Day convolution


Monoidal categories

Category theory



The category of presheaves over the opposite of a monoidal category canonically inherits itself a monoidal category structure via a categorified convolution product. This holds generally in the context of enriched category theory. This was first observed by (Day 70) and accordingly these monoidal structures are called Day convolutions products.

More in detail, just as there is convolution of functions f:Gβ†’β„‚f : G \to \mathbb{C} whenever GG carries the structure of a group, or more generally just the structure of a monoid, so there is convolution of functors f:𝒒→Setf \colon \mathcal{G} \to Set whenever the category 𝒒\mathcal{G} carries the structure of a monoidal category.

This may be generalized by replacing Set with a more general cocomplete symmetric monoidal category VV. The technical condition is that the tensor product uβŠ—vu \otimes v preserves colimits in its two arguments separately; hence that the functors uβŠ—βˆ’u \otimes - and βˆ’βŠ—v- \otimes v preserve colimits. This occurs notably when VV is symmetric closed monoidal (so that these functors are left adjoints).


For monoidal categories

For (C,βŠ—)(C, \otimes) a small monoidal category and for F,G:C opβ†’SetF, G \colon C^{op} \to Set two presheaves on CC, their Day convolution product F⋆GF \star G is the presheaf given by the coend

F⋆Gβ‰”βˆ« c,d∈CF(c)Γ—G(d)Γ—Hom C(βˆ’,cβŠ—d). F \star G \coloneqq \int^{c,d \in C} F(c) \times G(d) \times Hom_C(-, c \otimes d) \,.

More generally:

Let VV be a good symmetric monoidal category for VV-enriched category theory (in particular having all small colimits, e.g. a Benabou cosmos). For π’ž\mathcal{C} a VV-enriched category, write [π’ž,V][\mathcal{C},V] for the VV-enriched functor category to VV, etc.


For (π’ž,βŠ—,I)(\mathcal{C}, \otimes, I) a small monoidal VV-enriched category, the Day convolution product on [π’ž,V][\mathcal{C},V] is the functor

βŠ— Day:[π’ž,V]Γ—[π’ž,V]⟢[π’ž,V] \otimes_{Day} \;\colon\; [\mathcal{C},V] \times [\mathcal{C},V] \longrightarrow [\mathcal{C},V]

given by the coend

XβŠ— DayY:cβ†¦βˆ« c 1,c 2βˆˆπ’žX(c 1)βŠ—Y(c 2)βŠ—[c 1βŠ—c 2,c]. X \otimes_{Day} Y \;\colon\; c \mapsto \int^{c_1, c_2 \in \mathcal{C}} X(c_1) \otimes Y(c_2) \otimes [c_1 \otimes c_2, c] \,.

We may think of this equivalently as a Kan extension:


For (π’ž,βŠ—)(\mathcal{C}, \otimes) a monoidal VV-enriched category, its external tensor product is the VV-functor

βŠ—Λœβ‰”:[π’ž,V]Γ—[π’ž,V]⟢[π’žΓ—π’ž,V] \tilde \otimes \coloneqq \;\colon\; [\mathcal{C},V] \times [\mathcal{C},V] \longrightarrow [\mathcal{C}\times \mathcal{C}, V]

given by

XβŠ—ΛœYβ‰”βŠ—βˆ˜(X,Y). X \tilde \otimes Y \coloneqq \otimes \circ (X,Y) \,.

The Day convolution product, def. 1, is equivalently given by left Kan extension Lan βŠ—Lan_\otimes (along the tensor product βŠ—\otimes) of the result of the external tensor product, def. 2: there is a natural isomorphism

XβŠ— DayY≃Lan βŠ—(XβŠ—ΛœY). X \otimes_{Day} Y \simeq Lan_\otimes (X \tilde \otimes Y) \,.

This perspective is highlighted in (MMSS 00, p. 60).


The general formula for pointwise Kan extension via coends (here) says that left Kan extension of any F:π’Ÿβ†’VF \colon \mathcal{D} \to V along some p:π’Ÿβ†’β„°p \colon \mathcal{D} \to \mathcal{E} is given by

Lan pF:eβ†¦βˆ« dβˆˆπ’Ÿβ„°(p(d),e)βŠ—F(d). Lan_p F \;\colon\; e \mapsto \int^{d\in \mathcal{D}} \mathcal{E}(p(d), e) \otimes F(d) \,.

In our case

  • π’Ÿ=π’žΓ—π’ž\mathcal{D} = \mathcal{C}\times \mathcal{C};

  • β„°=π’ž\mathcal{E} = \mathcal{C};

  • p=βŠ—p = \otimes;

  • F=XβŠ—ΛœXF = X \tilde \otimes X

and hence the general formula here becomes

(Lan βŠ—XβŠ—ΛœY)(c) β‰ƒβˆ« (c 1,c 2)βˆˆπ’žΓ—π’žπ’ž(c 1βŠ—c 2,c)βŠ—((XβŠ—ΛœY)(c 1,c 2)) β‰ƒβˆ« (c 1,c 2)βˆˆπ’žΓ—π’žπ’ž(c 1βŠ—c 2,c)βŠ—(X(c 1)βŠ—Y(c 2)). \begin{aligned} (Lan_{\otimes} X \tilde \otimes Y)(c) & \simeq \int^{(c_1,c_2) \in \mathcal{C}\times \mathcal{C}} \mathcal{C}( c_1 \otimes c_2, c ) \otimes ( (X \tilde \otimes Y)(c_1,c_2) ) \\ & \simeq \int^{(c_1,c_2) \in \mathcal{C}\times \mathcal{C}} \mathcal{C}( c_1 \otimes c_2, c ) \otimes ( X(c_1) \otimes Y(c_2) ) \end{aligned} \,.

Day convolution βŠ— Day\otimes_{Day}, def. 1, is universally characterized by the property that there are natural isomorphisms

[π’ž,V](XβŠ— DayY,Z)≃[π’žΓ—π’ž,V](XβŠ—ΛœY,Zβˆ˜βŠ—), [\mathcal{C},V](X \otimes_{Day} Y, Z) \simeq [\mathcal{C}\times \mathcal{C},V](X \tilde{\otimes} Y, Z \circ \otimes) \,,

where βŠ—Λœ\tilde \otimes is the external product of def. 2.


By prop. 1 and since left Kan extension along any ff is the left adjoint to precomposition with ff.


y:π’ž op⟢[π’ž,V] y \colon \mathcal{C}^{op} \longrightarrow [\mathcal{C}, V]

for the VV-Yoneda embedding, so that for cβˆˆπ’žc\in \mathcal{C} any object, y(c)y(c) is the corepresented functor y(c):c′↦[c,cβ€²]y(c)\colon c' \mapsto [c,c'].


For (π’ž,βŠ—,I)(\mathcal{C}, \otimes, I) a small monoidal VV-enriched category, the Day convolution product βŠ— Day\otimes_{Day} of def. 1 makes

([π’ž,V],βŠ—,y(I)) ( [\mathcal{C}, V], \otimes, y(I))

a closed monoidal category with tensor unit y(I)y(I) co-represented by the tensor unit II of π’ž\mathcal{C}.


To see that y(I)y(I) is the tensor unit for βŠ— Day\otimes_{Day}, use the Fubini theorem for coends and then twice the co-Yoneda lemma to get for any X∈[π’ž,V]X \in [\mathcal{C},V] that

XβŠ— Dayy(I) =∫ c 1,c 2βˆˆπ’žX(c 1)βŠ—[I,c 2]βŠ—[c 1βŠ—c 2,βˆ’] β‰ƒβˆ« c 1βˆˆπ’žX(c 1)βŠ—βˆ« c 2βˆˆπ’ž[I,c 2]βŠ—[c 1βŠ—c 2,βˆ’] β‰ƒβˆ« c 1βˆˆπ’žX(c 1)βŠ—[c 1βŠ—I,βˆ’] β‰ƒβˆ« c 1βˆˆπ’žX(c 1)βŠ—[c 1,βˆ’] ≃X(βˆ’) ≃X. \begin{aligned} X \otimes_{Day} y(I) & = \int^{c_1,c_2 \in \mathcal{C}} X(c_1) \otimes [I,c_2] \otimes [c_1\otimes c_2,-] \\ & \simeq \int^{c_1\in \mathcal{C}} X(c_1) \otimes \int^{c_2 \in \mathcal{C}} [I,c_2] \otimes [c_1\otimes c_2,-] \\ & \simeq \int^{c_1\in \mathcal{C}} X(c_1) \otimes [c_1 \otimes I, -] \\ & \simeq \int^{c_1\in \mathcal{C}} X(c_1) \otimes [c_1, -] \\ & \simeq X(-) \\ & \simeq X \end{aligned} \,.

For promonoidal categories

In the original article (Day 70), a stronger form of the convolution is discussed, in which AA is assumed only to be a promonoidal category.

Let VV be a Benabou cosmos, and AA a small VV-enriched category.


There is an equivalence of categories between the category of pro-monoidal structures on AA with strong pro-monoidal functors between them and the category of biclosed monoidal structures on V A opV^{A^{op}} with strong monoidal functors between them.

This is claimed without proof in (Day 70).


Basic properties

Let j:C→PSh(C)j \colon C \to PSh(C) be the Yoneda embedding.


With I∈CI \in C the tensor unit of CC, then the presheaf j(I)j(I) that it represents is a tensor unit for the Day convolution product.


Using the co-Yoneda lemma on the two coends we have

F⋆j(I) β‰ƒβˆ« c,d∈CF(c)Γ—Hom C(d,I)Γ—Hom C(βˆ’,cβŠ—d) β‰ƒβˆ« c∈CF(c)Γ—Hom C(βˆ’,cβŠ—I) β‰ƒβˆ« c∈CF(c)Γ—Hom C(βˆ’,c) ≃F(βˆ’). \begin{aligned} F \star j(I) & \simeq \int^{c,d \in C} F(c) \times Hom_C(d,I) \times Hom_C(-, c\otimes d) \\ & \simeq \int^{c \in C} F(c) \times Hom_C(-, c \otimes I) \\ & \simeq \int^{c \in C} F(c) \times Hom_C(-, c) \\ & \simeq F(-) \end{aligned} \,.

For CC a small monoidal category, regard the category of presheaves (PSh(C),⋆,j(I))(PSh(C), \star, j(I)) as a monoidal category with tensor product the Day convolution product and unit the unit of CC under the Yoneda embedding j:Cβ†ͺPSh(C)j : C \hookrightarrow PSh(C).


  1. (PSh(C),⋆,j(I))(PSh(C), \star, j(I)) is a closed monoidal category;

  2. the Yoneda embedding constitutes a strong monoidal functor (C,βŠ—,I)β†ͺ(PSh(C),⋆,j(I))(C,\otimes, I) \hookrightarrow (PSh(C), \star, j(I)).


In analogy to the cartesian closed monoidal structure on presheaves we see that if the internal hom in PSh(C)PSh(C) exists at all, (with [F,βˆ’][F,-] right adjoint to (βˆ’)⋆F(-) \star F) then by the Yoneda lemma it has to be given by

[F,G](c) ≃Hom C(j(c),[F,G]) ≃Hom C(j(c)⋆F,G). \begin{aligned} [F,G](c) & \simeq Hom_C(j(c), [F,G]) \\ &\simeq Hom_C(j(c)\star F, G) \end{aligned} \,.


Monoids with respect to Day convolution

Given any monoidal category then one may consider monoid objects and module objects inside it.


For (π’ž,βŠ—)(\mathcal{C}, \otimes) a small (symmetric) monoidal VV-enriched category, then (commutative) monoid objects in the Day convolution monoidal category ([π’ž,V],βŠ— Day,y(I))([\mathcal{C},V], \otimes_{Day}, y(I)) of prop. 2 are equivalent to (symmetric) lax monoidal functors π’žβ†’V\mathcal{C} \to V:

Mon([π’ž,V],βŠ— Day,y(I))≃MonFunc(π’ž,V) Mon([\mathcal{C},V], \otimes_{Day}, y(I)) \simeq MonFunc(\mathcal{C},V)
CMon([π’ž,V],βŠ— Day,y(I))≃SymMonFunc(π’ž,V). CMon([\mathcal{C},V], \otimes_{Day}, y(I)) \simeq SymMonFunc(\mathcal{C},V) \,.

Moreover, module objects over these monoid objects are equivalent to the corresponding modules over monoidal functors.

This is stated in some form in (Day 70, example 3.2.2). It was highlighted again in (MMSS 00, prop. 22.1). See also MO discussion here.


A lax monoidal functor F:π’žβ†’VF \colon \mathcal{C} \to V is given by natural transformations

I V⟢F(I π’ž) I_V \longrightarrow F(I_{\mathcal{C}})
F(c 1)βŠ— VF(c 2)⟢F(c 1βŠ— π’žc 2) F(c_1) \otimes_V F(c_2) \longrightarrow F(c_1 \otimes_{\mathcal{C}} c_2)

satisfying compatibility conditions. Under the natural isomorphism of corollary 1 these are identified with natural transformations

y(I)β†’F y(I) \to F
FβŠ— DayF⟢F F \otimes_{Day} F \longrightarrow F

satisfying analogous conditions. This is just the structure of a monoid object on FF under βŠ— Day\otimes_{Day}.

Similarly for module objects and modules over monoidal functors.


In the case that VV is pointed topological spaces or pointed simplicial sets equipped with the smash product of pointed objects and that π’ž\mathcal{C} is a diagram category for spectra, then monoids in prop. 5 are known as ring spectra and the lax monoidal functors in prop. 5 are known as the incarnation of ring spectra as β€œfunctors with smash product”.

(MMSS 00, section 22).

Modules with respect to Day convolution


For (π’ž,βŠ—,I)(\mathcal{C},\otimes, I) a small monoidal VV-enriched category, and for R∈Mon([π’ž,V],βŠ— Day)R \in Mon([\mathcal{C}, V],\otimes_{Day}) a monoid object with respect to Day convolution over RR, write

RFreeModβ†ͺRMod R FreeMod \hookrightarrow R Mod

for the full subcategory of the category of modules over RR on those that are free modules. Hence the objects of RFreeModR FreeMod are those of π’ž\mathcal{C} and the hom-objects are

RFreeMod(c 1,c 2) ≔RMod(y(c 1)βŠ—R,y(c 2)βŠ—R) ≃[π’ž,V](y(c 1),y(c 2)βŠ—R) ≃(y(c 2)βŠ—R)(c 1) ≃lim⟢c 3βŠ—c 4β†’c 1π’ž(c 2,c 3)βŠ—R(c 4). \begin{aligned} R FreeMod(c_1,c_2) & \coloneqq R Mod( y(c_1) \otimes R , y(c_2)\otimes R) \\ & \simeq [\mathcal{C},V](y(c_1), y(c_2) \otimes R) \\ & \simeq (y(c_2) \otimes R)(c_1) \\ & \simeq \underset{\underset{c_3 \otimes c_4\to c_1}{\longrightarrow}}{\lim} \mathcal{C}(c_2, c_3) \otimes R(c_4) \end{aligned} \,.

For (π’ž,βŠ—,I)(\mathcal{C},\otimes, I) a small monoidal VV-enriched category, and for R∈Mon([π’ž,V],βŠ— Day)R \in Mon([\mathcal{C}, V],\otimes_{Day}) a monoid object with respect to Day convolution over π’ž\mathcal{C}, then there is an equivalence of categories

RMod≃[RFreeMod op,V] R Mod \simeq [R FreeMod^{op}, V]

between the category of modules over RR and the enriched functor category out of the opposite category of that of free RR-modules from def. 3.

(MMSS 00, theorem 2.2)

Proof idea

Use the identification from prop. 5 of RR with a lax monoidal functor and of any RR-module object NN as a functor with the structure of a module over a monoidal functor, given by natural transformations

N(c 1)βŠ—R(c 2)⟢N(c 1βŠ—c 2). N(c_1) \otimes R(c_2) \longrightarrow N(c_1 \otimes c_2) \,.

These transformations have just the same structure as those of the enriched functoriality of NN of the form

N(c 1)βŠ—π’ž(c 1,c 2)⟢N(c 2). N(c_1) \otimes \mathcal{C}(c_1,c_2) \longrightarrow N(c_2) \,.

Hence we may unify these two kinds of transformarmations in a single kind of the form

N(c 1)βŠ—(π’ž(c 1,c 3)βŠ—R(c 4)⟢N(c 2)forc 2=c 3βŠ—c 4 N(c_1) \otimes ( \mathcal{C}(c_1, c_3) \otimes R(c_4) \longrightarrow N(c_2) \;\;\; for c_2 = c_3 \otimes c_4

and subject to certain identifications.

By comparison with def. 3, this is just the form of the functoriality of an enriched functor over RFreeMod opR FreeMod^{op}.



Let CC be a discrete category over a set, which is hence a monoid (for instance a group) with product β‹…\cdot.

Then the Day convolution product is

F⋆G:eβ†¦βŠ• cβ‹…d=eF(c)Γ—G(d). F \star G : e \mapsto \oplus_{c \cdot d = e} F(c) \times G(d) \,.

Notice that if we regard the presheaves FF and GG here, assuming they take values in finite sets, as categorifications of ℕ\mathbb{N}-valued functions |F|,|G|:C→ℕ|F|, |G| : C \to \mathbb{N}, where |⋅|:Set→ℕ|\cdot| : Set \to \mathbb{N} is the cardinality operation on finite sets, then this reproduces precisely the ordinary convolution product of these ℕ\mathbb{N}-valued functions

|F⋆G|:e β†¦βˆ‘ c,d∈C|F(c)|Γ—|G(d)|Γ—Ξ΄(e,cβŠ—d) =βˆ‘ cβ‹…d=e|F(c)|β‹…|F(d)| \begin{aligned} |F \star G| : e &\mapsto \sum_{c,d \in C} |F(c)| \times |G(d)| \times \delta(e, c \otimes d) & = \sum_{c \cdot d = e} |F(c)| \cdot |F(d)| \end{aligned}

This uses in particular that for every object c∈Cc \in C the functor

Hom C(c,βˆ’)=Ξ΄ c Hom_C(c,-) = \delta_c

is in this sense the Kronecker delta-function on the set CC supported at c∈Cc \in C. Precisely because by assumption CC has only identity morphisms.

Hom C(c,d)={* ifc=d βˆ… ifcβ‰ d Hom_C(c,d) = \left\{ \array{ * & if c = d \\ \emptyset & if c \neq d } \right.

Further examples:


The symmetric smash product of spectra on, in particular, symmetric spectra and orthogonal spectra is the Day convolution product for Top-enriched functors on monoidal categories of symmetric groups of orthogonal groups, respectively (MMSS 00, theorem 1.7 and section 21.).

Similarly the symmetric smash product of spectra on the model structure for excisive functors is Day convolution for sSet-enriched functors on the plain smash product of finite pointed simplicial sets (Lydakis 98).

See also at functor with smash products.


The concept originates in

  • Brian Day, On closed categories of functors, Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics Vol. 137. Springer-Verlag, 1970, pp 1-38 (pdf)

General discussion includes

The application of Day convolution to the construction of symmetric smash products of spectra for highly structured spectra is due to

and for excisive functors due to

  • Lydakis, Simplicial functors and stable homotopy theory Preprint, available via Hopf archive, 1998 (pdf)

(see also at functors with smash product).

Day convolution for (∞,1)-categories is discussed in

Revised on April 14, 2016 07:36:03 by David Corfield (