Bornological Algebras in Exotic Derived Categories and Condensed Mathematics

Abstract

This thesis is divided into three parts. In the first part, we examine the quasi-abelian category of complete bornological spaces and its contraderived categories. We classify all projective complete bornological spaces and show that the category has infinite global dimension. Furthermore, we demonstrate that nuclear Fréchet spaces have finite projective dimension, provided a specific cardinality condition holds. Assuming the continuum hypothesis, we establish the existence of a symmetric monoidal quasi-abelian subcategory of all complete bornological spaces, which includes Fréchet spaces, for which the contraderived category can be defined and identified with the homotopy category of projectives. This result is extended to complete bornological modules over a nuclear Fréchet algebra, with applications to smooth functions and the de Rham algebra on real smooth manifolds. In the second part, we construct categories of ℳ-complete and liquid condensed vector spaces, drawing on niche notions from classical functional analysis. We demonstrate that Waelbroeck's compactological sets form a category equivalent to quasi-separated condensed sets. Building on this, we extend the idea to vector spaces, utilizing the theory of Smith spaces and compactologies to construct the category of compactological spaces and demonstrate that it is equivalent to ℳ-complete condensed vector spaces. Additionally, we introduce the concept of p-lensed spaces, defined via non-locally convex Smith spaces, and show that their category is equivalent to that of quasi-separated p-liquid spaces. Finally, we prove that the left heart of the quasi-abelian category of p-lensed spaces coincides with the category of p-liquid vector spaces. In the third part, we investigate relative versions of dualizability designed for relative versions of topological field theories (TFTs), also called twisted TFTs, or quiche TFTs in the context of symmetries. In even dimensions we show an equivalence between lax and oplax fully extended framed relative topological field theories valued in an (∞,N)-category in terms of adjunctibility. Motivated by this, we systematically investigate higher adjunctibility conditions and their implications for relative TFTs. Finally, for fun we explore a tree version of adjunctibility and compute the number of equivalence classes thereof.