Strict monadic topology I: First separation axioms and reflections

Abstract

Given a monad T on the category of sets, we consider reflections of Alg(T) into its full subcategories formed by algebras satisfying natural counterparts of topological separation axioms T_0, T_1, T_2, T_ts, and T_ths; here ts stands for totally separated and ths for what we call totally homomorphically separated, which coincides with ts in the (compact Hausdorff) topological case. We ask whether these reflections satisfy simple conditions useful in categorical Galois theory, and give some partial answers in easy cases.