Nonexistence of colimits in naive discrete homotopy theory

Abstract

We show that the quasicategory defined as the localization of the category of (simple) graphs at the class of A-homotopy equivalences does not admit colimits. In particular, we settle in the negative the question of whether the A-homotopy equivalences in the category of graphs are part of a model structure.

Appl. Categ. Structures (to appear); 5 pages; v2 reorganized following the referee's suggestion