We show finite 2-colimits exist in extensive 1-categories, indicating a rich structure of internal categories.
We show that for an extensive $1$-category E with pullback-stable coequalisers admitting free internal categories over internal graphs, the $2$-category Cat(E) of internal categories, functors and natural transformations has finite $2$-colimits. In addition, Cat(E) is extensive and codescent coequalisers are stable under pullback along discrete Conduché fibrations. Moreover, we give converse results to this.
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Hughes et al. (2026) studied this question.