From Simplicial Homotopy Theory to Higher categories
Simona Paoli (University of Aberdeen)
Monday 6th December 16:00-17:00 Online
Topological spaces can be studied by breaking them into building blocks, called n-types, using a classical construction in homotopy theory, the Postnikov decomposition. The desire to model algebraically the building blocks of spaces is one of the motivations for the development of higher groupoids, generalizing the fundamental groupoid of a space. In this talk I will first illustrate how this naturally leads to the need to encode weakly associative and unital compositions of ‘higher morphisms’ in a higher groupoid and I will discuss the challenges that this poses. I will then introduce one of the combinatorial machineries to formalize this, the one of multi-simplicial sets, and I will illustrate how one can obtain from a space a ‘fundamental n-groupoid’ using this machinery. I will then briefly discuss how some of the structures emerging from this simplicial homotopy-theoretic approach can be used to build models of higher categories.
The talk will be preceded by a tea time at 3:45pm. The Zoom link for the seminar is https://uofglasgow.zoom.us/j/98078798957 and the passcode is the genus of the two-dimensional sphere (4 letters, all lowercase).