nLab Day convolution

Context

Monoidal categories

monoidal categories

category theory

Contents

Idea

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 $f : G \to \mathbb{C}$ whenever $G$ carries the structure of a group, or more generally just the structure of a monoid, so there is convolution of functors $f \colon \mathcal{G} \to Set$ whenever the category $\mathcal{G}$ carries the structure of a monoidal category.

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

Definition

For monoidal categories

For $(C, \otimes)$ a small monoidal category and for $F, G \colon C^{op} \to Set$ two presheaves on $C$, their Day convolution product $F \star G$ is the presheaf given by the coend

$F \star G \coloneqq \int^{c,d \in C} F(c) \times G(d) \times Hom_C(-, c \otimes d) \,.$

More generally:

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

Definition

For $(\mathcal{C}, \otimes, I)$ a small monoidal $V$-enriched category, the Day convolution product on $[\mathcal{C},V]$ is the functor

$\otimes_{Day} \;\colon\; [\mathcal{C},V] \times [\mathcal{C},V] \longrightarrow [\mathcal{C},V]$

given by the coend

$X \otimes_{Day} Y \;\colon\; c \mapsto \int^{c_1, c_2 \in \mathcal{C}} X(c_1) \otimes Y(c_2) \otimes [c_1 \otimes c_2, c] \,.$

We may think of this equivalently as a Kan extension:

Definition

For $(\mathcal{C}, \otimes)$ a monoidal $V$-enriched category, its external tensor product is the $V$-functor

$\tilde \otimes \coloneqq \;\colon\; [\mathcal{C},V] \times [\mathcal{C},V] \longrightarrow [\mathcal{C}\times \mathcal{C}, V]$

given by

$X \tilde \otimes Y \coloneqq \otimes \circ (X,Y) \,.$
Proposition

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

$X \otimes_{Day} Y \simeq Lan_\otimes (X \tilde \otimes Y) \,.$

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

Proof

The general formula for pointwise Kan extension via coends (here) says that left Kan extension of any $F \colon \mathcal{D} \to V$ along some $p \colon \mathcal{D} \to \mathcal{E}$ is given by

$Lan_p F \;\colon\; e \mapsto \int^{d\in \mathcal{D}} \mathcal{E}(p(d), e) \otimes F(d) \,.$

In our case

• $\mathcal{D} = \mathcal{C}\times \mathcal{C}$;

• $\mathcal{E} = \mathcal{C}$;

• $p = \otimes$;

• $F = X \tilde \otimes X$

and hence the general formula here becomes

\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} \,.
Corollary

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

$[\mathcal{C},V](X \otimes_{Day} Y, Z) \simeq [\mathcal{C}\times \mathcal{C},V](X \tilde{\otimes} Y, Z \circ \otimes) \,,$

where $\tilde \otimes$ is the external product of def. 2.

Proof

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

Write

$y \colon \mathcal{C}^{op} \longrightarrow [\mathcal{C}, V]$

for the $V$-Yoneda embedding, so that for $c\in \mathcal{C}$ any object, $y(c)$ is the corepresented functor $y(c)\colon c' \mapsto [c,c']$.

Proposition

For $(\mathcal{C}, \otimes, I)$ a small monoidal $V$-enriched category, the Day convolution product $\otimes_{Day}$ of def. 1 makes

$( [\mathcal{C}, V], \otimes, y(I))$

a closed monoidal category with tensor unit $y(I)$ co-represented by the tensor unit $I$ of $\mathcal{C}$.

Proof

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

\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 $A$ is assumed only to be a promonoidal category.

Let $V$ be a Benabou cosmos, and $A$ a small $V$-enriched category.

Proposition

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

This is claimed without proof in (Day 70).

Properties

Basic properties

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

Lemma

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

Proof

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

\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} \,.
Proposition

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

Then

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

2. the Yoneda embedding constitutes a strong monoidal functor $(C,\otimes, I) \hookrightarrow (PSh(C), \star, j(I))$.

Proof

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

\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.

Proposition

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

$Mon([\mathcal{C},V], \otimes_{Day}, y(I)) \simeq MonFunc(\mathcal{C},V)$
$CMon([\mathcal{C},V], \otimes_{Day}, y(I)) \simeq SymMonFunc(\mathcal{C},V) \,.$

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.

Proof

A lax monoidal functor $F \colon \mathcal{C} \to V$ is given by natural transformations

$I_V \longrightarrow F(I_{\mathcal{C}})$
$F(c_1) \otimes_V F(c_2) \longrightarrow F(c_1 \otimes_{\mathcal{C}} c_2)$

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

$y(I) \to F$
$F \otimes_{Day} F \longrightarrow F$

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

Similarly for module objects and modules over monoidal functors.

Example

In the case that $V$ is pointed topological spaces or pointed simplicial sets equipped with the smash product of pointed objects and that $\mathcal{C}$ 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 “functors with smash product”.

Modules with respect to Day convolution

Definition

For $(\mathcal{C},\otimes, I)$ a small monoidal $V$-enriched category, and for $R \in Mon([\mathcal{C}, V],\otimes_{Day})$ a monoid object with respect to Day convolution over $R$, write

$R FreeMod \hookrightarrow R Mod$

for the full subcategory of the category of modules over $R$ on those that are free modules. Hence the objects of $R FreeMod$ are those of $\mathcal{C}$ and the hom-objects are

\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} \,.
Proposition

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

$R Mod \simeq [R FreeMod^{op}, V]$

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

Proof idea

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

$N(c_1) \otimes R(c_2) \longrightarrow N(c_1 \otimes c_2) \,.$

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

$N(c_1) \otimes \mathcal{C}(c_1,c_2) \longrightarrow N(c_2) \,.$

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

$N(c_1) \otimes ( \mathcal{C}(c_1, c_3) \otimes R(c_4) \longrightarrow N(c_2) \;\;\; for c_2 = c_3 \otimes c_4$

and subject to certain identifications.

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

Examples

Example

Let $C$ be a discrete category over a set, which is hence a monoid (for instance a group) with product $\cdot$.

Then the Day convolution product is

$F \star G : e \mapsto \oplus_{c \cdot d = e} F(c) \times G(d) \,.$

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

\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 $c \in C$ the functor

$Hom_C(c,-) = \delta_c$

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

$Hom_C(c,d) = \left\{ \array{ * & if c = d \\ \emptyset & if c \neq d } \right.$

Further examples:

Example

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).

References

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)