A Complete Classification of Mono Preservation under Day Convolution

A Complete Classification of Mono Preservation under Day Convolution Joaquim Reizi Higuchi The Open University of Japan December 17, 2025 Abstract Day convolution transports a monoidal structure to presheaf categories and is widely used in applied categorical settings. However, it does not preserve monomorphisms in general, even for set-valued presheaves. A recent analysis identified coend collisions as the unique mechanism responsible for this phenomenon. In this preprint we complete the resulting stability theory for Day convolution by giving an if and only if classification of mono preservation in terms of collision– free (mono-stable) promonoidal weights, and by providing checkable sufficient criteria. These criteria strictly extend flatness (for example, finite coproducts of representables are mono-stable but typically not flat), yielding a practical checklist for applications. Keywords. Day convolution; coend; monomorphisms; promonoidal structures; flatness. MSC 2020. 18D10; 18A25; 18F20. Conflict of Interest. The author declares no conflict of interest. 1 Introduction Day convolution is a standard construction for inducing monoidal structures on functor categories. Given a small monoidal category ( M,⊗, I ), Day’s formula equips the presheaf category SetMop with a monoidal product (F ⋆ G)(c) := Za,b∈M M(c, a ⊗b)×F(a)×G(b),(1.1) 1 see [ 2 , 3 ]. In many applications, monomorphisms encode subobjects or inclusions and one informally expects tensoring to preserve subobjects. Nevertheless, Day convolution does not preserve monomorphisms in general. In [ 1 ] we gave a precise structural explanation of this failure. The key observation is that each component of (1.1) is a coend quotient, and mono failure occurs exactly when that quotient forces new identifications. We formalized this as the existence of coend collisions and proved that collisions are the unique cause of mono failure for coend tensor products. The present preprint develops a stability theory specialized to Day convolution. Our main goals are: (i) to package the collision criterion into an intrinsic mono-stability property of the promonoidal weights Wc(a, b) = M(c, a ⊗b), (ii) to exhibit collision-free classes beyond flatness (hence sharpening the standard “flatness implies stability” principle), and (iii) to provide a concise checklist that can be applied in concrete monoidal settings. Roadmap. Section 2 reviews coends and Day convolution in the form needed for the collision analysis. Section 3 introduces mono-stable weights and collects checkable sufficient conditions, including a family of non-flat mono-stable examples. Section 4 states and proves the main collision criterion for Day convolution. Section 5 summarizes the results as a practical checklist, and Section 6 indicates applications and directions. 2 Preliminaries and conventions Throughout, C denotes a small category. A monomorphism in SetCop means a natural transformation that is injective at each component. 2.1 Coends as tensor products Let P∈SetCop be a presheaf and W∈SetC a copresheaf. Their coend tensor product is P⊗CW:= Zc∈C P(c)×W(c).(2.1) 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, (2.2) 2 where the parallel arrows are induced by ( x, w ) 7→ ( P ( u )( x ) , w )and ( x, w ) 7→ ( x, W ( u )( w )). Equivalently, P⊗CW is the quotient of `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)).(2.3) Given a monomorphism ι : P ,→P′ in SetCop and W∈SetC , there is an induced map ι⊗CW:P⊗CW−→ P′⊗CW. (2.4) 2.2 Day convolution as a coend tensor Let ( M,⊗, I )be a small monoidal category. For c∈ M , define a copresheaf on M×M Wc(a, b) := M(c, a ⊗b).(2.5) If F, G ∈SetMop , view their external product as a presheaf F⊠G∈ Set(M×M)op by ( F⊠G )( a, b ) := F ( a ) ×G ( b ). Then Day’s formula (1.1) rewrites pointwise as (F ⋆ G)(c)∼ =(F⊠G)⊗M×M Wc.(2.6) Consequently, for a monomorphism ι : F ,→F′ in SetMop the component map (ι⋆G)(c) : (F ⋆ G)(c)−→ (F′⋆ G)(c)(2.7) identifies with (ι⊠idG)⊗M×M Wc.(2.8) Thus mono preservation for Day convolution reduces pointwise to mono preservation for coend tensor products. 3 Mono-stable weights and collision-free classes The results of [ 1 ] show that mono failure for coend tensor products is completely characterized by the presence of coend collisions. We package this into an intrinsic stability property of the weight. 3 3.1 Relative collisions and mono-stability Definition 3.1 (Relative collision).Let C be a small category, W∈SetC , and ι : P ,→P′ a monomorphism in SetCop . An ( ι, W )-collision is a pair of elements (x, w)∈P(c)×W(c),(x′, w′)∈P(c′)×W(c′) with qP,W (x, w)=qP,W (x′, w′)but qP′,W (ι(x), w) = qP′,W (ι(x′), w′). Definition 3.2 (Mono-stable (collision-free) weight).A copresheaf W∈ SetC is mono-stable if − ⊗CW : SetCop →Set preserves monomorphisms; equivalently, if for every monomorphism ι : P ,→P′ the induced map ι⊗CW is injective. Remark 3.3. By [ 1 , Theorem 4.3], ι⊗CW fails to be injective if and only if an ( ι, W )-collision exists. Thus W is mono-stable if and only if it admits no collisions against any monomorphism. 3.2 Representables and finite coproducts For c∈ C let y( c ) = C ( c, − ) ∈SetC denote the covariant representable. By the co-Yoneda lemma [4], there is a natural isomorphism P⊗Cy(c)∼ =P(c). Proposition 3.4 (Representables are mono-stable).For every c∈ C , the weight y( c )is mono-stable. More generally, any finite coproduct of representables `n i=1 y(ci)is mono-stable. Proof. Evaluation P7→ P ( c )preserves monomorphisms, hence y( c )is monostable. For a finite coproduct W = `n i=1 y( ci ), coends preserve coproducts in the weight variable, so P⊗CW∼ =`n i=1 P ( ci )naturally in P . A finite coproduct of injective maps in Set is injective. Remark 3.5 (Non-flat but mono-stable).If n≥ 2, the category of elements of `n i=1 y( ci )is a nontrivial discrete category and hence is not filtered. Therefore `n i=1 y( ci )is typically not flat. This already shows that mono-stability is strictly weaker than flatness. 4 3.3 Flatness (sufficient, not necessary) We recall the standard notion of flat Set-valued functors. Definition 3.6 (Flat weight).A copresheaf W∈SetC is flat if its category of elements el ( W )is filtered. Equivalently, W is a filtered colimit of representables. Theorem 3.7 (Flatness implies mono-stability).If W∈SetC is flat, then Wis mono-stable. Proof. This is proved in [1, Theorem 5.5]. Corollary 3.8 (Flatness is sufficient but not necessary).Mono-stability is strictly weaker than flatness. Proof. By Proposition 3.4,y(c1)`y(c2)is mono-stable. As noted above, it is not flat in general. 3.4 Day convolution: collision-free weights yield mono preservation We now translate mono-stability of the weights Wc into mono preservation for Day convolution. Proposition 3.9 (Mono-stable weights imply mono preservation).Let ( M,⊗, I )be a small monoidal category. Assume that for each c∈ M the weight Wc of (2.5) is mono-stable as a functor M × M → Set . Then for every G∈SetMop the functor −⋆ G : SetMop →SetMop preserves monomorphisms. Proof. Let ι : F ,→F′ be monic in SetMop . Then ι⊠idG : F⊠G ,→F′⊠G is monic in Set(M×M)op . For each c∈ M , by (2.8) the component map ( ι ⋆ G )( c )identifies with ( ι⊠idG ) ⊗M×M Wc . Since Wc is mono-stable, this map is injective. Thus ι⋆Gis pointwise injective and hence monic. 4 Main collision criterion for Day convolution We now state the collision criterion for mono failure under Day convolution. Theorem 4.1 (Collision criterion for Day convolution).Let ( M,⊗, I )be a small monoidal category. Let ι : F ,→F′ be a monomorphism in SetMop 5 and let G∈SetMop . For c∈ M , the component map ( ι⋆G )( c )fails to be injective if and only if there exists an (ι⊠idG, Wc)-collision in the coend (F⊠G)⊗M×M Wc. Equivalently, ι⋆G is monic if and only if no such collision exists for any c∈ M. Proof. Fix c∈ M . By (2.6) and (2.8) ,( ι ⋆ G )( c )identifies with ( ι⊠ idG ) ⊗M×M Wc . By [ 1 , Theorem 4.3], this induced map fails to be injective if and only if an ( ι⊠idG, Wc )-collision exists. The final statement follows by pointwise reasoning. Corollary 4.2 (A global criterion).If every weight Wc is mono-stable, then Day convolution preserves monomorphisms in the first argument: for all G , the functor −⋆ G preserves monomorphisms. Conversely, if for some c the weight Wc admits a collision, then there exist F ,→F′ and G such that Day convolution fails to preserve that monomorphism at component c. Proof. The forward direction is Proposition 3.9. For the converse, apply Theorem 4.1 to any witnessed collision. 5 Checklist and practical criteria We summarize the preceding results as a concrete checklist. Step 1. Compute the weights. For each c∈ M , compute Wc ( a, b ) = M(c, a ⊗b). Step 2. Choose a mono-stability certificate. To prove mono preservation of −⋆ G for all G , it suffices to prove that each Wc is mono-stable. Any of the following certificates works: (a) Wcis flat (Theorem 3.7); (b) Wcis a finite coproduct of representables (Proposition 3.4); (c) Wc is built from mono-stable weights via finite coproducts and filtered colimits. Step 3. If in doubt, look for collisions. If mono preservation fails, Theorem 4.1 guarantees the existence of an explicit collision in the coend defining ( F ⋆ G )( c ). This provides a diagnostic tool: identify the zig– zag in the coend relation responsible for the unintended identification. 6 6 Applications and outlook The collision viewpoint clarifies when “tensor stability of subobjects” can be expected in functorial monoidal settings. • Monoidal localization. In monoidal localizations one often requires stability of subobjects under the induced tensor product. The collision criterion pinpoints precisely where such stability breaks. • Sheaf-theoretic settings. In sheaf semantics, convolution-like constructions interact with exactness and mono/epi stability. Collision-free weights provide a clean sufficient hypothesis for mono stability in these contexts. • Beyond monomorphisms. An important direction is to extend the collision analysis to exactness and to classes of regular monomorphisms. A Boundary examples This appendix records a basic family of mono-stable but non-flat weights. Example A.1 (Coproduct of representables).Let C be any small category and let c1, c2∈ C . Then W := y( c1 ) ` y( c2 )is mono-stable by Proposition 3.4. If c1∼ =c2 , the category of elements el ( W )is a discrete category with two objects, hence not filtered. Therefore W is not flat, yet tensoring with W preserves monomorphisms. Remark A.2. This example shows that flatness is not the “minimal” stability condition. It is nevertheless valuable because it interacts well with other exactness properties. References [1] J. R. Higuchi, Coend Collisions and Mono Preservation in Day Convolution, Zenodo, 2025. https://doi.org/10.5281/zenodo.17964488 [2] B. Day, On closed categories of functors, in Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics 137, Springer, 1970. [3] G. M. Kelly, Basic Concepts of Enriched Category Theory, Cambridge University Press, 1982. 7 [4] S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer, 1998. 8