University of Chicago
Department of Mathematics

Homotopy n-types are an important class of topological spaces: they amount to CW complexes whose homotopy groups vanish in dimension higher than n. The problem of modeling homotopy types is relevant both in higher category theory and homotopy theory and received contributions from both areas. There is a particularly simple model of homotopy types in the path connected case, consisting of n-fold categories internal to groups, also called catⁿ-groups. This model, however, has the disadvantage that is it does not have an algebraic description of the Postnikov decomposition nor it is easy to establish algebraically when a map of catⁿ-groups is a weak equivalence. In this talk we introduce a new model of connected n-types through a subcategory of catⁿ-groups, which we call weakly globular, for which the above issues are resolved in transparent way. We also describe other homotopical properties of this model, and discuss the relevance of these structures for higher category theory. In the second part of this talk we discuss the use of the notion of weak globularity to deal with the non-path connected case in low dimension (n=2) and we illustrate an application. The second part of this talk is joint work with David Blanc.