category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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).
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
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.
For $(\mathcal{C}, \otimes, I)$ a small monoidal $V$-enriched category, the Day convolution product on $[\mathcal{C},V]$ is the functor
given by the coend
We may think of this equivalently as a Kan extension:
For $(\mathcal{C}, \otimes)$ a monoidal $V$-enriched category, its external tensor product is the $V$-functor
given by
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
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 $F \colon \mathcal{D} \to V$ along some $p \colon \mathcal{D} \to \mathcal{E}$ is given by
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
Day convolution $\otimes_{Day}$, def. 1, is universally characterized by the property that there are natural isomorphisms
where $\tilde \otimes$ is the external product of def. 2.
By prop. 1 and since left Kan extension along any $f$ is the left adjoint to precomposition with $f$.
Write
for the $V$-Yoneda embedding, so that for $c\in \mathcal{C}$ any object, $y(c)$ is the corepresented functor $y(c)\colon c' \mapsto [c,c']$.
For $(\mathcal{C}, \otimes, I)$ a small monoidal $V$-enriched category, the Day convolution product $\otimes_{Day}$ of def. 1 makes
a closed monoidal category with tensor unit $y(I)$ co-represented by the tensor unit $I$ of $\mathcal{C}$.
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
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.
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).
Let $j \colon C \to PSh(C)$ be the Yoneda embedding.
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.
Using the co-Yoneda lemma on the two coends we have
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
$(PSh(C), \star, j(I))$ is a closed monoidal category;
the Yoneda embedding constitutes a strong monoidal functor $(C,\otimes, I) \hookrightarrow (PSh(C), \star, j(I))$.
In analogy to the cartesian closed monoidal structure on presheaves we see that if the internal hom in $PSh(C)$ exists at all, (with $[F,-]$ right adjoint to $(-) \star F$) then by the Yoneda lemma it has to be given by
β¦
Given any monoidal category then one may consider monoid objects and module objects inside it.
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 $\mathcal{C} \to V$:
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 $F \colon \mathcal{C} \to V$ is given by natural transformations
satisfying compatibility conditions. Under the natural isomorphism of corollary 1 these are identified with natural transformations
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.
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 βfunctors with smash productβ.
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
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
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
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.
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
These transformations have just the same structure as those of the enriched functoriality of $N$ of the form
Hence we may unify these two kinds of transformarmations in a single kind of the form
and subject to certain identifications.
By comparison with def. 3, this is just the form of the functoriality of an enriched functor over $R FreeMod^{op}$.
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
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
This uses in particular that for every object $c \in C$ the functor
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.
Further examples:
There is an obvious monoidal structure on the cube category. By Day convolution this induces a monoidal structure on cubical sets. This in turn induces a monoidal structure on strict omega-categories.
There is a monoidal structure on the augmented simplex category which by Day convolution induces a monoidal structure on the category of augmented simplicial sets, which by restriction induces the join operation on simplicial sets.
If $C$ is a large category in one universe, then its universe enlargement to a bigger universe can be given a closed monoidal structure via Day convolution.
The semantics of linear logic obtained from Girardβs βphase spacesβ, or more generally from ternary frames, is essentially Day convolution for posets.
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
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
(see also at functors with smash product).
Day convolution for (β,1)-categories is discussed in