# Does categorical equivalence commute with colimit?

Lurie defines $$\mathfrak{C}:Set_\Delta \rightarrow Cat_\Delta$$ the functor from simplicial sets to simplicially enriched categories. And defines:

Simplicial sets $$X$$ are considered to be categorically equivalent to $$Y$$, if $$\mathfrak{C}[X]$$ and $$\mathfrak{C}[Y]$$ is an equivalence of $$H$$ enriched category.

Here $$H$$ denotes the homotopy category of simplicial sets. Does this notion commute with colimit? I.e. if $$\{X_\alpha \rightarrow Y_\alpha\}$$ are all categorically equivalent, then the induced map on their colimit is categorically equivalent.

This may seem like a direct consequence of the fact that $$\mathfrak{C}$$ is a left adjoint : but only of the underlying category. So this doesn't seem to apply.

My main concern is how Lurie shows that class of "covariant equivalences" is perfect. Or that of push out preserves covariant equivalences. (which is a special type of categorical equivalence). These are 2.1.4.6, and 2.1.4.7.

• As I suspected, there is already an issue in that even for ordinary categories, a colimit of equivalences is not an equivalence. For instance, take $\tilde{G}$ to be the universal covering groupoid of the one-object groupoid $BG$ for some group $G$. Then we have a functor $BG\to \mathbf{Cat}$ which sends the object to $\tilde{G}$ and the maps act in the obvious way; and there is a map from this functor to the constant $*$ functor, which is a pointwise equivalence, but on colimits this is $BG\to *$ : not an equivalence. So I think it must be related to the specific colimits in question Oct 11, 2019 at 12:17
• Incidentally, a covariant equivalence need not be a categorical equivalence. If I may give some unasked for advice: it is probably not possible to read Lurie if you aren't already quite experienced in category theory and abstract homotopy theory. I'd encourage you to incorporate some reading from Cisinski's new book on higher categories for a much more compact and streamlined approach to the subject. Riehl and Verity's work also makes many notions from Lurie far more intuitive. Oct 12, 2019 at 17:10

## 1 Answer

Categorical equivalences are the weak equivalences in a model structure. Such a class is essentially never closed under colimits. It is closed under homotopy colimits, and it is an important technical activity to describe classes of ordinary colimits which coincide with the corresponding homotopy colimits. A simple example is that a pushout of two cofibrations between cofibrant objects is always a homotopy pushout.

• Thanks a lot! If you could recall, what sections of Cisinski, and Riehl and Verity's work (you mentioned above) should I look into? Oct 12, 2019 at 20:26
• Though, for Lemma 2.1.4.6 in Lurie's, we do not have any model structures (including Joyal), how does he deduce the pushout of an inner anodyne inclusion would be a categorical equivalence? Oct 12, 2019 at 20:40
• @BryanShih The pushout of an inner anodyne map is still inner anodyne, and in particular a categorical equivalence, because inner anodyne maps are the left half of a weak factorization system. Regarding the other references, you seem to be learning $\infty$-category theory, and I was suggesting those as better places to start than Lurie. You should ready sections having to do with whatever you're trying to learn. Oct 12, 2019 at 21:30
• Thanks so much. Yes I am learning infty category theory. Last question, I don't recall Lurie has proven anywhere that inner ano => cat equiv? Oct 12, 2019 at 22:33
• This is a fundamental part of the construction of the Joyal model structure, so I don't have a more exact reference for the result than to look at any development thereof. Oct 13, 2019 at 2:03