nLab Day convolution



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In higher category theory

Category theory



The category of functors on 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 convolution products.

In more 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 VV is a monoidally cocomplete category: i.e. 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

Let VV be a closed symmetric monoidal category with all small limits and colimits.

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.

We discuss two equivalent ways of defining Day convolution

  1. In terms of coends

  2. In terms of profunctors

In terms of coends


Let (π’ž,βŠ—,1)(\mathcal{C}, \otimes, 1) be a small VV-enriched monoidal category.

Then the Day convolution tensor product on [π’ž,V][\mathcal{C},V]

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]

is given by the following coend

XβŠ— DayY:cβ†¦βˆ«(c 1,c 2)βˆˆπ’žΓ—π’žπ’ž(c 1βŠ—c 2,c)βŠ— VX(c 1)βŠ— VY(c 2). X \otimes_{Day} Y \;\colon\; c \;\mapsto\; \overset{(c_1,c_2)\in \mathcal{C}\times \mathcal{C}}{\int} \mathcal{C}(c_1 \otimes c_2, c) \otimes_V X(c_1) \otimes_V Y(c_2) \,.

We observe now that Day convolution is equivalently a left Kan extension. This will be key for understanding monoids and modules with respect to Day convolution.


Let π’ž\mathcal{C} be a small VV-monoidal category. Its external tensor product is

βŠ—Β―:[π’ž,V]Γ—[π’ž,V]⟢[π’žΓ—π’ž,V] \overline{\otimes} \;\colon\; [\mathcal{C}, V] \times [\mathcal{C}, V] \longrightarrow [\mathcal{C}\times \mathcal{C}, V]

given by

XβŠ—Β―Yβ‰”βŠ— V∘(X,Y), X \overline{\otimes} Y \;\coloneqq\; \otimes_V \circ (X,Y) \,,


(XβŠ—Β―Y)(c 1,c 2)=X(c 1)βŠ— VY(c 2). (X \overline\otimes Y)(c_1,c_2) = X(c_1)\otimes_V Y(c_2) \,.

The Day convolution product (def. ) of two functors is equivalently the left Kan extension of their external tensor product (def. ) along the tensor product βŠ— π’ž\otimes_{\mathcal{C}}: there is a natural isomorphism

XβŠ— DayY≃Lan βŠ— π’ž(XβŠ—Β―Y). X \otimes_{Day} Y \simeq Lan_{\otimes_{\mathcal{C}}} (X \overline{\otimes} Y) \,.

Hence the adjunction unit is a natural transformation of the form

π’žΓ—π’ž ⟢XβŠ—Β―Y V βŠ—β†˜ ⇓ β†— XβŠ— DayY π’ž. \array{ \mathcal{C} \times \mathcal{C} && \overset{X \overline{\otimes} Y}{\longrightarrow} && V \\ & {}^{\mathllap{\otimes}}\searrow &\Downarrow& \nearrow_{\mathrlap{X \otimes_{Day} Y}} \\ && \mathcal{C} } \,.

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


By prop. we may compute the left Kan extension as the following coend:

Lan βŠ— π’ž(XβŠ—Β―Y)(c) β‰ƒβˆ«(c 1,c 2)π’ž(c 1βŠ— π’žc 2,c)∧(XβŠ—Β―Y)(c 1,c 2) =∫(c 1,c 2)π’ž(c 1βŠ— π’žc 2,c)∧X(c 1)βŠ— VY(c 2). \begin{aligned} Lan_{\otimes_{\mathcal{C}}} (X\overline{\otimes} Y)(c) & \simeq \overset{(c_1,c_2)}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_2, c ) \wedge (X\overline{\otimes}Y)(c_1,c_2) \\ & = \overset{(c_1,c_2)}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_2, c) \wedge X(c_1) \otimes_V Y(c_2) \end{aligned} \,.

Proposition implies the following fact, which is the key for the identification of β€œfunctors with smash product”.


The operation of Day convolution βŠ— Day\otimes_{Day} (def. ) is universally characterized by the property that there are natural isomorphisms

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

where βŠ—Β―\overline{\otimes} is the external product of def. .

In terms of profunctors

The Day convolution can also be expressed in terms of profunctors. The tensor product βŠ—:π’žβŠ—π’žβ†’π’ž\otimes :\mathcal{C}\otimes \mathcal{C}\to \mathcal{C} induces a representable profunctor π’ž(βŠ—,1):π’žβ‡Έπ’žβŠ—π’ž\mathcal{C}(\otimes,1): \mathcal{C} ⇸ \mathcal{C}\otimes \mathcal{C}. The above definition can be interpreted to say that if X,Y∈[π’ž,V]X,Y\in [\mathcal{C},V] are regarded as profunctors π’žβ‡ΈI\mathcal{C} ⇸ I, where II is the unit VV-category, then XβŠ— DayYX\otimes_{Day} Y is the composite of profunctors

π’žβ†’π’ž(βŠ—,1)π’žβŠ—π’žβ†’XβŠ—YIβŠ—Iβ‰…I. \mathcal{C}\xrightarrow{\mathcal{C}(\otimes,1)} \mathcal{C}\otimes \mathcal{C}\xrightarrow{X\otimes Y} I\otimes I \cong I.

A more β€œglobal” way to say the same thing is to consider the β€œevaluation” functor [π’ž,V]βŠ—π’žβ†’V[\mathcal{C},V] \otimes \mathcal{C} \to V to be a profunctor E:π’žβ‡Έ[π’ž,V] opE:\mathcal{C}⇸[\mathcal{C},V]^{op}. Then the profunctor composite

π’žβ†’π’ž(βŠ—,1)π’žβŠ—π’žβ†’EβŠ—E[π’ž,V] opβŠ—[π’ž,V] op \mathcal{C}\xrightarrow{\mathcal{C}(\otimes,1)} \mathcal{C}\otimes \mathcal{C}\xrightarrow{E\otimes E} [\mathcal{C},V]^{op} \otimes [\mathcal{C},V]^{op}

is a functor π’žβŠ—[π’ž,V]βŠ—[π’ž,V]β†’V\mathcal{C}\otimes [\mathcal{C},V] \otimes [\mathcal{C},V] \to V, which by exponential transpose gives a functor [π’ž,V]βŠ—[π’ž,V]β†’[π’ž,V][\mathcal{C},V] \otimes [\mathcal{C},V] \to [\mathcal{C},V]; this is the Day convolution product.

For promonoidal categories

The above description in terms of profunctors makes it clear that the construction only depends on the representable profunctor induced by βŠ—:π’žβŠ—π’žβ†’π’ž\otimes : \mathcal{C}\otimes \mathcal{C}\to \mathcal{C}, i.e. on the underlying promonoidal category of π’ž\mathcal{C}. In the original article (Day 70), a stronger form of the convolution is discussed, in which π’ž\mathcal{C} is assumed only to be a promonoidal category.

Before continuing our discussion, we comment a bit on a convention adopted in (Day’s thesis). To define promonoidal structures, Day used functors of the form P:π’œ opΓ—π’œ opΓ—π’œβ†’π’±P\colon\mathcal{A}^\mathrm{op}\times\mathcal{A}^\mathrm{op}\times\mathcal{A} \to \mathcal{V}, whereas the nLab convention is that a profunctor P:π’œΓ—π’œP\colon\mathcal{A}\times\mathcal{A}β‡Έπ’œ\mathcal{A} is a functor P:π’œ opΓ—π’œΓ—π’œβ†’π’±P\colon\mathcal{A}^\mathrm{op}\times\mathcal{A}\times\mathcal{A} \to \mathcal{V}. Following modern usage and (Corner 2016), instead of defining Day convolution for 𝒱\mathcal{V}-enriched functors, we do so for 𝒱\mathcal{V}-presheaves.

Let 𝒱\mathcal{V} be a BΓ©nabou cosmos, (π’ž,P,I,Ξ»,ρ,Ξ±)(\mathcal{C},P,I,\lambda,\rho,\alpha) be a small 𝒱\mathcal{V}-enriched category equipped with the structure of a promonoidal category, and write P B,C AP^A_{B,C} for P(A,B,C)P(A,B,C) (this is called the Einstein notation for profunctors; see (Loregian 2019, Definition 5.1.10)).


The Day convolution tensor product on [π’ž op,𝒱][\mathcal{C}^\mathsf{op},\mathcal{V}] is the bifunctor

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

defined on objects by the coend

FβŠ— DayG=∫A,Bβˆˆπ’žF(A)βŠ— VG(B)βŠ— VP A,B (βˆ’).F\otimes_{Day}G=\overset{A,B\in\mathcal{C}}{\int}F(A)\otimes_V G(B)\otimes_V P^{(-)}_{A,B}.

There is an equivalence of categories between the category of pro-monoidal structures on π’ž\mathcal{C} with strong pro-monoidal functors between them and the category of biclosed monoidal structures on [π’ž op,V][\mathcal{C}^{op},V] with strong monoidal functors between them.

This is claimed without proof in (Day 70).

Via multicategories

The monoidal structure on [C,D][C, D] for CC promonoidal and DD monoidally cocomplete may be seen as the multicategory structure arising from viewing CC and DD as multicategories, since CC is exponentiable in this case. See multicategory and Proposition 2.12 of Pisani 2014 for more details.


Closed monoidal structure


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

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

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

This may be deduced fairly abstractly from the above description of Day convolution in terms of profunctors, using the associativity of the promonoidal structure on π’ž\mathcal{C}. It of course also holds when π’ž\mathcal{C} is only promonoidal, as shown in Day70 and Day’s thesis.


For (π’ž,βŠ— π’ž,I)(\mathcal{C}, \otimes_{\mathcal{C}}, I) a small monoidal VV-enriched category, the monoidal category with Day convolution ([π’ž,V],βŠ— Day,y(I))([\mathcal{C},V], \otimes_{Day}, y(I)) from def. is a closed monoidal category. Its internal hom [βˆ’,βˆ’] Day[-,-]_{Day} is given by the end

[X,Y] Day(c)β‰ƒβˆ«c 1,c 2V(π’ž(cβŠ— π’žc 1,c 2),V(X(c 1),Y(c 2))). [X,Y]_{Day}(c) \simeq \underset{c_1,c_2}{\int} V\left( \mathcal{C}(c \otimes_{\mathcal{C}} c_1,c_2), V(X(c_1), Y(c_2)) \right) \,.

or equivalently by the end

[X,Y] Day(c)=∫ c 1V(X(c 1),Y(cβŠ—c 1)) [X,Y]_{Day}(c) = \int_{c_1} V\left(X(c_1),Y(c\otimes c_1)\right)

First note that the equivalence between the two formulas follows from the Yoneda lemma. (We mention them both, even though the second is undoubtedly simpler, because the more general case of a promonoidal π’ž\mathcal{C} this simplification is unavailable.)

In analogy to the cartesian closed monoidal structure on presheaves we see that if the internal hom in [π’ž,V][\mathcal{C},V] exists at all, (with [X,βˆ’] Day[X,-]_{Day} right adjoint to (βˆ’)βŠ— DayX(-) \otimes_{Day} X) then by the enriched Yoneda lemma and by the end-expression for the hom-objects in the enriched functor category [π’ž,V][\mathcal{C},V] it has to be given by

[X,Y] Day(c) ≃[π’ž,V](y(c),[X,Y]) ≃[π’ž,V](y(c)βŠ— DayX,Y) β‰ƒβˆ«c 1V((y(c)βŠ— DayX)(c 1),Y(c 1)) β‰ƒβˆ«c 1V(∫d 2∫d 1π’ž(d 1βŠ— π’žd 2,c 1)βŠ— Vπ’ž(c,d 1)βŠ— VX(d 2),Y(c 1)) β‰ƒβˆ«c 1∫d 2V(∫d 1π’ž(d 1βŠ— π’žd 2,c 1)βŠ— Vπ’ž(c,d 1)βŸβ‰ƒπ’ž(cβŠ— π’žd 2,c 1)βŠ— VX(d 2),Y(c 1)) β‰ƒβˆ«c 1,d 2V(π’ž(cβŠ— π’žd 2,c 1),V(X(d 2),Y(c 1))) =∫c 1,c 2V(π’ž(cβŠ— π’žc 1,c 2),V(X(c 1),Y(c 2))). \begin{aligned} [X,Y]_{Day}(c) & \simeq [\mathcal{C},V](y(c), [X,Y]) \\ & \simeq [\mathcal{C},V](y(c) \otimes_{Day} X, Y) \\ & \simeq \underset{c_1}{\int} V((y(c) \otimes_{Day} X)(c_1), Y(c_1)) \\ &\simeq \underset{c_1}{\int} V\left( \overset{d_2}{\int} \overset{d_1}{\int} \mathcal{C}(d_1 \otimes_{\mathcal{C}} d_2, c_1) \otimes_V \mathcal{C}(c,d_1) \otimes_V X(d_2) , Y(c_1) \right) \\ & \simeq \underset{c_1}{\int} \underset{d_2}{\int} V\left( \underset{\simeq \mathcal{C}(c \otimes_{\mathcal{C}} d_2, c_1 )}{ \underbrace{ \overset{d_1}{\int} \mathcal{C}(d_1 \otimes_{\mathcal{C}} d_2, c_1) \otimes_V \mathcal{C}(c,d_1) } } \otimes_V X(d_2) , Y(c_1) \right) \\ & \simeq \underset{c_1,d_2}{\int} V\left( \mathcal{C}(c \otimes_{\mathcal{C}} d_2,c_1), V(X(d_2), Y(c_1)) \right) \\ & = \underset{c_1,c_2}{\int} V\left( \mathcal{C}(c \otimes_{\mathcal{C}} c_1,c_2), V(X(c_1), Y(c_2)) \right) \end{aligned} \,.

This exists, by the assumption that π’ž\mathcal{C} is small and that VV has all small limits. Now to check that this really gives a right adjoint:

[π’ž,V](X,[Y,Z] Day) β‰ƒβˆ«cV(X(c),∫c 1,c 2V(π’ž(cβŠ— π’žc 1,c 2),V(Y(c 1),Z(c 2)))) β‰ƒβˆ«c∫c 1,c 2V(π’ž(cβŠ— π’žc 1,c 2)βŠ— VX(c)βŠ— VY(c 1),Z(c 2)) β‰ƒβˆ«c 2V(∫c,c 1π’ž(cβŠ— π’žc 1,c 2)βŠ— VX(c)βŠ— VY(c 1),Z(c 2)) β‰ƒβˆ«c 2V((XβŠ— DayY)(c 2),Z(c 2)) ≃[π’ž,V](XβŠ— DayY,Z). \begin{aligned} [\mathcal{C},V]( X, [Y,Z]_{Day} ) & \simeq \underset{c}{\int} V\left( X(c), \underset{c_1,c_2}{\int} V\left( \mathcal{C}(c \otimes_{\mathcal{C}} c_1 , c_2), V(Y(c_1), Z(c_2)) \right) \right) \\ & \simeq \underset{c}{\int} \underset{c_1,c_2}{\int} V\left( \mathcal{C}(c \otimes_{\mathcal{C}} c_1, c_2) \otimes_V X(c) \otimes_V Y(c_1) ,\; Z(c_2) \right) \\ & \simeq \underset{c_2}{\int} V\left( \overset{c,c_1}{\int} \mathcal{C}(c \otimes_{\mathcal{C}} c_1, c_2) \otimes_V X(c) \otimes_V Y(c_1) ,\; Z(c_2) \right) \\ &\simeq \underset{c_2}{\int} V\left( (X \otimes_{Day} Y)(c_2), Z(c_2) \right) \\ &\simeq [\mathcal{C},V](X \otimes_{Day} Y, Z) \end{aligned} \,.

While most of this page discusses the covariant Day convolution, the contravariant one interacts conveniently with the Yoneda embedding.


The Yoneda embedding constitutes a strong monoidal functor (π’ž,βŠ— π’ž,I)β†ͺ([π’ž op,V],βŠ— Day,y(I))(\mathcal{C},\otimes_{\mathcal{C}}, I) \hookrightarrow ([\mathcal{C}^{\mathrm{op}},V], \otimes_{Day}, y(I)).


The proof that the tensor unit is respected is dual to the argument in prop. . To see that the tensor product is respected, apply the co-Yoneda lemma twice to get the following natural isomorphism

(y(c 1)βŠ— Dayy(c 2))(c) β‰ƒβˆ«d 1,d 2π’ž(c,d 1βŠ— π’žd 2)βŠ— Vπ’ž(d 1,c 1)βŠ— Vπ’ž(d 2,c 2) β‰ƒπ’ž(c,c 1βŠ— π’žc 2) =y(c 1βŠ— π’žc 2)(c). \begin{aligned} (y(c_1) \otimes_{Day} y(c_2))(c) & \simeq \underset{d_1, d_2}{\int} \mathcal{C}( c ,d_1 \otimes_{\mathcal{C}} d_2) \otimes_V \mathcal{C}(d_1,c_1) \otimes_V \mathcal{C}(d_2,c_2) \\ & \simeq \mathcal{C}( c ,c_1\otimes_{\mathcal{C}}c_2 ) \\ & = y(c_1 \otimes_{\mathcal{C}} c_2 )(c) \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. 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) \,.

In functional programming, these monoids give rise to the notion of Applicative.

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 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.


This page primarily addresses the Day convolution monoidal structure on [π’ž op,𝒱][\mathcal{C}^{op},\mathcal{V}] when π’ž\mathcal{C} is monoidal or promonoidal and enriched over 𝒱\mathcal{V}. This monoidal structure β€œconvolves” the respective structures of π’ž\mathcal{C} and 𝒱\mathcal{V}. One might ask if one can convolve the monoidal structures of two promonoidal categories in general. In other words, is it possible to equip the functor category [π’ž,π’Ÿ][\mathcal{C},\mathcal{D}], for promonoidal categories π’ž\mathcal{C} and π’Ÿ\mathcal{D}, with a convolution monoidal structure? The literature does not seem to give a definite answer to this question in general. However, in the case that π’Ÿ\mathcal{D} is biclosed and cocomplete there is a monoidal convolution structure on [π’ž,π’Ÿ][\mathcal{C},\mathcal{D}] as shown in Proposition 5.4 of Day74. In this special situation, the monoid objects of [π’ž,π’Ÿ][\mathcal{C},\mathcal{D}] are precisely the promonoidal functors π’žβ†’π’Ÿ\mathcal{C}\to\mathcal{D} (Remark 2.2 of Day77), where π’Ÿ\mathcal{D} obtains a promonoidal structure from its monoidal structure by Theorem 4.1 of Day70.


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. are known as ring spectra and the lax monoidal functors in prop. 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 π’ž\mathcal{C}, write

RFree π’žModβ†ͺRMod R Free_{\mathcal{C}}Mod \hookrightarrow R Mod

for the full subcategory of the category of modules over RR on those that are free modules and moreover free on objects in π’ž\mathcal{C} (under the Yoneda embedding). Hence the objects of RFree π’žModR Free_{\mathcal{C}}Mod are those of π’ž\mathcal{C} and the hom-objects are

RFree π’žMod(c 1,c 2)≔RMod(y(c 1)βŠ— DayR,y(c 2)βŠ— DayR). R Free_{\mathcal{C}}Mod(c_1,c_2) \;\coloneqq\; R Mod( y(c_1) \otimes_{Day} R , y(c_2) \otimes_{Day} R) \,.

For (π’ž,βŠ—,I)(\mathcal{C},\otimes, I) a small 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

Mod R≃[RFree π’žMod op,V] Mod_R \simeq [R Free_{\mathcal{C}}Mod^{op}, V]

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

(MMSS 00, theorem 2.2)

Proof idea

Use the identification from prop. 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) \,.

Notice that these transformations have just the same structure as those of the enriched functoriality of NN of the form

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

Hence we may unify these two kinds of transformations into a single kind of the form

π’ž(c 1βŠ—c 4,c 2)βŠ—R(c 4)βŠ—N(c 1)βŸΆπ’ž(c 1βŠ—c 4,c 2)βŠ—N(c 1βŠ—c 4)⟢N(c 2) \mathcal{C}(c_1 \otimes c_4, c_2) \otimes R(c_4) \otimes N(c_1) \longrightarrow \mathcal{C}(c_1 \otimes c_4, c_2) \otimes N(c_1 \otimes c_4) \longrightarrow N(c_2)

and subject to certain identifications.

Now observe that the hom-objects of RFree π’žModR Free_{\mathcal{C}}Mod (def. ) have just this structure:

RFree π’žMod(c 2,c 1) =RMod(y(c 2)βŠ— DayR,y(c 1)βŠ— DayR) ≃[π’ž,V](y(c 2),y(c 1)βŠ— DayR) ≃(y(c 1)βŠ— DayR)(c 2) β‰ƒβˆ«c 3,c 4π’ž(c 3βŠ—c 4,c 2)βŠ— Vπ’ž(c 1,c 3)βŠ— VR(c 4) β‰ƒβˆ«c 4π’ž(c 1βŠ—c 4,c 2)βŠ— VR(c 4). \begin{aligned} R Free_{\mathcal{C}}Mod(c_2,c_1) & = R Mod( y(c_2) \otimes_{Day} R , y(c_1) \otimes_{Day} R) \\ & \simeq [\mathcal{C},V](y(c_2), y(c_1) \otimes_{Day} R) \\ & \simeq (y(c_1) \otimes_{Day} R)(c_2) \\ & \simeq \overset{c_3,c_4}{\int} \mathcal{C}(c_3 \otimes c_4,c_2) \otimes_V \mathcal{C}(c_1, c_3) \otimes_V R(c_4) \\ & \simeq \overset{c_4}{\int} \mathcal{C}(c_1 \otimes c_4,c_2) \otimes_V R(c_4) \end{aligned} \,.

We claim that under this identification, composition in RFree π’žModR Free_{\mathcal{C}}Mod is given by

RFree π’žMod(c 2,c 1)βŠ— VRFree π’žMod(c 3,c 2) =(∫c 4π’ž(c 1βŠ— π’žc 4,c 2)βŠ— VR(c 4))βŠ— V(∫c 5π’ž(c 2βŠ— π’žc 5,c 3)βŠ— VR(c 5)) β‰ƒβˆ«c 4,c 5π’ž(c 1βŠ— π’žc 4,c 2)βŠ— Vπ’ž(c 2βŠ— π’žc 5,c 3)βŠ— VR(c 4)βŠ— VR(c 5) ⟢∫c 4,c 5π’ž(c 1βŠ— π’žc 4βŠ— π’žc 5,c 2βŠ— π’žc 5)βŠ— Vπ’ž(c 2βŠ—c 5,c 3)βŠ— VR(c 4βŠ— π’žc 5) ⟢∫c 4,c 5π’ž(c 1βŠ— π’žc 4βŠ— π’žc 5,c 3)βŠ— VR(c 4βŠ— π’žc 5) ⟢∫c 4π’ž(c 1βŠ— π’žc 4,c 3)βŠ— VR(c 4), \begin{aligned} R Free_{\mathcal{C}}Mod(c_2, c_1) \otimes_V R Free_{\mathcal{C}}Mod(c_3, c_2) & = \left( \overset{c_4}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2) \otimes_V R(c_4) \right) \otimes_V \left( \overset{c_5}{\int} \mathcal{C}(c_2 \otimes_{\mathcal{C}} c_5, c_3 ) \otimes_V R(c_5) \right) \\ & \simeq \overset{c_4, c_5}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4 , c_2 ) \otimes_V \mathcal{C}(c_2 \otimes_{\mathcal{C}} c_5, c_3) \otimes_V R(c_4) \otimes_V R(c_5) \\ & \longrightarrow \overset{c_4,c_5}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_5 , c_2 \otimes_{\mathcal{C}} c_5 ) \otimes_V \mathcal{C}(c_2 \otimes c_5, c_3) \otimes_V R(c_4 \otimes_{\mathcal{C}} c_5 ) \\ & \longrightarrow \overset{c_4, c_5}{\int} \mathcal{C}(c_1\otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_5 , c_3) \otimes_V R(c_4 \otimes_{\mathcal{C}} c_5 ) \\ & \longrightarrow \overset{c_4}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4 , c_3) \otimes_V R(c_4 ) \end{aligned} \,,


  1. the first morphism is, in the integrand, the tensor product of

    1. forming the tensor product of hom-objects of π’ž\mathcal{C} with the identity of c 5c_5

      π’ž(c 1βŠ— π’žc 4,c 2)β‰ƒπ’ž(c 1βŠ— π’žc 4,c 2)βŠ— V1 VβŸΆπ’ž(c 1βŠ— π’žc 4,c 2)βŠ—π’ž(c 5,c 5)βŸΆπ’ž(c 1βŠ— π’žc 4βŠ— π’žc 5,c 2βŠ— π’žc 5) \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2 ) \simeq \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2 ) \otimes_V 1_V \overset{}{\longrightarrow} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2 ) \otimes \mathcal{C}(c_5,c_5) \longrightarrow \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_5, c_2 \otimes_{\mathcal{C}} c_5)
    2. the monoidal functor incarnation R(c 4)βŠ— VR(c 5)⟢R(c 4βŠ— π’žc 5)R(c_4) \otimes_V R(c_5)\longrightarrow R(c_4 \otimes_{\mathcal{C}} c_5 ) of the monoid structure on RR;

  2. the second morphism is, in the integrand, given by composition in π’ž\mathcal{C};

  3. the last morphism is the morphism induced on coends by regarding extranaturality in c 4c_4 and c 5c_5 separately as a special case of extranaturality in c 6≔c 4βŠ—c 5c_6 \coloneqq c_4 \otimes c_5 (and then renaming).

It is fairly straightforward to see that, under the above identifications, functoriality under this composition is equivalently functoriality in π’ž\mathcal{C} together with the action property over RR.



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, and many of its basic properties, originates in several works of Brian day, including:

  • Brian Day, Construction of Biclosed Categories, PhD thesis. School of Mathematics of the University of New South Wales, September 1970. Link.

  • 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),

  • Brian Day, On closed categories of functors II, Category Seminar, Sydney 1972/73, Springer Lecture Notes, Vol. 420, 20-53.

  • Brian Day, Promonoidal functor categories, J. Austral. Math. Soc. 23 (Series A) (1977), 312-328.

(Note that some unproven statements in (Day’s report) are proved in (Day’s thesis) and vice versa.)

The universal property of the Day convolution, in the sense of free monoidal cocompletion, is discussed in

  • Geun Bin Im and G. M. Kelly, A universal property of the convolution monoidal structure, J. Pure Appl. Algebra 43 (1986), no. 1, 75-88.

A generalisation to Ξ¦\Phi-cocompletions for some class Ξ¦\Phi of weights is discussed in Β§3 of:

  • S. R. Johnson, Monoidal Morita equivalence, Journal of Pure and Applied Algebra 59.2 (1989): 169-177.

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 monoidal bicategories is developed in

  • Alexander Corner?, Day convolution for monoidal bicategories, School of Mathematics and Statistics of the University of Sheffield. Available through the White Rose theses database.

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

Other references cited in this page:

On ∞ \infty -colimits and Day convolution in the context of enriched ∞ \infty -categories:

Various generalisations are discussed in:

  • Brian Day and Ross Street, Lax monoids, pseudo-operads, and convolution, Contemporary Mathematics 318 (2003): 75-96.

The link with multicategories is observed in:

Last revised on April 11, 2024 at 18:39:34. See the history of this page for a list of all contributions to it.