Exact categories, Koszul duality, and derived analytic algebra

Thesis

Recent work of Bambozzi, Ben-Bassat, and Kremnitzer suggests that derived analytic geometry over a valued field k can be modelled as geometry relative to the quasi-abelian category of Banach spaces, or rather its completion Ind(Ban_k). In this thesis we develop a robust theory of homotopical algebra in Ch(E) for E any sufficiently 'nice' quasi-abelian, or even exact, category.

Firstly we provide sufficient conditions on weakly idempotent complete exact categories E such that various categories of chain complexes in E are equipped with projective model structures. In particular we show that as soon as E has enough projectives, the category Ch₊(E) of bounded below complexes is equipped with a projective model structure. In the case that E also admits all kernels we show that it is also true of Ch_≥0(E), and that a generalisation of the Dold-Kan correspondence holds. Supplementing the existence of kernels with a condition on the existence and exactness of certain direct limit functors guarantees that the category of unbounded chain complexes Ch(E) also admits a projective model structure. When E is monoidal we also examine when these model structures are monoidal.

We then develop the homotopy theory of algebras in Ch(E). In particular we show, under very general conditions, that categories of operadic algebras in Ch(E) can be equipped with transferred model structures. Specialising to quasi-abelian categories we prove our main theorem, which is a vast generalisation of Koszul duality. We conclude by defining analytic extensions of the Koszul dual of a Lie algebra in Ind(Ban_k).

Authors: Kelly, J

• DOI: 10.5287/ora-rv7jee17e
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Category Theory; Homotopy Theory; Algebra; Koszul Duality
• Subjects: Mathematics