nLab internal category in a monoidal category

Idea

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.

Definition

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).

Examples

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.

Related concepts

• quantum group

References

The main historical reference:

• Marcelo Aguiar, Internal categories and quantum groups, PhD thesis, Cornell 1997 (pdf, pdf)