  Condensed mathematics, developed by Clausen and Scholze over the last few
years, proposes a generalization of topology with better categorical
properties. It replaces the concept of a topological space by that of a
condensed set, which can be defined as a sheaf for the coherent topology on a
certain category of compact Hausdorff spaces. In this case, the sheaf condition
has a fairly simple explicit description, which arises from studying the
relationship between the coherent, regular and extensive topologies. In this
paper, we establish this relationship under minimal assumptions on the
category, going beyond the case of compact Hausdorff spaces. Along the way, we
also provide a characterizations of sheaves and covering sieves for these
categories. All results in this paper have been fully formalized in the Lean
proof assistant.
