Next we describe the main outcomes of the project.
In the categorical approach to geometry put forth by Van den Bergh, Kontsevich, Kaledin and others, categories represent spaces in the absence of actual points. The result is sometimes called a noncommutative space, because it typically involves noncommutative algebra. On the one hand, quantum mechanics can famously be described through noncommutative “deformation quantisation” of classical mechanics. On the other hand, the
categorical structures studied in our project occur as models in string theory, and our goal is precisely to study their deformation theory, or, in physical terms, how they can be quantised. Our main results involve several particular categorical models, and can be summarised as follows:
(1) Prestacks are structures in which the local geometry of a space can be organised algebraically. Joint with Hoang Dinh Van and Lander Hermans, we established the complete higher combinatorial structure on their deformation complexes, a question that has been under investigation in particular cases since the 1980s. Our solution makes use of certain patchworks of rectangles (see picture0).
(2) The most widely studied and applicable contemporary model for noncommutative spaces is that of differential graded categories. However, their deformation theory faces the “curvature problem” (Keller - Lowen, 2009; Lurie, 2011) by which not only points, but also sheaves may disappear from the classical picture. Joint with Alessandro Lehmann, we completely solved this problem for first order deformations, hereby introducing a novel conception of “categorical deformation”.
(3) Joint with Arne Mertens, we proposed the candidate new model of “quasi-categories in modules” to overcome certain difficulties in the homotopy theory of differential graded categories. To this end we established the novel framework of templicial modules, which are linear analogues of simplicial sets (see picture1). Joint with Violeta Borges Marques and Arne Mertens, we initiated the deformation theory of these objects.
