Monoid actions, their categorification and applications

Abstract

We study actions of monoids and monoidal categories, and their relations with (co)homology theories. We start by discussing actions of monoids via bi-actions. We show that there is a well-defined functorial reverse action when a monoid action is given, which corresponds to acting by the inverses for group actions. We use this reverse actions to construct a homotopical structure on the category of monoid actions, which allow us to build the Burnside ring of a monoid. Then, we study categorifications of the previously introduced notions. In particular, we study actions of monoidal categories on categories and show that the ideas of action reversing of monoid actions extends to actions of monoidal categories. We use the reverse action for actions of monoidal categories, along with homotopy theory, to define homology, cohomology, homotopy and cohomotopy theories graded over monoidal categories. We show that most of the existing theories fits into our setting; and thus, we unify the existing definitions of these theories. Finally, we construct the spectral sequences for the theories graded over monoidal categories, which are the strongest tools for computation of cohomology and homotopy theories in existence.