$\DeclareMathOperator\Cat{Cat}$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 $X \amalg A'$.

7more comments