Higher linear topoi and curved noncommutative spaces. 01/10/2018 - 30/09/2020

Abstract

Broadly, this project can be summarized as looking for connections between noncommutative algebraic geometry (NCAG) and higher category theory. NCAG is the modern understanding, and a drastic abstract generalization, of classical geometry. To known geometrical spaces, one can associate commutative (i.e. x*y = y*x) algebraic structures. However, in algebra, noncommutative structures are just as common. The idea of NCAG is to study new ``geometric spaces'' associated to these noncommutative algebraic structures. Higher category theory and in particular so-called infinity-topoi generalize the following idea. Consider the familiar example of sets, and maps that describe relations between those sets. Further, we can also describe relations between the maps, which we could call "2-maps". We then have 3-maps between 2-maps and so on, yielding an infinite hierarchy of maps. In relation to NCAG, the most important abelian categories correspond to linear topoi, in which the "maps" have some additional structure. One goal of the project is to establish a suitable notion of linear infinity-topoi, using ideas from NCAG. Another goal is to use ideas from higher categories to investigate the so-called "curvature problem" from NCAG. This involves "curved objects", which are slightly tweaked versions of some original object, that turn out difficult to grasp using the familiar tools of homological algebra.

Researcher(s)

Research team(s)