# 2-adjunctions between categories and homotopical categories

A homotopical category is a category $\mathsf{C}$ equipped with a subcategory $\mathrm{core}(\mathsf{C}) \hookrightarrow \mathsf{W} \hookrightarrow \mathsf{C}$ such that the arrows in $\mathsf{W}$ satisfy the 2-out-of-6 rule. The arrows in $\mathsf{W}$ are called weak equivalences, and a homotopical functor between homotopical categories is a functor which preserves the weak equivalences in the obvious way. This gives a category $\mathsf{HomoCat}$ of homotopical categories and homotopical functors. This should naturally inherit the structure of a 2-category from $\mathsf{Cat}$.

By taking the localization with respect to the weak equivalences, we get a functor $\mathrm{Ho}: \mathsf{HomoCat} \to \mathsf{Cat}$.

Given any category $\mathsf{D}$, we can take the minimal weak equivalences $\mathsf{W}= \mathrm{core}(\mathsf{D})$, and this gives us a fully faithful functor $\mathsf{Cat} \hookrightarrow \mathsf{HomoCat}$ that makes each category into a minimal homotopical category.

We can also define a fully faithful "maximal" homotopical category functor $\mathsf{Cat} \hookrightarrow \mathsf{HomoCat}$ which makes every morphism in a category $\mathsf{D}$ into a weak equivalence: $\mathsf{W}= \mathsf{D}$.

Since these are 2-categories, are these functors part of 2-adjunctions? I'm not very experienced with higher categories so I'm not completely sure what exactly a 2-adjunction would entail. Anyway, here's my conjecture: by analogy with the discrete, forgetful, and codiscrete adjunctions in topology, I would guess that $\mathrm{Ho}$ is right adjoint to the "minimal" functor and left adjoint to the "maximal" functor.

Is my conjecture correct? I am just starting to read Emily Riehl's book on categorical homotopy theory, and I was thinking about this after seeing the definition of homotopical categories in chapter 2.