Colimits of internal categories

Abstract

We show that for an extensive $1$-category $\mathcal{E}$ with pullbacks and pullback stable coequalisers in which the forgetful functor $\mathcal{U}:\mathbf{Cat}(\mathcal{E}{)}_1\to \mathbf{Gph}(\mathcal{E})$ has left adjoint, the $2$-category $\mathbf{Cat}(\mathcal{E})$ of internal categories, functors and natural transformations has finite $2$-colimits. In addition, $\mathbf{Cat}(\mathcal{E})$ is extensive, has pullbacks and codescent coequalisers are stable under pullback along discrete Conduché fibrations. Moreover, we give converse results to this.

30 pages, version 5. Typos and a couple of small errors fixed. Version published in the Bulletin for the Belgian Mathematical Society--- Simon Stevin