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?
1

$\begingroup$ "injective on objects" is an evil condition. Don't you want something like $F(X)\cong F(Y)\implies X\cong Y$ or (stronger) "F reflects isomorphisms" ? $\endgroup$ – Johannes Hahn Mar 22 '14 at 18:56

3$\begingroup$ @JohannesHahn: "injective on objects" is evil, but so is "pushout"; if the question involves evil concepts, then the answer may do as well. $\endgroup$ – Neil Strickland Mar 22 '14 at 19:08

$\begingroup$ @NeilStrickland Why is a pushout evil? It is a limit... $\endgroup$ – Johannes Hahn Mar 22 '14 at 19:17

1$\begingroup$ @JohannesHahn: limits and colimits of categories are usually evil. That is why people consider 2(co)limits or homotopy (co)limits instead. $\endgroup$ – Neil Strickland Mar 22 '14 at 19:51

$\begingroup$ Do all these "evil" connotations apply to groupoids, and higher groupoids? I like colimits of groupoids and higher groupoids, and have used them for specific homotopy invariant calculations. The higher case led to the notion of nonabelian tensor product of groups, in work with Loday. Can someone please give me an example of an algebraic calculation using homotopy 2colimits? $\endgroup$ – Ronnie Brown Mar 23 '14 at 18:40
5
$\begingroup$
$\endgroup$
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.

$\begingroup$ I guess you are taking $B$ to be a oneobject category rather than a posetcategory? $\endgroup$ – Zhen Lin Mar 23 '14 at 12:21

$\begingroup$ Yes, one object category. You can take any monoid which is not contractible. $\endgroup$ – Werner Thumann Mar 23 '14 at 13:41