Categorical Bass-Serre Theory for Tree Quotients

Abstract

We develop a categorical 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, a group object acting on a 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.

28 pages, v2: removed the previous 2.22, added Appendix B, other minor changes