Equivalence of categories and colimits

Question

Consider functors $F,G:J\to Cat$ which admit colimits and $\eta:F\to G$ a natural transformation such that $\forall i\in J$ $\eta_i:F(i)\to G(i)$ is an equivalence of categories. Is it true that then the induced map from colim$_JF$ to colim$_JG$ is an equivalence of categories ?

Answer 1

Here is an example of a pushout in $\textbf{Grpd}$ (or $\textbf{Cat}$ – there is no difference with respect to colimits) that is not equivalence-invariant.

Consider the inclusion of the discrete groupoid $\{ 0, 1 \}$ into the "interval" $\mathbf{I} = \{ 0 \cong 1 \}$, which is the groupoid with two objects and a unique morphism between any two objects. Let $\mathbf{S}$ be the groupoid with two objects $0$ and $1$, with $\textrm{Hom}_\mathbf{S} (x, y) = \mathbb{Z}$ for any objects $x$ and $y$ in $\mathbf{S}$, and composition given by integer addition. Then we have the following pushout square in $\textbf{Grpd}$: $$\require{AMScd} \begin{CD} \{ 0, 1 \} @>>> \mathbf{I} \\ @VVV @VVV \\ \mathbf{I} @>>> \mathbf{S} \end{CD}$$ (Incidentally, the above is also a bicategorical pushout square.)

On the other hand, $\mathbf{I}$ is equivalent to the "point" $\{ \bullet \}$, and the following is also a pushout square in $\textbf{Grpd}$: $$\begin{CD} \{ 0, 1 \} @>>> \{ \bullet \} \\ @VVV @VVV \\ \{ \bullet \} @>>> \{ \bullet \} \end{CD}$$ Yet, $\mathbf{S}$ is not equivalent to $\{ \bullet \}$. This demonstrates that pushouts are not equivalence-invariant.

(The "reason" is that the second square is not a bicategorical pushout square. The bicategorical pushout of $\{ \bullet \} \leftarrow \{ 0, 1 \} \rightarrow \{ \bullet \}$ "is" $\mathbf{S}$, or if you insist on a model making the diagram strictly commute, the one-object groupoid $\mathbf{B} \mathbb{Z}$ with $\mathbb{Z}$ as the group of automorphisms of the unique object. Of course, $\mathbf{B} \mathbb{Z}$ is equivalent to $\mathbf{S}$.)