# How do the various homotopy 2-categories compare?

There are various models of $$\infty$$-categories floating around, so there are as many models of the associated homotopy 1- and 2-categories. Because the relations between the former are worked out in great detail, it feels like the relations between the latter should be well-understood as well. But since I was unable to find much about the 2-dimensional case I'd like to ask for references here.

Specifically I am interested in the following questions.

1. Given a Kan-enriched category $$\mathcal{C}$$ we can apply the homotopy coherent nerve to get a quasicategory $$N^\Delta(\cal{C})$$. We can then apply [HTT Prop. 2.3.4.12] to pass to a simplicial set $$h_2(N^\Delta(\cal{C}))$$, which Lurie calls the underlying 2-category. On the other hand we can take $$\cal{C}$$ and apply hom-wise the homotopy-category functor and obtain a Grpd-enriched category, say $$H_2(\cal{C})$$. We can now apply the Duskin-nerve to get another simplicial set $$N^D(H_2(\cal{C}))$$, which ought to be a 2-category in the sense of HTT. Are both sSets equivalent in an appropriate sense (I guess it means Joyal-equivalent)?

Even more important to me is the following variation:

1. Note that $$H_2(\cal{C})$$ is a Grpd-enriched hence a strict (2,1)-category. In the light of Kerodon Rem. 2.3.6 it seems like the Duskin-nerve has a left adjoint $$|-|^D:\mathsf{sSet} \rightarrow (2,1)\mathsf{Cat}_{str}$$. So are the strict (2,1)-categories $$H_2(\cal{C})$$ and $$|h_2(N^\Delta(\cal{C}))|^D$$ equivalent as strict (2,1)-categories or at least as weak (2,1)-categories?

Out of curiosity I'd like to add a third variation on the question:

1. Using the Cordier-realization $$|-|^C:\mathsf{sSet} \rightarrow \mathsf{sSet}\text{-}\mathsf{Cat}$$ and the fibrant replacement of simplicial categories $$R:\mathsf{sSet}\text{-}\mathsf{Cat} \rightarrow \mathsf{Kan}\text{-}\mathsf{Cat}$$ we can turn a quasicategory $$C$$ into a Kan-enriched category $$R(|C|^C)$$. So how are $$|h_2(C)|^D$$ and $$H_2(R(|C|^C))$$ related?

Thank you very much for your time and considerations!

1. The simplicial sets $$h_2(N^\Delta(\mathcal{C}))$$ and $$N^D(H_2(\mathcal{C}))$$ are isomorphic. To prove this, observe that the universal property of $$h_2(N^\Delta(\mathcal{C}))$$ applied to the image under $$N^{\Delta}$$ of the quotient simplicial functor $$\mathcal{C} \to H_2(\mathcal{C})$$ yields a map of simplicial sets $$h_2(N^\Delta(\mathcal{C})) \to N^D(H_2(\mathcal{C}))$$, which one can easily check is an isomorphism of simplicial sets (it suffices to check that it's a bijection on 0-, 1-, and 2-simplices).
2. The strict (2,1)-categories $$|h_2(N^\Delta(\mathcal{C}))|^{\mathcal{D}}$$ and $$H_2(\mathcal{C})$$ are biequivalent; in fact, there is a strict 2-functor $$|h_2(N^\Delta(\mathcal{C}))|^{\mathcal{D}} \to H_2(\mathcal{C})$$ which is a bijective-on-objects biequivalence. To prove this, note that the composite of the Duskin nerve functor (for strict (2,1)-categories) with its left adjoint sends a strict (2,1)-category $$\mathcal{A}$$ to its "normal pseudofunctor classifier" $$Q\mathcal{A}$$, which is a strict (2,1)-category with the universal property that strict 2-functors $$Q\mathcal{A} \to \mathcal{B}$$ are in natural bijection with normal pseudofunctors $$\mathcal{A} \to \mathcal{B}$$, for $$\mathcal{B}$$ a strict (2,1)-category. Moreover, by this universal property, there is a "counit" strict 2-functor $$Q\mathcal{A} \to \mathcal{A}$$ which one can show is bijective on objects and an equivalence on hom-categories, and hence a biequivalence.
3. The strict (2,1)-categories $$|h_2(C)|^D$$ and $$H_2(R(|C|^C))$$ are isomorphic (if you make a good choice of $$R$$, e.g. change-of-base along $$Ex^\infty$$). Indeed, the two functors $$|h_2(-)|^D$$ and $$H_2(|-|^C)$$ are naturally isomorphic, and the functor $$H_2$$ sends the "unit" map $$\mathcal{E} \to R(\mathcal{E})$$ to an isomorphism (for a good choice of $$R$$ as above).