n-Fold models of the building blocks of spaces

Simona Paoli, University of Leicester. Part of the algebra seminar series.

A classical way to study topological spaces is by breaking them into smaller components, the n-types, via the so called Postnikov decomposition. It is desirable in view of applications to find models of such a Postnikov decomposition built out of purely algebraic and categorical tools. These are called algebraic-categorical models of n-types. In joint work with Blanc we built a functor which associates to a space an algebraic structure called a weakly globular n-fold groupoid, which has desirable properties in view of applications. We showed that this induces a functor from the homotopy category of n-types to a suitable localization of weakly globular n-fold groupoids, and this functor is essentially surjective on objects. I will explain how to enlarge the category of weakly globular n-fold groupoids so that this functor becomes an equivalence of categories. The resulting algebraic model of n-types is called groupoidal weakly globular n-fold categories. I will briefly explain how this appears as the homotopy hypothesis of a new model of weak n-categories. Part of this talk is based on joint work with David Blanc.

After each talk at 4:15pm there will be tea/coffee in the Common Room at level 9 in Maths building. The current seminar organiser is Eleonore Faber. 

Simona Paoli, University of Leicester