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.

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 $\otimes$ 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.

References

The main historical reference is Marcelo Aguiar’s 1997 Cornell thesis (pdf), under the guidance of S. Chase. George Janelidze calls such generalization noncartesian internal category, because if the tensor product is the cartesian product the notion reduces to the traditional internal category.

David Roberts: I think Ross Street and the other Australian category theorists call this a quantum category - I did go to a talk once, but my notes are elsewhere.

David Roberts: It has been pointed out to me by Jeff Egger that this is incorrect, in that quantum categories as defined by the Australian school are generalisations of bimonoids/bialgebras, whereas internal categories generalise monoids (via horizontal categorification). Hmm, I wasn’t paying attention in that talk as much as I thought I was.

Last revised on April 26, 2018 at 16:10:22.
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