Internal Bass-Serre Theory for Tree Quotients

We develop an internal-category-theoretic framework for Bass-Serre theory, which recovers parts of classical and profinite Bass-Serre theories for tree quotients. More precisely, we show that in what we call a Bass-Serre category, an (internal) group object acting on an (internal) graph satisfying some assumptions can be recovered as the fundamental group of the associated graph of groups acting on the standard graph. Examples of Bass-Serre categories include the category of sets, the category of profinite spaces, and any Grothendieck topos.