Characterizations of modalities and lex modalities

A reflective subuniverse in homotopy type theory is an internal version of the notion a localization topology or ∞-categories. Working theory, we give new characterizations following conditions on L: (1) associated L′ L-separated types modality; (2) L (3) lex and (4) cotopological modality. In each case, several necessary sufficient conditions. Our involve various families maps to L, such as L-étale maps, L-equivalences, L-local L-connected unit ηX, their left and/or right orthogonal complements. More generally, our main theorem gives overview how all these classes related other. We also examples that show inclusions describe between can be strict.