# A question on the stability of $\operatorname{Cat}$ in $\operatorname{Cat}_\infty$

$$\DeclareMathOperator\Cat{Cat}$$Suppose we have a span in $$\Cat$$ $$\require{AMScd} \begin{CD} A @> G>> X \\ @VVFV \\ B \end{CD}$$ We can view this as a span in $$\Cat_\infty$$. What useful conditions can we impose to ensure the pushout is still a 1-category?

As a specific example, is either of the following conditions sufficient?

• $$F$$ is injective on objects and arrows
• $$F$$ is injective on objects and arrows, and every isomorphism of the form $$F(X) \cong F(Y)$$ is in the image of $$F : A(X,Y) \to B(F(X), F(Y))$$
• Both $$F$$ and $$G$$ satisfy the property above

Remark: This second proposition is the property $$F$$ is a monomorphism in $$Cat_\infty$$ together with the proposition that $$F$$ is an isocofibration in $$Cat$$ so that if the pushout in $$Cat_\infty$$ is a 1-category, it's given by taking the the pushout in $$Cat$$. For the question as asked we can drop the isocofibration condition.

Being injective on objects is not sufficient, since we have a pushout square in $$\Cat_\infty$$ $$\require{AMScd} \begin{CD} S^1 @>>> 1 \\ @VVV @VVV \\ 1 @>>> S^2 \end{CD}$$ and $$S^1 \to 1$$ can be given by a functor between 1-object categories.

As @AchimKrause points out in the comments, injective on objects and arrows is not sufficient either.

An example of a sufficient condition that does work (but is too restrictive for me) is if $$A$$, $$B$$, $$X$$ are all free categories and $$F$$ is obtained from an inclusion of the generating graphs.

In this case, we can compute this in the Bergner model structure on simplicially enriched categories. The map $$A \to B$$, when viewed in simplicial categories, is a cofibration between cofibrant objects (it is $$\mathfrak{C}[-]$$ applied to the inclusion of the generating graphs viewed as simplicial sets), and $$X$$ is cofibrant as well, and thus the pushout (which is obviously a 1-category) is a homotopy pushout, and thus computes the pushout in $$\Cat_\infty$$.

Another case that works, as described in the comments, is when $$A$$ and $$B$$ are groupoids and $$F$$ is a monomorphism in $$Cat_\infty$$; in this case, $$B \cong A \amalg A'$$, and thus the pushout in $$Cat_\infty$$ is $$X \amalg A'$$.

• If F is an equivalence then I imagine the pushout is still a 1-category, but this is too restrictive. Oct 25, 2020 at 2:07
• Note that being injective on objects or on morphisms is not a condition stable under equivalence, and so it is unlikely to be of help. Maybe asking for one leg to be a replete inclusion will work, but I'm honestly skeptical there's a sensible condition for this. Oct 25, 2020 at 6:39
• For groupoids, something akin to your first condition should indeed work, namely that the functors are injective on $\pi_0$ and faithful. This should reduce to the corresponding statement for classifying spaces of groups. For general categories, condition 1 is definitely not enough. You can consider a pushout where the upper right corner is an arbitrary category, the upper left corner is a disjoint union of multiple $\Delta^1$, and the l.l. corner is the localisation of the u.l. corner. Then the pushout is a localisation of the category you started with, and not generally a $1$-category. Oct 25, 2020 at 7:37
• @DenisNardin I was thinking about computing the pushout in the canonical model structure on Cat which is why I was focusing on the "injective on objects" condition. IIRC, the second of the two conditions I list the proposition "$F$ is a monomorphism in $Cat_\infty$" restricted to the case $F$ is injective on objects. So the equivalence-respecting condition would be to consider $F$ being a monomorphism in $Cat_\infty$. Oct 25, 2020 at 9:04
• @PushoutOfCategories "F is a monomorphism in $Cat_∞$" is precisely the condition of being (equivalent to) the inclusion of a replete subcategory. Unfortunately Achim's example shows that this is of course not nearly enough (every localization can be realized as a pushout along a replete subcategory!). Oct 25, 2020 at 9:09

You have to specify how to look at a category as a particular relative category, and the natural way is to assign to a category $$\mathcal{A}$$ the pair $$(\mathcal{A}, \mathrm{iso}(\mathcal{A}))$$. However, I think this functor does not take Dwyer morphisms in categories to Dwyer morphisms in relative categories. Indeed, I think that already the inclusion of the object $$a$$ into the category $${a is not a Dwyer morphism of relative categories. It seems that checking Barwick-Kan §§3.2-3.5 shows that you would need your homotopy to be a relative functor $$(a which maps $$b0$$ to $$a$$ and $$b1$$ to $$b$$, so that the weak equivalence $$b0\to b1$$ would map to a map which is not a weak equivalence.