Stacks in derived bornological geometry

Thesis

In this thesis, we describe a higher categorical framework for discussing derived analytic and derived smooth geometry. Analogous to the Toën and Vezzosi model of derived algebraic geometry which uses simplicial commutative rings as its building blocks, in our model we use simplicial commutative complete bornological rings. This work builds upon foundational work of Ben-Bassat, Kelly, and Kremnizer. This general framework allows us to prove several results about derived stacks, in particular we develop the obstruction theory of stacks and use it to prove a representability theorem. This theorem cements this new theory of derived bornological geometry as strong and versatile, and gives differential and analytic geometers a new perspective on their own representability problems. In this thesis, we begin by studying a generalisation of the Koszul duality theory of Beilinson, Ginzburg, and Soergel to the setting of algebra objects in a bicomplete closed symmetric monoidal exact category E with enough flat projectives. Examples include the category CBorn_R of complete bornological spaces and the derived equivalent category Ind(Ban_R) of formal filtered colimits of Banach modules over a Banach ring R. We then define a general categorical context we call a derived geometry context. In these contexts we obtain our representability theorem. If a derived stack has a geometric truncation, is compatible with Postnikov towers, and has a well defined obstruction theory, our theorem shows that it is representable by a derived geometric stack. Working relative to CBorn_R for an appropriately chosen Banach ring R, we can define suitable derived geometry contexts modelling derived complex analytic and derived smooth geometry. In the derived smooth geometry setting, we develop a theory of C∞-bornological rings extending the theory of C∞-rings. Finally, we show that the derived moduli stack of non-linear elliptic PDEs is representable by a derived C∞-bornological affine scheme.

Authors: Savage, R

• DOI: 10.5287/ora-amzzvymdm
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: derived stacks; representability theorem; derived geometry; Koszul duality; infinity categories; moduli stacks; bornological rings
• Subjects: Geometry, Algebraic; Categories (Mathematics)