30/09/2016, 10:30 — 11:30 — Room P4.35, Mathematics Building
Simon Henry, Collège de France
On the homotopy hypothesis and new algebraic model for higher groupoids and homotopy types.
In his unpublished manuscript "Pursuing stacks" Grothendieck gave a definition of infinity groupoids and conjectured that the homotopy category of his infinity groupoids is equivalent to the homotopy category of spaces.
This conjecture (called the homotopy hypothesis) is still an open problem, and in fact there is a lot of expected basic results concerning his definition of infinity groupoids that are open problems. For these reasons, one prefers nowadays to use less problematic definitions of infinity groupoids, typically involving simplicial sets, as a starting point for higher category theory.
But Grothendieck's definition also has a lot of good properties not shared by the simplicial approaches: it is considerably closer to the intuitive notion of infinity category, it has a more general universal property, it is considerably simpler to extend to infinity categories, etc. And most recently realized, it can be defined within the framework of the Homotopy type theory program, while the definition of simplicial objects in this framework is considered to be one of the most important open problems of this program.
In this talk I will discuss a new sort of definitions of infinity groupoids that are inspired from Grothendieck's definition but that do not share any of its problems while retaining most of its advantages. We will also state a precise and simple looking technical conjecture which implies that Grothendieck definition is a special case of our framework, and hence also implies Grothendieck's homotopy hypothesis and most of the conjectures related to Grothendieck's definition.
