# Weak enrichment and bicategories

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.

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.
• @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. Jun 19 '19 at 1:06
• It's true that things get subtler in higher dimensions, as always, but the OP only asked about $n=2$. Jun 19 '19 at 13:22