nLab Day convolution

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Contents

Context

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Category theory

Contents

1. Idea

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:Gโ†’โ„‚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:๐’ขโ†’Setf \colon \mathcal{G} \to Set whenever the category ๐’ข\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 uโŠ—vu \otimes v preserves colimits in its two arguments separately; hence that the functors uโŠ—โˆ’u \otimes - and โˆ’โŠ—v- \otimes v preserve colimits. This occurs notably when VV is symmetric closed monoidal (so that these functors are left adjoints).

2. Definition

For monoidal categories

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

For ๐’ž\mathcal{C} a VV-enriched category, write [๐’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

Definition 2.1. Let (๐’ž,โŠ—,I)(\mathcal{C}, \otimes, I) be a small VV-enriched monoidal category.

Then the Day convolution tensor product on [๐’ž,V][\mathcal{C},V]

โŠ— Day:[๐’ž,V]ร—[๐’ž,V]โŸถ[๐’ž,V] \otimes_{Day} \;\colon\; [\mathcal{C},V] \times [\mathcal{C},V] \longrightarrow [\mathcal{C},V]

is given by the following coend

XโŠ— DayY:cโ†ฆโˆซ(c 1,c 2)โˆˆ๐’žร—๐’ž๐’ž(c 1โŠ—c 2,c)โŠ— VX(c 1)โŠ— 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) \,.

Its unit is I Day(c)=๐’ž(1,โˆ’)I_{Day}(c) = \mathcal{C}(1,-)

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.

Definition 2.2. Let ๐’ž\mathcal{C} be a small VV-monoidal category. Its external tensor product is

โŠ—ยฏ:[๐’ž,V]ร—[๐’ž,V]โŸถ[๐’žร—๐’ž,V] \overline{\otimes} \;\colon\; [\mathcal{C}, V] \times [\mathcal{C}, V] \longrightarrow [\mathcal{C}\times \mathcal{C}, V]

given by

XโŠ—ยฏYโ‰”โŠ— Vโˆ˜(X,Y), X \overline{\otimes} Y \;\coloneqq\; \otimes_V \circ (X,Y) \,,

i.e.

(XโŠ—ยฏY)(c 1,c 2)=X(c 1)โŠ— VY(c 2). (X \overline\otimes Y)(c_1,c_2) = X(c_1)\otimes_V Y(c_2) \,.

Proposition 2.3. The Day convolution product (def. 2.1) of two functors is equivalently the left Kan extension of their external tensor product (def. 2.2) along the tensor product โŠ— ๐’ž\otimes_{\mathcal{C}}: there is a natural isomorphism

XโŠ— DayYโ‰ƒLan โŠ— ๐’ž(XโŠ—ยฏY). X \otimes_{Day} Y \simeq Lan_{\otimes_{\mathcal{C}}} (X \overline{\otimes} Y) \,.

Hence the adjunction unit is a natural transformation of the form

๐’žร—๐’ž โŸถXโŠ—ยฏY V โŠ—โ†˜ โ‡“ โ†— XโŠ— DayY ๐’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).

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

Lan โŠ— ๐’ž(XโŠ—ยฏY)(c) โ‰ƒโˆซ(c 1,c 2)๐’ž(c 1โŠ— ๐’žc 2,c)โˆง(XโŠ—ยฏY)(c 1,c 2) =โˆซ(c 1,c 2)๐’ž(c 1โŠ— ๐’žc 2,c)โˆงX(c 1)โŠ— 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} \,.

ย ย โ–ฎ

Proposition 2.3 implies the following fact, which is the key for the identification of โ€œfunctors with smash productโ€.

Corollary 2.4. The operation of Day convolution โŠ— Day\otimes_{Day} (def. 2.1) is universally characterized by the property that there are natural isomorphisms

[๐’ž,V](XโŠ— DayY,Z)โ‰ƒ[๐’žร—๐’ž,V](XโŠ—ยฏY,Zโˆ˜โŠ— 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 โŠ—ยฏ\overline{\otimes} is the external product of def. 2.2.

In terms of profunctors

The Day convolution can also be expressed in terms of profunctors. The tensor product โŠ—:๐’žโŠ—๐’žโ†’๐’ž\otimes :\mathcal{C}\otimes \mathcal{C}\to \mathcal{C} induces a representable profunctor ๐’ž(โŠ—,1):๐’žโ‡ธ๐’žโŠ—๐’ž\mathcal{C}(\otimes,1): \mathcal{C} ⇸ \mathcal{C}\otimes \mathcal{C}. The above definition can be interpreted to say that if X,Yโˆˆ[๐’ž,V]X,Y\in [\mathcal{C},V] are regarded as profunctors ๐’žโ‡ธI\mathcal{C} ⇸ I, where II is the unit VV-category, then XโŠ— DayYX\otimes_{Day} Y is the composite of profunctors

๐’žโ†’๐’ž(โŠ—,1)๐’žโŠ—๐’žโ†’XโŠ—YIโŠ—Iโ‰…I. \mathcal{C}\xrightarrow{\mathcal{C}(\otimes,1)} \mathcal{C}\otimes \mathcal{C}\xrightarrow{X\otimes Y} I\otimes I \cong I.

A more โ€œglobalโ€ way to say the same thing is to consider the โ€œevaluationโ€ functor [๐’ž,V]โŠ—๐’žโ†’V[\mathcal{C},V] \otimes \mathcal{C} \to V to be a profunctor E:๐’žโ‡ธ[๐’ž,V] opE:\mathcal{C}⇸[\mathcal{C},V]^{op}. Then the profunctor composite

๐’žโ†’๐’ž(โŠ—,1)๐’žโŠ—๐’žโ†’EโŠ—E[๐’ž,V] opโŠ—[๐’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 ๐’žโŠ—[๐’ž,V]โŠ—[๐’ž,V]โ†’V\mathcal{C}\otimes [\mathcal{C},V] \otimes [\mathcal{C},V] \to V, which by exponential transpose gives a functor [๐’ž,V]โŠ—[๐’ž,V]โ†’[๐’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 โŠ—:๐’žโŠ—๐’žโ†’๐’ž\otimes : \mathcal{C}\otimes \mathcal{C}\to \mathcal{C}, i.e. on the underlying promonoidal category of ๐’ž\mathcal{C}. In the original article (Day 70), a stronger form of the convolution is discussed, in which ๐’ž\mathcal{C} is assumed only to be a promonoidal category.

Before continuing our discussion, we comment a bit on a convention adopted in (Dayโ€™s thesis). To define promonoidal structures, Day used functors of the form P:๐’œ opร—๐’œ opร—๐’œโ†’๐’ฑP\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:๐’œร—๐’œP\colon\mathcal{A}\times\mathcal{A}โ‡ธ๐’œ\mathcal{A} is a functor P:๐’œ opร—๐’œร—๐’œโ†’๐’ฑP\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 ๐’ฑ\mathcal{V}-enriched functors, we do so for ๐’ฑ\mathcal{V}-presheaves.

Let ๐’ฑ\mathcal{V} be a Bรฉnabou cosmos, (๐’ž,P,I,ฮป,ฯ,ฮฑ)(\mathcal{C},P,I,\lambda,\rho,\alpha) be a small ๐’ฑ\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)).

Definition 2.5. The Day convolution tensor product on [๐’ž op,๐’ฑ][\mathcal{C}^\mathsf{op},\mathcal{V}] is the bifunctor

โŠ— Day:[๐’ž op,๐’ฑ]ร—[๐’ž op,๐’ฑ]โŸถ[๐’ž op,๐’ฑ]\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

FโŠ— DayG=โˆซA,Bโˆˆ๐’žF(A)โŠ— VG(B)โŠ— VP A,B (โˆ’).F\otimes_{Day}G=\overset{A,B\in\mathcal{C}}{\int}F(A)\otimes_V G(B)\otimes_V P^{(-)}_{A,B}.

Proposition 2.6. There is an equivalence of categories between the category of pro-monoidal structures on ๐’ž\mathcal{C} with strong pro-monoidal functors between them and the category of biclosed monoidal structures on [๐’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.

3. Properties

Closed monoidal structure

Proposition 3.1. For (๐’ž,โŠ— ๐’ž,I)(\mathcal{C}, \otimes_{\mathcal{C}}, I) a small monoidal VV-enriched category, the Day convolution product โŠ— Day\otimes_{Day} of def. makes

([๐’ž,V],โŠ— 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 ๐’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 ๐’ž\mathcal{C}. It of course also holds when ๐’ž\mathcal{C} is only promonoidal, as shown in Day70 and Dayโ€™s thesis.

Proposition 3.2. For (๐’ž,โŠ— ๐’ž,I)(\mathcal{C}, \otimes_{\mathcal{C}}, I) a small monoidal VV-enriched category, the monoidal category with Day convolution ([๐’ž,V],โŠ— Day,y(I))([\mathcal{C},V], \otimes_{Day}, y(I)) from def. 3.1 is a closed monoidal category. Its internal hom [โˆ’,โˆ’] Day[-,-]_{Day} is given by the end

[X,Y] Day(c)โ‰ƒโˆซc 1,c 2V(๐’ž(cโŠ— ๐’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)=โˆซ c 1V(X(c 1),Y(cโŠ—c 1)) [X,Y]_{Day}(c) = \int_{c_1} V\left(X(c_1),Y(c\otimes c_1)\right)

Proof. 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 ๐’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 [๐’ž,V][\mathcal{C},V] exists at all, (with [X,โˆ’] Day[X,-]_{Day} right adjoint to (โˆ’)โŠ— DayX(-) \otimes_{Day} X) then by the enriched Yoneda lemma and by the end-expression for the hom-objects in the enriched functor category [๐’ž,V][\mathcal{C},V] it has to be given by

[X,Y] Day(c) โ‰ƒ[๐’ž,V](y(c),[X,Y]) โ‰ƒ[๐’ž,V](y(c)โŠ— DayX,Y) โ‰ƒโˆซc 1V((y(c)โŠ— DayX)(c 1),Y(c 1)) โ‰ƒโˆซc 1V(โˆซd 2โˆซd 1๐’ž(d 1โŠ— ๐’žd 2,c 1)โŠ— V๐’ž(c,d 1)โŠ— VX(d 2),Y(c 1)) โ‰ƒโˆซc 1โˆซd 2V(โˆซd 1๐’ž(d 1โŠ— ๐’žd 2,c 1)โŠ— V๐’ž(c,d 1)โŸโ‰ƒ๐’ž(cโŠ— ๐’žd 2,c 1)โŠ— VX(d 2),Y(c 1)) โ‰ƒโˆซc 1,d 2V(๐’ž(cโŠ— ๐’žd 2,c 1),V(X(d 2),Y(c 1))) =โˆซc 1,c 2V(๐’ž(cโŠ— ๐’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 ๐’ž\mathcal{C} is small and that VV has all small limits. Now to check that this really gives a right adjoint:

[๐’ž,V](X,[Y,Z] Day) โ‰ƒโˆซcV(X(c),โˆซc 1,c 2V(๐’ž(cโŠ— ๐’žc 1,c 2),V(Y(c 1),Z(c 2)))) โ‰ƒโˆซcโˆซc 1,c 2V(๐’ž(cโŠ— ๐’žc 1,c 2)โŠ— VX(c)โŠ— VY(c 1),Z(c 2)) โ‰ƒโˆซc 2V(โˆซc,c 1๐’ž(cโŠ— ๐’žc 1,c 2)โŠ— VX(c)โŠ— VY(c 1),Z(c 2)) โ‰ƒโˆซc 2V((XโŠ— DayY)(c 2),Z(c 2)) โ‰ƒ[๐’ž,V](XโŠ— 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} \,.

ย ย โ–ฎ

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

Proposition 3.3. The Yoneda embedding constitutes a strong monoidal functor (๐’ž,โŠ— ๐’ž,I)โ†ช([๐’ž op,V],โŠ— Day,y(I))(\mathcal{C},\otimes_{\mathcal{C}}, I) \hookrightarrow ([\mathcal{C}^{\mathrm{op}},V], \otimes_{Day}, y(I)).

Proof. The proof that the tensor unit is respected is dual to the argument in prop. 3.1. To see that the tensor product is respected, apply the co-Yoneda lemma twice to get the following natural isomorphism

(y(c 1)โŠ— Dayy(c 2))(c) โ‰ƒโˆซd 1,d 2๐’ž(c,d 1โŠ— ๐’žd 2)โŠ— V๐’ž(d 1,c 1)โŠ— V๐’ž(d 2,c 2) โ‰ƒ๐’ž(c,c 1โŠ— ๐’žc 2) =y(c 1โŠ— ๐’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} \,.

ย ย โ–ฎ

Monoids with respect to Day convolution

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

Proposition 3.4. For (๐’ž,โŠ—)(\mathcal{C}, \otimes) a small (symmetric) monoidal VV-enriched category, then (commutative) monoid objects in the Day convolution monoidal category ([๐’ž,V],โŠ— Day,y(I))([\mathcal{C},V], \otimes_{Day}, y(I)) of prop. 3.1 are equivalent to (symmetric) lax monoidal functors ๐’žโ†’V\mathcal{C} \to V:

Mon([๐’ž,V],โŠ— Day,y(I))โ‰ƒMonFunc(๐’ž,V) Mon([\mathcal{C},V], \otimes_{Day}, y(I)) \simeq MonFunc(\mathcal{C},V)
CMon([๐’ž,V],โŠ— Day,y(I))โ‰ƒSymMonFunc(๐’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.

Proof. A lax monoidal functor F:๐’žโ†’VF \colon \mathcal{C} \to V is given by natural transformations

I VโŸถF(I ๐’ž) I_V \longrightarrow F(I_{\mathcal{C}})
F(c 1)โŠ— VF(c 2)โŸถF(c 1โŠ— ๐’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 2.4 these are identified with natural transformations

y(I)โ†’F y(I) \to F
FโŠ— DayFโŸถF F \otimes_{Day} F \longrightarrow F

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

Similarly for module objects and modules over monoidal functors.ย ย โ–ฎ

Remark 3.5. This page primarily addresses the Day convolution monoidal structure on [๐’ž op,๐’ฑ][\mathcal{C}^{op},\mathcal{V}] when ๐’ž\mathcal{C} is monoidal or promonoidal and enriched over ๐’ฑ\mathcal{V}. This monoidal structure โ€œconvolvesโ€ the respective structures of ๐’ž\mathcal{C} and ๐’ฑ\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 [๐’ž,๐’Ÿ][\mathcal{C},\mathcal{D}], for promonoidal categories ๐’ž\mathcal{C} and ๐’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 ๐’Ÿ\mathcal{D} is biclosed and cocomplete there is a monoidal convolution structure on [๐’ž,๐’Ÿ][\mathcal{C},\mathcal{D}] as shown in Proposition 5.4 of Day74. In this special situation, the monoid objects of [๐’ž,๐’Ÿ][\mathcal{C},\mathcal{D}] are precisely the promonoidal functors ๐’žโ†’๐’Ÿ\mathcal{C}\to\mathcal{D} (Remark 2.2 of Day77), where ๐’Ÿ\mathcal{D} obtains a promonoidal structure from its monoidal structure by Theorem 4.1 of Day70.

Example 3.6. In the case that VV is pointed topological spaces or pointed simplicial sets equipped with the smash product of pointed objects and that ๐’ž\mathcal{C} is a diagram category for spectra, then monoids in prop. 3.4 are known as ring spectra and the lax monoidal functors in prop. 3.4 are known as the incarnation of ring spectra as โ€œfunctors with smash productโ€.

(MMSS 00, section 22).

Modules with respect to Day convolution

Definition 3.7. For (๐’ž,โŠ—,I)(\mathcal{C},\otimes, I) a small monoidal VV-enriched category, and for RโˆˆMon([๐’ž,V],โŠ— Day)R \in Mon([\mathcal{C}, V],\otimes_{Day}) a monoid object with respect to Day convolution over ๐’ž\mathcal{C}, write

RFree ๐’žModโ†ชRMod 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 ๐’ž\mathcal{C} (under the Yoneda embedding). Hence the objects of RFree ๐’žModR Free_{\mathcal{C}}Mod are those of ๐’ž\mathcal{C} and the hom-objects are

RFree ๐’žMod(c 1,c 2)โ‰”RMod(y(c 1)โŠ— DayR,y(c 2)โŠ— 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) \,.

Proposition 3.8. For (๐’ž,โŠ—,I)(\mathcal{C},\otimes, I) a small VV-enriched category, and for RโˆˆMon([๐’ž,V],โŠ— Day)R \in Mon([\mathcal{C}, V],\otimes_{Day}) a monoid object with respect to Day convolution over ๐’ž\mathcal{C}, then there is an equivalence of categories

Mod Rโ‰ƒ[RFree ๐’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. 3.7.

(MMSS 00, theorem 2.2)

Proof idea. Use the identification from prop. 3.4 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)โŠ—R(c 2)โŸถN(c 1โŠ—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

๐’ž(c 1,c 2)โŠ—N(c 1)โŸถN(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

๐’ž(c 1โŠ—c 4,c 2)โŠ—R(c 4)โŠ—N(c 1)โŸถ๐’ž(c 1โŠ—c 4,c 2)โŠ—N(c 1โŠ—c 4)โŸถN(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 ๐’žModR Free_{\mathcal{C}}Mod (def. 3.7) have just this structure:

RFree ๐’žMod(c 2,c 1) =RMod(y(c 2)โŠ— DayR,y(c 1)โŠ— DayR) โ‰ƒ[๐’ž,V](y(c 2),y(c 1)โŠ— DayR) โ‰ƒ(y(c 1)โŠ— DayR)(c 2) โ‰ƒโˆซc 3,c 4๐’ž(c 3โŠ—c 4,c 2)โŠ— V๐’ž(c 1,c 3)โŠ— VR(c 4) โ‰ƒโˆซc 4๐’ž(c 1โŠ—c 4,c 2)โŠ— 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 ๐’žModR Free_{\mathcal{C}}Mod is given by

RFree ๐’žMod(c 2,c 1)โŠ— VRFree ๐’žMod(c 3,c 2) =(โˆซc 4๐’ž(c 1โŠ— ๐’žc 4,c 2)โŠ— VR(c 4))โŠ— V(โˆซc 5๐’ž(c 2โŠ— ๐’žc 5,c 3)โŠ— VR(c 5)) โ‰ƒโˆซc 4,c 5๐’ž(c 1โŠ— ๐’žc 4,c 2)โŠ— V๐’ž(c 2โŠ— ๐’žc 5,c 3)โŠ— VR(c 4)โŠ— VR(c 5) โŸถโˆซc 4,c 5๐’ž(c 1โŠ— ๐’žc 4โŠ— ๐’žc 5,c 2โŠ— ๐’žc 5)โŠ— V๐’ž(c 2โŠ—c 5,c 3)โŠ— VR(c 4โŠ— ๐’žc 5) โŸถโˆซc 4,c 5๐’ž(c 1โŠ— ๐’žc 4โŠ— ๐’žc 5,c 3)โŠ— VR(c 4โŠ— ๐’žc 5) โŸถโˆซc 4๐’ž(c 1โŠ— ๐’žc 4,c 3)โŠ— 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} \,,

where

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

    1. forming the tensor product of hom-objects of ๐’ž\mathcal{C} with the identity of c 5c_5

      ๐’ž(c 1โŠ— ๐’žc 4,c 2)โ‰ƒ๐’ž(c 1โŠ— ๐’žc 4,c 2)โŠ— V1 VโŸถ๐’ž(c 1โŠ— ๐’žc 4,c 2)โŠ—๐’ž(c 5,c 5)โŸถ๐’ž(c 1โŠ— ๐’žc 4โŠ— ๐’žc 5,c 2โŠ— ๐’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)โŠ— VR(c 5)โŸถR(c 4โŠ— ๐’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 ๐’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 6โ‰”c 4โŠ—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 ๐’ž\mathcal{C} together with the action property over RR.ย ย โ–ฎ

4. Examples

Example 4.1. Let CC be a discrete category over a set, which is hence a monoid (for instance a group) with product โ‹…\cdot.

Then the Day convolution product is

Fโ‹†G:eโ†ฆโŠ• cโ‹…d=eF(c)ร—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 โ„•\mathbb{N}-valued functions |F|,|G|:Cโ†’โ„•|F|, |G| : C \to \mathbb{N}, where |โ‹…|:Setโ†’โ„•|\cdot| : Set \to \mathbb{N} is the cardinality operation on finite sets, then this reproduces precisely the ordinary convolution product of these โ„•\mathbb{N}-valued functions

|Fโ‹†G|:e โ†ฆโˆ‘ c,dโˆˆC|F(c)|ร—|G(d)|ร—ฮด(e,cโŠ—d) =โˆ‘ cโ‹…d=e|F(c)|โ‹…|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 cโˆˆCc \in C the functor

Hom C(c,โˆ’)=ฮด c Hom_C(c,-) = \delta_c

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

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

Further examples:

Example 4.2. 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.

6. References

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 (Dayโ€™s report) are proved in (Dayโ€™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 ฮฆ\Phi-cocompletions for some class ฮฆ\Phi of weights is discussed in ยง3 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 (โˆž,1)-categories is discussed in

Other references cited in this page:

On โˆž \infty -colimits and Day convolution in the context of enriched โˆž \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 January 10, 2025 at 23:56:14. See the history of this page for a list of all contributions to it.