The classical notion of an internal category in a category with pullbacks, can be generalized by replacing pullbacks with cotensor products in a monoidal category.
One typically starts with a monoidal category which is regular in the sense that it has equalizers which are preserved under the tensor products from the left and from the right. The monoidal category does NOT need be symmetric. For simplicity we will treat the monoidal category as a strict one.
The internal category in ( is a tuple…
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The main historical reference is Marcelo Aguiar’s 1997 Cornell thesis (pdf), under the guidance of S. Chase.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 the notion reduces to the traditional internal category.
There are two kind of morphisms of noncartesian internal categories: functors and cofunctors.
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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.