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

Question

$\DeclareMathOperator\Cat{Cat}$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.

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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 $X \amalg A'$.

Answer 1

Martina Rovelli and I have indeed thought about the case of Dwyer morphisms before. Originally, we were also trying to employ Barwick-Kan, but I think there is the following subtle point there.

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 ${a<b}$ 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 $$ (a<b, \mathrm{id}) \times (0<1, \mathrm{max}) \to (a<b, \mathrm{id}) $$ 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.

Edit May 2022: Instead, we believe to have found an explicit proof using anodyne extensions now The question turned out to be far more subtle. As we were trying to use it in a joint work with Philip Hackney and Emily Riehl (https://arxiv.org/abs/2106.03660), the referee was pointing out that our proof did not work in a full generality. We have been thinking about these pushouts for a while since then, and we still believe that the statement holds true, although the proof is much more involved now (https://arxiv.org/abs/2205.02353).