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I'm trying to find examples where the following perspective on bicategories is developed.

We can define a 2-category as being enriched in Cat, where Cat is treated as a monoidal category using the Cartesian product. It seems like you can treat a bicategory as a weak enrichment, where the necessary associativity diagrams are weakened and replaced with the associator and unitor 2-cells.

This seems to make certain things much simpler, particularly if you wanted to work with a bicategory whose hom-categories were categories with structure and the composition and coherent 2-cells were to respect that structure. That also makes me think that something goes wrong when you take this approach, because I can't seem to find any references where this is developed.

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It is difficult to axiomatize a "weak enrichment" in general. You have an enrichment base $\mathcal V$, hom-objects $C(a,b)\in \mathcal V$, a composition morphism $C(b,c)\otimes C(a,b)\to C(a,c)$ in $\mathcal V$, then an associator isomorphism in $\mathcal V$...So $\mathcal V$ must be a 2-category! In fact this line of thought can be pushed through, see here. But such a $C$ is certainly no easier to define than a plain bicategory, any more than an enriched category is easier to define than a category. The weakness of the enrichment forces us to abandon the dimensional drop, in which an $n$-category is a category enriched in $n-1$-categories, and in particular makes this approach unmanageably complex for higher dimensions.

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    $\begingroup$ @AlexanderCampbell Yes, that was definitely tongue-in-cheek. Since you're here, do you happen to have a reference for an equivalence with a fully non-algebraic model? $\endgroup$
    – Kevin Carlson
    Commented Jun 19, 2019 at 0:55
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    $\begingroup$ @KevinCarlson In Lack & Paoli's paper 2-nerves for bicategores they prove a biequivalence between 2-categories of bicategories and Tamsamani's 2-categories (i.e. Cat-enriched Segal categories). In my draft paper A homotopy coherent cellular nerve for bicategories, I prove (among other things) Quillen equivalences between Lack's model category of bicategories, a Bousfield localisation of Ara's model structure for 2-quasi-categories, and Rezk's model structure for (2,2)-$\Theta$-spaces. $\endgroup$
    – Alexander Campbell
    Commented Jun 19, 2019 at 1:06
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    $\begingroup$ I'm also not quite sure what you mean by "difficult to define a weak enrichment". Defining an enriched bicategory is totally straightforward, and does indeed make things much easier when you want to talk about bicategories whose hom-categories have extra structure. Moreover, an ordinary bicategory is a bicategory weakly enriched over the strict 2-category Cat, so you don't actually need to have already defined an ordinary bicategory is before defining an enriched one. $\endgroup$
    – Mike Shulman
    Commented Jun 19, 2019 at 13:21
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    $\begingroup$ It's true that things get subtler in higher dimensions, as always, but the OP only asked about $n=2$. $\endgroup$
    – Mike Shulman
    Commented Jun 19, 2019 at 13:22
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    $\begingroup$ No, I think it's equally possible to define a bicategory in an explicit way with a collection of 0-cells, a collection of 1-cells, a collection of 2-cells, with various operations satisfying various axioms, without mentioning any hom-categories off the bat. $\endgroup$
    – Mike Shulman
    Commented Jun 19, 2019 at 20:42

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