o iangula ed ca ego ies along wi h canonical -s uc u es on he iangu-
la ed ca ego ies o he equi alences and induced equi alences o Tannakian ca ego ies
DMAT(Spec(k),Q)c≃Ho(ModAk(Sp⊗(BGal(ksep/k))⊗D⊗(Ab)Z)c
Q)≃D
Gal(ksep/k).
P oo . This ollows om Theo em 4.121 and Theo em 4.73 and Co olla y 2.14.
Theo em 4.123 (J. Scholbach):Le kbe a numbe ield and le S⊂Ok. The iangu-
la ed ca ego y DMAT(S, Q)cadmi s a non-degene a e -s uc u e. In pa icula , he e
exis s a symme ic monoidal equi alence be ween he iangula ed ca ego y o compac a-
ional mixed A in-Ta e mo i es DMAT(S, Q)co e Sand he bounded de i ed ca ego y
o he hea o his -s uc u e.
P oo . See [50, Theo em 0.1].
4.3 G aded No m Func o s in S able Equi a ian Homo opy Theo y
Rema k 4.124: The ollowing Sec ion 4.3 is a sine qua non o Sec ion 4.4. We con-
s uc he unc o Sp⊗(−)(−), see also P oposi ion 4.146, whe eupon we cons uc he
na u al ans o ma ion cΓ, see Co olla y 4.160, ia Rema k 4.154 and Rema k 4.155.
The cons uc ion o he unc o Sp⊗(−)(−)is es ablished by way o he Rema k 4.136.
Rema k 4.125: The ollowing De ini ion 4.126 and De ini ion 4.127 a e he s a ing
poin o he cons uc ion o he unc o Sp⊗(−)(−), see also P oposi ion 4.146. The
unc o is essen ial o ou second main esul , see Theo em 4.177. The las one sha pens
ou main esul , see Theo em 4.73.
De ini ion 4.126: Le Z∈Ob(P o(FinG pd)). Le Γbe a disc e e commu a i e monoid.
Le also ΓZ-FinSe be he ollowing ca ego y:
(1) Le (X, γ)∈Ob(ΓZ-FinSe )be X∈Ob(Z-FinSe )along wi h a locally cons an
unc ion on he ollowing p o ini e se
Z-FinSe ≃FC/Z, X ∈Ob(FC/Z), π0(X)∈Ob(P o(FinSe )) , γ ∶π0(X)→Γ.

4.3 G aded No m Func o s in S able Equi a ian Homo opy Theo y 150
(2) Le ( ∶(X, γ)→(Y, δ)) ∈Mo (ΓZ-FinSe )be ( ∶X→Y)∈Mo (Z-FinSe )such
ha he locally cons an unc ions γand δcoincide on X.
Re e ence. See also De ini ion 3.87.
De ini ion 4.127: Le Z∈Ob(P o(FinG pd)). Le Γbe a disc e e commu a i e monoid.
Le also ΓZ-FinSe +be he ollowing ca ego y:
(1) Le (X, γ)∈Ob(ΓZ-FinSe +)be X∈Ob(Z-FinSe )along wi h a locally cons an
unc ion on he ollowing p o ini e se
Z-FinSe ≃FC/Z, X ∈Ob(FC/Z), π0(X)∈Ob(P o(FinSe )) , γ ∶π0(X)→Γ.
(2) Le ( ∶(X, γ)→(Y, δ)) ∈Mo (ΓZ-FinSe +)be ( ∶X+→Y+)∈Mo (Z-FinSe +)
such ha he locally cons an unc ions γand δcoincide on −1(Y)⊂X.
Re e ence. C . De ini ion 3.210.
Rema k 4.128: The ollowing De ini ion 4.129 is used in Rema k 4.134, which i sel
is used in Rema k 4.136, whe e he unc o Sp⊗(−)(−)is cons uc ed, see also P oposi-
ion 4.146. The unc o Sp⊗(−)(−)is essen ial o P oposi ion 4.170 and Theo em 4.177.
De ini ion 4.129: Le Z∈Ob(P o(FinG pd)). Le Γbe a disc e e commu a i e monoid.
(1) Le (ΓZ,+)be he commu a i e monoid o locally cons an unc ions and le
γ∶π0(Z)→Γ,( ∶X→Y)∈Mo (Z-FinSe ), ∗∶ΓY→ΓX, γ ↦γ○ .
(2) Le p∶X→Zbe a ini e co e ing o p o ini e g oupoids and γlocally cons an and
γ∶π0(X)→Γ, p∗(γ)∶π0(Z)→Γ, p∗(γ)(z)∶=∑
x∈p−1(z)
γ(x).
Re e ence. C . De ini ion 3.212.
Rema k 4.130: The ollowing De ini ion 4.131 is used in Rema k 4.134, which is hen
used in Rema k 4.136 o he cons uc ion o hen unc o in P oposi ion 4.146, i.e.
Sp⊗(−)(−)∶CAlg(Se )×Co (P o(FinG pd),all, co )→CAlg(Ca si
∞),
(Γ, X)↦Sp⊗(X)Γ.
4.3 G aded No m Func o s in S able Equi a ian Homo opy Theo y 151
De ini ion 4.131 (Weil Res ic ion):Le p∶X→Zbe a ini e co e ing mo phism o
p o ini e g oupoids. Le Y∈Ob(FC/X)and le hY∈Ob(PSh(FC/X,Spc))be he p eshea
ep esen ed by Y. Le p∗∶PSh(FC/X,Spc)→PSh(FC/Z,Spc). I he p eshea p∗(hY)
is ep esen able by a p o ini e g oupoid o e Z hen his p o ini e g oupoid is called he
Weil es ic ion Rp(Y)o Yalong p.
Re e ence. C . De ini ion 3.30.
Rema k 4.132: Le p∶X→Zbe a ini e co e ing mo phism in he ca ego y o p o ini e
g oupoids and le also Y∈Ob(FC/X). The Weil es ic ion Rp(Y)o Yalong pexis s.
This ollows om he exis ence o he ollowing adjunc ion, see P oposi ion 3.89, whe e
p∗∶X-FinSe ⇆Y-FinSe ∶p∗.
Rema k 4.133 (No a ion):Le p∶X→Zbe a ini e co e ing mo phism o p o ini e
g oupoids. Le h∶U→Xbe a be a ini e co e ing mo phism o p o ini e g oupoids. Le
qand gbe he canonical p ojec ion mo phisms. Le ebe he couni o he adjunc ion.
Le =Rp(h). Le also he ollowing diag am in he ca ego y o p o ini e g oupoids be
U
h
↘↘
Rp(U)×ZX
e
←←
g
↓↓
q→→Rp(U)
↓↓
Xp→→Z
P oo . C . De ini ion 3.174.
Rema k 4.134: Le Z∈Ob(P o(FinG pd)). Le Γbe a disc e e commu a i e monoid.
(1) The e exis s he ollowing unc o
Co (P o(FinG pd),all, co )→CAlg(Se ), Z ↦ΓZ,(Y
←Xp
→Z)↦p∗○ ∗.
(2) The unc o abo e ex ends o he ollowing unc o
ΓFinSe ⊗
+(−)∶Co (P o(FinG pd),all, co )→CAlg(Ca 1),
Z↦ΓZ-FinSe +,(Y
←Xp
→Z)↦p⊗○ ∗,
∗(U, γ)∶=(U×YX, ∗
U(γ)) , p⊗(U, γ)∶=(Rp(U), q∗(e∗(γ))),
Ue
←Rp(U)×ZXq
→Rp(U).
4.3 G aded No m Func o s in S able Equi a ian Homo opy Theo y 152
(3) The alues o he unc o s ∗and p⊗on mo phisms as well as hei symme ic
monoidal s uc u es a e uniquely de e mined by he equi emen ha he ai h ul
unc o ΓZ-FinSe +→Z-FinSe +is na u al in Z∈Ob(Co (P o(FinG pd),all, co )).
P oo . C . Rema k 3.213.
Rema k 4.135: The ollowing Rema k 4.136 ou lines he cons uc ion o he unc-
o Sp⊗(−)(−), see also P oposi ion 4.146. In pa icula , in Rema k 4.141 we expose
he cons uc ion o he unc o H⊗
∗(−)Γ, see P oposi ion 4.143. Fu he mo e, in Re-
ma k 4.145 we expose he cons uc ion o he unc o Sp⊗(−)Γ, wi h espec o a disc e e
commu a i e monoid Γ.
Rema k 4.136: Le Z∈Ob(P o(FinG pd)). Le Γbe a disc e e commu a i e monoid.
(1) Fo e e y mo phism φ he ollowing unc o is na u al in Zwi h
φ∶Γ→Γ′,
ΓZ-FinSe +→Γ′Z-FinSe +.
(2) We ob ain he e o e he ollwing unc o
(−)FinSe ⊗
+(−)∶CAlg(Se )×Co (P o(FinG pd,all, co )→CAlg(Ca 1).
(3) By epea ing he cons uc ion o he unc o Sp⊗(−)we ob ain also he unc o
Sp⊗(−)(−)∶CAlg(Se )×Co (P o(FinG pd),all, co )→CAlg(Ca si
∞),
(Γ, X)↦Sp⊗(X)Γ.
P oo . C . Rema k 3.214.
Rema k 4.137: In he ollowing we expose he cons uc ion o he unc o Sp⊗(−)(−)
in he Rema k 4.136 (3), see also P oposi ion 4.146. The unc o is essen ial o P opo-
si ion 4.162 and P oposi ion 4.170 and also o he Theo em 4.177.
Rema k 4.138: Le Z∈Ob(P o(FinG pd)). Le Γbe a disc e e commu a i e monoid.
(1) The ca ego y ΓZ-FinSe +admi s all ini e cop oduc s. The ollowing canonical
unc o is he ini ial unc o ha p ese es ini e cop oduc s in i s second a gumen
and he e exis s an equi alence o ∞-ca ego ies
Γ×Z-FinSe +→ΓZ-FinSe +,
PShΣ(ΓZ-FinSe +,Spc)≃PShΣ∗(Z-FinSe ,Spc)Γ.
4.3 G aded No m Func o s in S able Equi a ian Homo opy Theo y 153
(2) Fo e e y mo phism o disc e e commu a i e monoids φ he unc o φ!is he le
Kan ex ension o he ollowing one
φ∶Γ→Γ′,
φ!∶PShΣ∗(Z-FinSe ,Spc)Γ→PShΣ∗(Z-FinSe ,Spc)Γ′,
ΓZ-FinSe +→Γ′Z-FinSe +,(X, γ)↦(X, φ ○γ).
(3) The commu a i e monoid s uc u e o Γinduces a symme ic monoidal s uc u e
on he ca ego y ΓZ-FinSe +. The Day con olu ion induces he e o e a symme ic
monoidal s uc u e on he ∞-ca ego y PShΣ∗(Z-FinSe ,Spc)Γ.
P oo . C . Rema k 3.211.
Rema k 4.139: The ollowing P oposi ion 4.140 expands P oposi ion 3.89, which i sel
elies on Rema k 3.84 and Rema k 3.86 and De ini ion 3.87 and P oposi ion 3.88. These
esul s a e oo ed on gene al conside a ions wi hin ∞-Ca ego y Theo y, as explained in
he o iginal sou ce o he e e ence in P oposi ion 3.89. By ollowing he a gumen a ion
o he o iginal sou ce in he con ex o De ini ion 4.126 we ob ain he P oposi ion 4.140.
P oposi ion 4.140: Le X, Y ∈Ob(P o(FinG pd)) be p o ini e g oupoids. Le Γbe a
disc e e commu a i e monoid.
(1) I p∶Y→Xis a ini ely p esen ed mo phism hen he e exis s an adjunc ion
p∗∶ΓX-FinSe ⇆ΓY-FinSe ∶p∗.
(2) I in addi ion pis a ini e co e ing hen he e is also he ollowing adjunc ion
p#∶ΓY-FinSe ⇆ΓX-FinSe ∶p∗.
P oo . C . P oposi ion 3.89.
Rema k 4.141 (Cons uc ion):Le Γbe a disc e e commu a i e monoid.
(1) The e exis s he ollowing unc o
ΓFinSe ⊗(−)∶Co (P o(FinG pd),all, co )→Ca lex
1,
X↦ΓX-FinSe ,(Y
←Xp
→Z)↦p∗○ ∗.