Colimits of internal categories
Calum Hughes, Adrian Miranda
TL;DR
The paper identifies conditions on an extensive base category $\mathcal{E}$ with pullbacks and pullback-stable coequalisers under which the $2$-category $\mathbf{Cat}(\mathcal{E})$ has finite $2$-colimits. It develops a constructive approach that reduces the problem to coproducts, copowers by $\mathbf{2}$, and coequalisers, and then builds coequalisers via a two-step process: first coequalisers that agree on objects, then coequalisers of arbitrary pairs, using free internal categories and coequifiers when available. A key contribution is a converse characterization: the existence of finite $2$-colimits in $\mathbf{Cat}(\mathcal{E})$ is equivalent to $\mathcal{E}$ being extensive with pullback-stable coequalisers and the forgetful functor $\mathcal{U}: \mathbf{Cat}(\mathcal{E})_1 \to \mathbf{Gph}(\mathcal{E})$ possessing a left adjoint, formulated through codescent theory. The paper provides concrete examples (e.g., elementary toposes with natural numbers objects, list-arithmetic pretoposes, locos) and outlines future work on $2$-toposes and internal model structures, highlighting the broad applicability of the construction. All mathematical notation is presented with proper $\LaTeX$-style delimiters.
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.