category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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 $f : G \to \mathbb{C}$ 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).
Let $V$ be a closed symmetric monoidal category with all small limits and colimits.
For $\mathcal{C}$ a $V$-enriched category, write $[\mathcal{C},V]$ for the $V$-enriched functor category to $V$, etc.
We discuss two equivalent ways of defining Day convolution
Let $(\mathcal{C}, \otimes, 1)$ be a small $V$-enriched monoidal category.
Then the Day convolution tensor product on $[\mathcal{C},V]$
is given by the following coend
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 $\mathcal{C}$ be a small $V$-monoidal category. Its external tensor product is
given by
i.e.
The Day convolution product (def. 1) of two functors is equivalently the left Kan extension of their external tensor product (def. 2) along the tensor product $\otimes_{\mathcal{C}}$: there is a natural isomorphism
Hence the adjunction unit is a natural transformation of the form
This perspective is highlighted in (MMSS 00, p. 60).
By prop. \ref{TopologicalLeftKanExtensionBCoend} we may compute the left Kan extension as the following coend:
Proposition 1 implies the following fact, which is the key for the identification of βfunctors with smash productβ.
The operation of Day convolution $\otimes_{Day}$ (def. 1) is universally characterized by the property that there are natural isomorphisms
where $\overline{\otimes}$ is the external product of def. 2.
The Day convolution can also be expressed in terms of profunctors. The tensor product $\otimes :\mathcal{C}\otimes \mathcal{C}\to \mathcal{C}$ induces a representable profunctor $\mathcal{C}(\otimes,1): \mathcal{C} ⇸ \mathcal{C}\otimes \mathcal{C}$. The above definition can be interpreted to say that if $X,Y\in [\mathcal{C},V]$ are regarded as profunctors $\mathcal{C} ⇸ I$, where $I$ is the unit $V$-category, then $X\otimes_{Day} Y$ is the composite of profunctors
A more βglobalβ way to say the same thing is to consider the βevaluationβ functor $[\mathcal{C},V] \otimes \mathcal{C} \to V$ to be a profunctor $E:\mathcal{C}⇸[\mathcal{C},V]^{op}$. Then the profunctor composite
is a functor $\mathcal{C}\otimes [\mathcal{C},V] \otimes [\mathcal{C},V] \to V$, which by exponential transpose gives a functor $[\mathcal{C},V] \otimes [\mathcal{C},V] \to [\mathcal{C},V]$; this is the Day convolution product.
The above description in terms of profunctors makes it clear that the construction only depends on the representable profunctor induced by $\otimes : \mathcal{C}\otimes \mathcal{C}\to \mathcal{C}$, i.e. on the underlying promonoidal category of $\mathcal{C}$. In the original article (Day 70), a stronger form of the convolution is discussed, in which $\mathcal{C}$ is assumed only to be a promonoidal category.
Let $V$ be a Benabou cosmos, and $\mathcal{C}$ a small $V$-enriched category.
There is an equivalence of categories between the category of pro-monoidal structures on $\mathcal{C}$ with strong pro-monoidal functors between them and the category of biclosed monoidal structures on $[\mathcal{C}^{op},V]$ with strong monoidal functors between them.
This is claimed without proof in (Day 70).
For $(\mathcal{C}, \otimes_{\mathcal{C}}, I)$ a small monoidal $V$-enriched category, the Day convolution product $\otimes_{Day}$ of def. \ref{DayConvolutionProduct} makes
a monoidal category with tensor unit $y(I)$ co-represented by the tensor unit $I$ of $\mathcal{C}$.
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 $\mathcal{C}$.
For $(\mathcal{C}, \otimes_{\mathcal{C}}, I)$ a small monoidal $V$-enriched category, the monoidal category with Day convolution $([\mathcal{C},V], \otimes_{Day}, y(I))$ from def. 3 is a closed monoidal category. Its internal hom $[-,-]_{Day}$ is given by the end
In analogy to the cartesian closed monoidal structure on presheaves we see that if the internal hom in $[\mathcal{C},V]$ exists at all, (with $[-,X]_{Day}$ right adjoint to $(-) \otimes_{Day} X$) then by the enriched Yoneda lemma and by the end-expression for the hom-objects in the enriched functor category $[\mathcal{C},V]$ it has to be given by
This exists, by the assumption that $\mathcal{C}$ is small and that $V$ has all small limits. Now to check that this really gives a right adjoint:
The Yoneda embedding constitutes a strong monoidal functor $(\mathcal{C},\otimes_{\mathcal{C}}, I) \hookrightarrow ([\mathcal{C},V], \otimes_{Day}, y(I))$.
That the tensor unit is respected is part of prop. 3. To see that the tensor product is respected, apply the co-Yoneda lemma twice to get the following natural isomorphism
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. 3 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. 6 are known as ring spectra and the lax monoidal functors in prop. 6 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 $\mathcal{C}$, write
for the full subcategory of the category of modules over $R$ on those that are free modules and moreover free on objects in $\mathcal{C}$ (under the Yoneda embedding). Hence the objects of $R Free_{\mathcal{C}}Mod$ are those of $\mathcal{C}$ and the hom-objects are
For $(\mathcal{C},\otimes, I)$ a small $V$-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 right 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. 6 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
Notice that 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 transformations into a single kind of the form
and subject to certain identifications.
Now observe that the hom-objects of $R Free_{\mathcal{C}}Mod$ (def. 3) have just this structure:
We claim that under this identification, composition in $R Free_{\mathcal{C}}Mod$ is given by
where
the first morphism is, in the integrand, the tensor product of
forming the tensor product of hom-objects of $\mathcal{C}$ with the identity of $c_5$
the monoidal functor incarnation $R(c_4) \otimes_V R(c_5)\longrightarrow R(c_4 \otimes_{\mathcal{C}} c_5 )$ of the monoid structure on $R$;
the second morphism is, in the integrand, given by composition in $\mathcal{C}$;
the last morphism is the morphism induced on coends by regarding extranaturality in $c_4$ and $c_5$ separately as a special case of extranaturality in $c_6 \coloneqq c_4 \otimes c_5$ (and then renaming).
It is fairly straightforward to see that, under the above identifications, functoriality under this composition is equivalently functoriality in $\mathcal{C}$ together with the action property over $R$.
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
The universal property of the Day convolution, in the sense of free monoidal cocompletion, is discussed 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