Abstract

The aim of this talk is to expose the natural foundations of non-abelian cohomology within higher category theory, as expounded by Grothendieck and Street among others. The fundamental principle is that higher stacks are the natural coefficients for non-abelian cohomology; the cohomology is the higher category of global sections of the higher stack. The familiar definitions of non-abelian cohomology of degree one and two in terms of torsors and gerbes are recovered by using a generalisation of Lawvere’s construction of the associated sheaf of a presheaf.

I will compare this theory with that presented in Street’s paper ‘Categorical and combinatorial aspects of descent theory’, and address the “future quest” proposed in the last sentence of that paper. I will also outline a method for developing the required category theory of bicategories and tricategories via the homotopy coherent category theory of model 2-categories and model Gray-categories.