Abstract: This talk will introduce the concept of orbifold as a generalization of that of manifold and some basic notions about groupoids. This alternate perspective will be helpful in answering some questions about orbifolds, as for example, what is a necessary condition for an orbifold to be a global quotient.

Abstract: This talk is a report on ongoing joint work with M. Hassanzadeh and I. Shapiro, where we extend the definition of Hopf cyclic cohomology with coefficients in a Yetter-Drinfeld type module to braided monoidal categories and fusion categories. There are many algebraic structures, e.g. Drinfeld's quasi-Hopf algebras, that share some of the axioms of Hopf algebras, but not all of them. In many cases these structures are objects of a monoidal category. I shall introduce a class of monoidal categories where one can in fact define a cyclic module for its objects.