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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.

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The 2-category of homotopical categories has morphisms the weak equivalence-preserving functors and all natural transformations. To find a 2-adjunction you need to say that the categories of functors, respectively, homotopical functors, on either side coincide (at least up to equivalence, but generally, and in this case, even isomorphism.) Indeed, every functor from the "discrete," or minimal, homotopical structure on a category is homotopical, so that the minimal structure is left 2-adjoint to the forgetful 2-functor, and dually for the "indiscrete," maximal, structure. All the 2-s are mostly window dressing in this case, since all functors in sight are "locally fully faithful," that is, induce bijections on 2-morphisms.

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