Coend Collisions and Mono Preservation in Day Convolution

Coend Collisions and Mono Preservation in Day Convolution Joaquim Reizi Higuchi The Open University of Japan December 17, 2025 Abstract Day convolution is a fundamental construction for inducing monoidal structures on functor categories, yet its interaction with basic categorical properties such as monomorphisms is subtle. In particular, Day convolution does not preserve monomorphisms in general, even in the category of set-valued presheaves. In this paper we give a precise structural explanation of this failure. We construct a small counterexample showing that mono preservation can break already in very simple monoidal settings, and we identify the unique mechanism responsible for this phenomenon. We prove that failure of mono preservation is completely characterized by the presence of coend collisions, i.e. nontrivial identifications forced by the coend quotient in Day convolution. Building on this characterization, we derive an explicit system of sufficient conditions ensuring mono preservation. These conditions are formulated in terms of flatness of the promonoidal weight and a reduction to representable copresheaves ensuring practical checkability. Together, our results clarify when and why Day convolution preserves monomorphisms, and provide a usable criterion applicable in concrete categorical settings. Keywords: Day convolution; coend; monomorphisms; presheaves; monoidal categories; tensor stability 1 Introduction Day convolution provides a systematic method for transporting a monoidal structure on a category Mto categories of functors Mop →Set and beyond. Since its introduction by Day [1], it has become a standard tool in category 1 theory, with applications ranging from enriched category theory [3] to geometric and logical contexts (e.g. sheaf semantics) [7]. Despite its ubiquity, the interaction of Day convolution with basic categorical properties remains delicate. One such property is the preservation of monomorphisms. In many applications, monomorphisms encode subobjects, inclusions, or injective structure maps, and their stability under monoidal operations is often tacitly assumed. However, Day convolution does not preserve monomorphisms in general. What is less clear from the existing literature is why this failure occurs, how small such failures can be, and under which explicit conditions mono preservation can be recovered. The present paper addresses these questions in a unified and structural way. Our starting point is a small explicit counterexample, showing that mono preservation can fail already for a one-object monoidal category. This example is deliberately simple, yet it captures the essential mechanism responsible for the failure. Rather than treating it as an isolated pathology, we use it as a guide to extract the underlying cause. Our first contribution is a structural characterization of mono failure in coend-based constructions. We introduce the notion of a coend collision, referring to a nontrivial identification forced by the coend quotient. We prove that such collisions are not merely sufficient but also necessary for mono failure: whenever Day convolution fails to preserve a monomorphism, the failure is witnessed by a collision in the underlying coend relation. Our second contribution is the derivation of explicit and checkable sufficient conditions for mono preservation. Using standard flatness technology for Set-valued functors (e.g. [6, 7]), we show that monomorphisms are preserved whenever the relevant promonoidal weights are flat, and we provide a reduction to representables ensuring practical checkability. The structure of the paper reflects this progression. After recalling the necessary preliminaries, we present the counterexample. We then develop the collision-based structural analysis and prove the characterization theorem. Finally, we formulate and prove sufficient conditions for mono preservation and briefly discuss their scope. By isolating the precise mechanism behind mono failure and by providing explicit criteria for its avoidance, this work clarifies an aspect of Day convolution that is often implicit but rarely analyzed in detail. We expect these results to be useful in contexts where tensor stability of subobjects plays a central role, such as monoidal localization and sheaf-theoretic settings. 2 2 Preliminaries 2.1 Coends and tensor products We use standard coend notation and basic properties as in [2]. Let Cbe a small category, P∈SetCop a presheaf and W∈SetCa copresheaf. The coend tensor product of Pand Wis the set P⊗CW:= Zc∈C P(c)×W(c). Concretely, it is the coequalizer in Set a u:c→d P(d)×W(c)⇒a c∈C P(c)×W(c)qP,W −−−→ P⊗CW, (2.1) where the parallel arrows are induced by (x, w)7→ (P(u)(x), w) and (x, w)7→ (x, W (u)(w)). Equivalently, P⊗CWis the quotient of the coproduct `cP(c)× W(c) by the equivalence relation generated by (P(u)(x), w)∼(x, W (u)(w)) (u:c→d, x ∈P(d), w ∈W(c)). (2.2) A monomorphism in SetCop is a natural transformation that is injective at each component. In particular, if ι:P ,→P′is monic and W∈SetC, there is an induced map ι⊗CW:P⊗CW−→ P′⊗CW obtained from the induced map on coproducts and the universal property of the coequalizer (2.1). 2.2 Day convolution as a coend tensor Let (M, ⊗, I) be a small monoidal category. The Set-valued Day convolution on SetMop is given by the coend (F ⋆ G)(c) := Za,b∈M M(c, a ⊗b)×F(a)×G(b),(2.3) see [1, 3]. For fixed c∈M, define a copresheaf Wc:M×M→Set, Wc(a, b) := M(c, a ⊗b). 3 Moreover, view Fand Gas a presheaf on M×Mvia (F⊠G)(a, b) := F(a)×G(b),(F⊠G)∈Set(M×M)op . Then (2.3) may be rewritten as a coend tensor product (F ⋆ G)(c)∼ =(F⊠G)⊗M×MWc.(2.4) Consequently, for any monomorphism ι:F ,→F′in SetMop and any G∈SetMop , the component map (ι⋆G)(c):(F ⋆ G)(c)−→ (F′⋆ G)(c) identifies with the coend-induced map (ι⊠idG)⊗M×MWc. Thus, questions about mono preservation for Day convolution reduce pointwise to mono preservation for coend tensor products. 3 A Small Counterexample We exhibit a small counterexample in the coend tensor product setting. As noted above, it captures the basic phenomenon responsible for mono failure in Day convolution, namely the appearance of new identifications in the coend quotient. 3.1 The indexing monoid Let T={1, α, β}be the monoid with unit 1 and multiplication determined by αt =α, βt =βfor all t∈T. Equivalently, αand βare left-zero elements. Let Cbe the one-object category associated to T. Then presheaves Cop →Set are right T-sets, and copresheaves C→Set are left T-sets. 3.2 Data Define a monomorphism ι:F ,→F′of right T-sets and a left T-set Was follows. •F={x0, x1}with the trivial right action: xi·t=xifor all t∈T. 4 •F′={x0, x1, y}, extending the trivial action on x0, x1and specifying y·1 = y, y ·α=x0, y ·β=x1. •ι:F→F′is the evident inclusion. •W={g}is the singleton left T-set with trivial action: t·g=gfor all t∈T. 3.3 Computation Proposition 3.1 (Failure of mono preservation).The induced map ι⊗CW:F⊗CW−→ F′⊗CW is not injective. Proof. Since Chas one object, the coend F⊗CWis the quotient of F×W by the relation generated by (x·t, g)∼(x, t ·g) (t∈T). Because the action on Fis trivial and the action on Wis trivial, this relation is trivial on F×W. Hence F⊗CW∼ =F×Whas two elements represented by (x0, g) and (x1, g). In F′⊗CW, the same relation applies but now involves the extra element y∈F′. In particular, (x0, g)=(y·α, g)∼(y, α·g)=(y, g),(x1, g)=(y·β, g)∼(y, β·g)=(y, g). Thus (x0, g) and (x1, g) represent the same element of F′⊗CW. Therefore ι⊗CWidentifies the two distinct elements of F⊗CWand is not injective. Proposition 3.2 (Minimality relative to submonoids).Let T′⊊Tbe a proper submonoid (so T′={1, α}or T′={1, β}), and view F ,→F′and Was T′-sets by restriction. Then the induced map F⊗C′W→F′⊗C′W is injective, where C′is the one-object category corresponding to T′. Proof. If T′={1, α}, then the only nontrivial relation in F′⊗C′Warises from αand identifies (y, g) with (y·α, g)=(x0, g), but there is no element of T′sending yto x1. Hence (x0, g) and (x1, g) remain distinct classes, so the induced map is injective. The case T′={1, β}is symmetric. Remark 3.3. In the language developed in Section 4, the proof of Theorem 3.1 exhibits a collision in the target coend which merges two distinct classes from the source. 5 4 Coend Collisions and Structural Failure Fix a small category C, a copresheaf W∈SetC, and a monomorphism ι:P ,→P′in SetCop . 4.1 Collisions Definition 4.1 (Relative coend collision).A (ι, W )-collision is a pair of elements (x, w)∈P(c)×W(c),(x′, w′)∈P(c′)×W(c′) such that qP,W (x, w)=qP,W (x′, w′) but qP′,W (ι(x), w) = qP′,W (ι(x′), w′). The key point is that collisions are measured relative to the inclusion ι: two source classes are distinct, but become identified after passing to the larger presheaf P′. 4.2 Zig–zags Lemma 4.2 (Zig–zag normal form).Two representatives in the coproduct `cP′(c)×W(c)have the same image in P′⊗CWif and only if they can be connected by a finite zig–zag of the generating relations (2.2) for (P′, W ). Proof. The coend quotient is obtained by taking the equivalence relation generated by (2.2). Thus equality in the quotient is equivalent to membership in the generated equivalence relation, which is witnessed by a finite composition of generating steps. 4.3 Structural characterization Theorem 4.3 (Collisions are the unique cause of mono failure).The induced map ι⊗CW:P⊗CW→P′⊗CWfails to be injective if and only if an (ι, W )-collision exists. Proof. (⇒) If ι⊗CWis not injective, there exist distinct elements α= β∈P⊗CWwith (ι⊗CW)(α)=(ι⊗CW)(β). Choose representatives (x, w) and (x′, w′) in the coproduct for P⊗CWwith qP,W (x, w) = αand qP,W (x′, w′) = β. Then qP,W (x, w)=qP,W (x′, w′), while equality of images implies qP′,W (ι(x), w) = qP′,W (ι(x′), w′), 6 so (x, w) and (x′, w′) form an (ι, W )-collision. (⇐) Conversely, if an (ι, W )-collision exists, then two distinct elements of P⊗CWhave the same image in P′⊗CW, hence ι⊗CWis not injective. Remark 4.4. Theorem 4.3 isolates the coend quotient as the unique mechanism by which mono preservation can fail: distinct source classes are merged only when the target coend relation identifies their images. In particular, any proof of mono preservation reduces to ruling out such collisions. 5 Sufficient Conditions for Mono Preservation We now give explicit conditions on Wensuring that − ⊗CWpreserves monomorphisms. 5.1 Representable weights Let y(c) = C(c, −)∈SetCbe the covariant representable. By the co-Yoneda lemma (see [2]), there is a natural isomorphism P⊗Cy(c)∼ =P(c). Proposition 5.1 (Representables are collision-free).For any c∈C, the functor − ⊗Cy(c) : SetCop →Set preserves monomorphisms (indeed, it is evaluation at c). 5.2 Flatness and filtered colimits We use the standard notion of flat Set-valued functors; see [6, 7]. Definition 5.2 (Flat copresheaf).A copresheaf W∈SetCis flat if its category of elements el(W) is filtered. Equivalently, Wis a filtered colimit of representables. Lemma 5.3 (Tensoring with a filtered colimit of representables).Suppose W∼ =colimi∈Iy(ci)in SetC. Then for every P∈SetCop there is a natural isomorphism P⊗CW∼ =colimi∈IP(ci). Proof. Since coends are colimits in each variable and Set is cocomplete, we may commute the coend with the colimit: P⊗CW∼ =Zc P(c)×colimiy(ci)(c)∼ =colimiZc P(c)×y(ci)(c). 7 By the co-Yoneda lemma, RcP(c)×y(ci)(c)∼ =P(ci), which gives the desired isomorphism. Lemma 5.4 (Filtered colimits preserve injections).Let (fi:Ai→Bi)i∈I be a diagram of injective maps in Set indexed by a filtered category I. Then colimIfi: colimIAi→colimIBiis injective. Proof. This is standard: filtered colimits in Set are exact, hence preserve monomorphisms; see e.g. [6]. Theorem 5.5 (Flatness implies mono preservation).If Wis flat, then the functor − ⊗CW:SetCop →Set preserves monomorphisms. Equivalently, for every monomorphism ι:P ,→P′, the induced map ι⊗CWis injective. Proof. By Theorem 5.2, write W∼ =colimi∈Iy(ci) with Ifiltered. For any monomorphism ι:P ,→P′, the component maps ιci:P(ci)→P′(ci) are injective. Using Theorem 5.3, the induced map ι⊗CWidentifies with colimi∈Iιci: colimiP(ci)→colimiP′(ci), which is injective by Theorem 5.4. 5.3 Consequences for Day convolution Corollary 5.6 (A checkable sufficient condition for Day convolution).Let (M, ⊗, I)be a small monoidal category. If for every c∈Mthe copresheaf Wc:M×M→Set, Wc(a, b) = M(c, a ⊗b) is flat (as a functor M×M→Set), then for every G∈SetMop the Day convolution functor −⋆ G :SetMop →SetMop preserves monomorphisms. Proof. Fix c∈M. By Equation (2.4), (F ⋆ G)(c)∼ =(F⊠G)⊗M×MWc. If F ,→F′is monic in SetMop , then F⊠G ,→F′⊠Gis monic in Set(M×M)op (pointwise product preserves injections). Flatness of Wcand Theorem 5.5 yield injectivity of (ι⋆G)(c) for each c, hence ι⋆Gis monic. Remark 5.7. The counterexample in Section 3 can be interpreted as a failure of flatness: the category of elements of the singleton T-set Wis not filtered (it contains parallel arrows induced by αand βwith no cocone identifying them). This obstruction is exactly what enables collisions in the target coend. 8 6 Applications and Outlook The results above isolate coend collisions as the unique source of nonpreservation of monomorphisms in coend-based constructions, including Day convolution. From an applied categorical perspective, this clarifies when tensor stability of subobjects can be expected in functorial monoidal settings. Two immediate directions are as follows. First, in contexts where monoidal localization is performed, one often requires stability of subobjects under the induced tensor product. Collision-based diagnostics provide a way to pinpoint where this stability breaks. Second, in sheaf-theoretic situations, convolution-like constructions interact with exactness and mono/epi stability; extending the present analysis to exactness or regular monos is a natural next step. References [1] B. Day. On closed categories of functors. In Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics 137. Springer, 1970. [2] S. Mac Lane. Categories for the Working Mathematician. Springer, 2nd edition, 1998. [3] G. M. Kelly. Basic Concepts of Enriched Category Theory. Cambridge University Press, 1982. [4] B. Day and R. Street. Monoidal bicategories and Hopf algebroids. Advances in Mathematics, 129(1):99–157, 1997. [5] P. Freyd. Abelian Categories. Harper & Row, 1964. [6] J. Ad´amek and J. Rosick´y. Locally Presentable and Accessible Categories. Cambridge University Press, 1994. [7] I. Moerdijk and S. Mac Lane. Sheaves in Geometry and Logic. Springer, 1992. 9