turkmath.org

This talk is based on a joint work with Özgün Ünlü. I will first discuss the monoid actions on sets, which is via action from both sides, but different then the usual biactions in terms of equivariance of maps. Using this view point, we will be able to associate a new action to a given monoid action on a set, so called inverse action, which generalizes the notion of inverse of group actions. In the second part I will discuss categorification of these definitions. I will define action of a monoidal categories on a category and the inverse of such action, analogues to the monoid case. As an application, ''homology and cohomology theories graded over a monoidal category'' will be introduced, which generalizes the usual definitions of (co)homology and (co)homotopy theories existing in the literature. This allow us to view (co)homology and (co)homotopy theories as fixed points of an action of monoidal category on a category.