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$$\def\Cat{\mathbf{Cat}} \def\qCat{\mathbf{qCat}} \def\Catinf{{\mathcal{C}at_\infty}} \def\Catone{{\mathcal{C}at_1}} \def\cC{\mathcal{C}}$$ Let $\Cat$ be the 1-category of small categories, $\qCat$ be the 1-category of small quasi-categories, $\Catinf$ be the $\infty$-category of small $\infty$-categories, and $\Catone$ be the $\infty$-category of small $1$-categories.

A typical construction of the "homotopy category" operation on $\infty$-categories looks starts off like the following construction on $\qCat$:

  • Take the objects of $\mathrm{h}(X)$ to be $X_0$
  • Define the hom-sets to be ...

One then shows that this defines a functor $\qCat \to \Cat$ that preserves equivalences, and thus induces a functor $\Catinf \to \Catone$.

However, this does not obviously induce a $\Cat$-valued functor, since the choice of objects was not made in an equivalence-preserving way. So, I ask

Question: Is there a "homotopy category" functor $\Catinf \to \Cat$?

To better emphasize the problem, consider another approach. the inclusion $\Catone \to \Catinf$ is an accessible, limit preserving functor between presentable $\infty$-categories, and so it has a left adjoint which is the aforementioned functor.

However, $\Cat \to \Catinf$ does not preserve limits, so this method does not work to produce a functor $\Catinf \to \Cat$.

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    $\begingroup$ Ignoring some (probably serious) set theoretic problems, the homotopy category functor can be defined, I suppose, as the right adjoint to the inclusion of categories in infinity categories given by the Nerve construction. This is because a functor from an infinity category to a 1-category factors canonically through the homotopy category. is this correct? $\endgroup$ – S. carmeli Dec 18 '19 at 19:59
  • $\begingroup$ @S.carmeli It's $\mathcal{C}at_1$ that embeds in $\mathcal{C}at_\infty$. The nerve construction, as a functor of $\infty$-categories $\mathbf{Cat} \to \mathcal{C}at_\infty$, doesn't preserve limits, so it can't have a left adjoint. For example, if $\mathbf{E}$ is the walking isomorphism, consider the cospan of the two points: $\mathbf{1} \to \mathbf{E} \leftarrow \mathbf{1}$. The pullback in $\mathbf{Cat}$ is the empty category, but the pullback in $\mathcal{C}at_{\infty}$ of their nerves is $\mathbf{1}$. $\endgroup$ – Hocat Dec 18 '19 at 21:26
  • $\begingroup$ Oh I see I swapped the notation in my mind sorry... $\endgroup$ – S. carmeli Dec 18 '19 at 21:57
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So by this standard, I don't think that the identity functor $Cat_1 \to Cat_1$ satisfies your desired constraints. But perhaps I should elaborate.

What you are describing might be the following. If $W$ is the class of equivalences in $Cat_\infty$, then you have shown that "taking the homotopy category" does not define a functor from $W^{-1} Cat_\infty$, the 1-category obtained by inverting equivalences, to the $1$-category $Cat_1$ of categories and functors: if it did, then it would take equivalences of quasicategories to isomorphisms of categories. The category $Cat_1$ embeds in $Cat_\infty$; a functor becomes an equivalence in $Cat_\infty$ if and only if it was an equivalence in $Cat_1$; the "homotopy category" functor is basically the identity on $Cat_1$; and we also have the result that the identity doesn't extend to a functor from the 1-category $V^{-1} Cat_1$, obtained by inverting equivalences, to the 1-category $Cat_1$.

In a sense, though, these 1-category localizations are ignoring higher structure. Namely, $V^{-1} Cat_1$ is the 1-category of categories and isomorphism classes of functors; similarly $W^{-1} Cat_\infty$ is the 1-category of quasicategories and equivalence classes of functors. We do have a functor $W^{-1} Cat_\infty \to V^{-1} Cat_1$: equivalent quasicategories have equivalent homotopy categories.

This is ... possibly? ... fixable. If we have access to some kind of global choice operator for proper classes, then we can construct a functor $sk: Cat_1 \to Cat_1$ that takes a category $C$ and sends it to a skeleton $sk(C) \subset C$, and construct a natural equivalence from $sk$ to the identity functor. (To do this, we need to choose one representative for every isomorphism class of object in every category $C$ simultaneously.) If we do that, then maybe we can compose the "homotopy category" functor with $sk$ and obtain a functor that does factor through $W^{-1} Cat_\infty$, because $sk$ factors through $V^{-1} Cat_1$. But I'm concerned that this is trying to provide a section of $Cat_1 \to V^{-1} Cat_1$ and I don't think that exists.

However, there are reasons why one might object to this. The idea that every category should be replaced by its skeleton, so that equivalent categories are isomorphic, requires hard set-theoretic theorems or is impossible. It is also, in some sense, solving a fake problem. We already have a well-established foundation for how to deal with equivalences of categories. Except in special cases, we are often not as interested in isomorphisms of categories. $Cat_1$, as a $1$-category, doesn't necessarily reflect our interests.

I would instead contend that the "homotopy" category functor $Cat_\infty \to Cat_1$ should at least reflect the 2-categorical nature of $Cat_1$. It takes quasicategories to categories and functors to functors, but it takes natural equivalences of functors to natural isomorphisms of functors. This means that equivalent quasicategories have equivalent homotopy categories, rather than isomorphic ones, and it seems likely that we shouldn't expect more.

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    $\begingroup$ For any choice of skeleton of the category $x\cong y$, there is a functor from $\{*\}$ not factoring through it on the nose. So you can’t make $sk$ into a 1-functor with a strictly natural transformation to the identity. $\endgroup$ – Kevin Carlson Dec 19 '19 at 5:37
  • $\begingroup$ To put it back in the nomenclature I was using, you're recounting the reflective embedding of Cat in qCat (or your favorite presentation) and that it induces the reflective embedding $\mathcal{C}at_1 \subseteq \mathcal{C}at_\infty$. I too am concerned whether $\mathbf{Cat} \to \mathcal{C}at_1$ has a section -- I think my question reduces to this -- but can't convince myself one does not exist. $\endgroup$ – Hocat Dec 21 '19 at 12:22
  • $\begingroup$ I agree that the $\mathcal{C}at_1$-valued homotopy category functor is the more natural notion, but the ambient context of my question is about highlighting the role that objects implicitly play in the theory; maybe the line of thought I'm going down could be described as taking the relative $\infty$-category of precategories (i.e. Segal spaces) and Dwyer-Kan equivalences as the correct analog of $\mathbf{Cat}$ and equivalence functors. $\endgroup$ – Hocat Dec 21 '19 at 12:28
  • $\begingroup$ Admittedly, a big reason for pursuing this line of thought was precisely the conscious acknowledgement that the canonical functor does not make Cat is a subcategory of $\mathcal{C}at_\infty$. (although it's not the entire reason; I'm also interested in the boundaries between 1-category theory and $\infty$-category theory and passing back and forth between them) $\endgroup$ – Hocat Dec 21 '19 at 12:32
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    $\begingroup$ @Hocat Here is a comment, then. We have the category $BD_{10}$, the one-object category with endomorphisms the dihedral group of size 10. In $\mathbf{Cat}$ the group of $BD_{10}$ is $Aut(D_{10}) = \Bbb F_5 \ltimes \Bbb F_5^\times$; in the homotopy category $h(Cat_1)$ the automorphism group is $Out(D_{10}) \cong \Bbb Z/2$. The functor $\mathbf{Cat} \to h(Cat_1)$ can't split because the map $Aut(D_{10}) \to Out(D_{10})$ doesn't split. (I learned this from Tom Goodwillie.) $\endgroup$ – Tyler Lawson Dec 24 '19 at 4:44

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