With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
The classical notion of an internal category in a category with pullbacks, can be generalized by replacing pullbacks with cotensor products of comodules in a monoidal category.
Applied to internal one-object groupoids this subsumes the notion of quantum groups.
George Janelidze calls this noncartesian internal category, because when the tensor product is the cartesian product then the notion reduces to the traditional of internal categories.
One typically starts with a monoidal category $M = (M, \otimes, 1)$ which is regular in the sense that it has equalizers which are preserved by the tensor product $\otimes$ on both sides. (The monoidal structure does not need be symmetric.)
There is a bicategory $Comod(M)$ of comonoids and bicomodules in $M$: this is a special case of the bicategory $Mod(K)$ of monads and bimodules in a bicategory $K$, where in this case $K = \mathbf{B} M^{op} = (\mathbf{B} M)^{co}$. Then an internal category in $M$ is a monad in $Comod(M)$.
There are two kind of morphisms of noncartesian internal categories: functors and cofunctors (which here are not the same as contravariant functors).
Because every set is canonically a comonoid with respect to the cartesian product, a comonoid in Set is just a set and a bicomodule is a span, and a monad in the bicategory of spans of sets is just a small category. More generally, an internal category in the above sense in any category with finite cartesian products (and equalizers, of course, hence a finitely complete category) is just an internal category in the usual sense.
The main historical reference:
Last revised on September 21, 2021 at 11:41:45. See the history of this page for a list of all contributions to it.