category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
Just as we can convolve functions $f : M \to \mathbb{C}$ where $M$ is a group, or more generally a monoid, we can convolve functors $f: M \to Set$ where $M$ is a monoidal category. So, for any monoidal category $M$, the functor category $Set^M$ becomes a monoidal category in its own right. The tensor product in $Set^M$ is called Day convolution, named after Brian Day.
We can generalize this idea by replacing Set with a more general cocomplete symmetric monoidal category $V$. The technical condition is that the tensor product $u \otimes v$ must preserve colimits in the separate arguments $u$ and $v$; that is, that the functors $u \otimes -$ and $- \otimes v$ must preserve colimits. This occurs when for instance $V$ is symmetric monoidal closed (so that these functors are left adjoints).
For $(C, \otimes)$ a monoidal category and $F, G : C^{op} \to Set$ two presheaves on $C$, their Day convolution product $F \star G$ is the presheaf given by the coend
Let $j : C \to PSh(C)$ be the Yoneda embedding.
With $I \in C$ the tensor unit of $C$, the presheaf $j(I)$ is a 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
…
In Day’s original paper, a stronger form of the Day 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.
…
Then the above 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.
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.
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 Girard’s “phase spaces”, or more generally from ternary frames, is essentially Day convolution for posets.
For (∞,1)-categories: