ON CATEGORIES OF MONOID-ACTIONS

Abstract

Given a monoidal category B and a category S of monoids in B we study the category MOD_S of all actions of monoids from S on B-objects. This is mainly done by investigation of the underlying functor V: MOD_S → SxB. In particular V creates limits; filtered colimits and arbitrary colimits are detected, provided the monoidal structure behaves nicely with respect to these constructions. Moreover MOD_S contains B as a full coreflective subcategory; S is contained as a full reflective (and coreflective) one provided B has a terminal (zero) object. Monadicity of MOD_S over B is discussed as well.