Consider the pushout of a diagram $A\leftarrow B\rightarrow C$ of categories and assume that at least one of the arrows is an embedding, i.e. injective on objects and arrows. When applying the nerve functor to the pushout diagram, do we get a homotopy pushout in simplicial sets?



Pushout of categories along embeddings gives homotopy pushout?
This is wrong. Let $B=\mathbb{N}$, $A$ the cone over $B$ and $C$ the cocone over $B$. The pushout category is contractible, since it has an initial and a terminal object. But the homotopy pushout on the nerve level is the homotopy type of a $2$-sphere.


