English

Colimits of internal categories

Category Theory 2025-11-07 v4

Abstract

We show that for an extensive 11-category E\mathcal{E} with pullbacks and pullback stable coequalisers in which the forgetful functor U:Cat(E)1Gph(E)\mathcal{U}: \mathbf{Cat}(\mathcal{E})_1 \to \mathbf{Gph}(\mathcal{E}) has left adjoint, the 22-category Cat(E)\mathbf{Cat}(\mathcal{E}) of internal categories, functors and natural transformations has finite 22-colimits. In addition, Cat(E)\mathbf{Cat}(\mathcal{E}) is extensive, has pullbacks and codescent coequalisers are stable under pullback along discrete Conduch\'{e} fibrations. Moreover, we give converse results to this.

Keywords

Cite

@article{arxiv.2501.17769,
  title  = {Colimits of internal categories},
  author = {Calum Hughes and Adrian Miranda},
  journal= {arXiv preprint arXiv:2501.17769},
  year   = {2025}
}

Comments

29 pages, comments welcome. V4, added important reference and alternative proof

R2 v1 2026-06-28T21:24:06.884Z