Universiteit Leiden Mathematisch Instituut
Geometry Seminar

Tuesday, 15 November 2005, 14:00 - 15:00, lecture room 403 of the Mathematisch Instituut.

Josh Nichols-Barrer (MIT): Higher Categories and Higher Stacks

Abstract:

Isomorphism classes of schemes are not merely sets; they carry (naturally) the extra information of a groupoid structure, which gets lost if we try to solve moduli problems in the category of schemes. We thus replace sheaves (schemes) with "sheaves in groupoids" (algebraic stacks), and find fine moduli stacks for all sorts of problems. If one instead would like to look for moduli of stacks (e.g. moduli of stacky curves), the functor of families no longer takes values in groupoids, but in 2-groupoids, and so we expect to see some kind of "2-stack" as the sort of object that a fine moduli space would be. Moreover, as the natural notion of pullback in this context is "weak," we should be looking at "sheaves in weak 2-groupoids."

In this talk I will motivate and present a definition of n-stack with the above remarks in mind, using A. Joyal's notion of "quasi-categories" as a model for the higher categories we will be interested in. Far from the sloppiness of the above description, the definition will have much more in common with the original notion of stack in groupoids written down by Deligne and Mumford. If there is time, I will try to discuss what an algebraic n-stack would look like in this formalism. As this is work in progress, proofs (if they are available) will be sketchy at best.