An approach to homological algebra up to $\epsilon$ -- Tobias Fritz (U. Innsbruck), PHK 04.10.2023 скачать в хорошем качестве

An approach to homological algebra up to $\epsilon$ -- Tobias Fritz (U. Innsbruck), PHK 04.10.2023

Title: An approach to homological algebra up to $\epsilon$ Speaker: Tobias Fritz (University of Innsbruck) Abstract: A theorem of Kazhdan on approximate representations of groups is based on a proof which seems to use cohomological methods "up to $\epsilon$". This means that being a cocycle or a coboundary is not a yes/no-property of a cochain $C$, but rather a quantitative statement where one measures how strongly $C$ deviates from being either. Based on Grandis's framework for nonabelian homological algebra, I will present a framework for such quantitative homological algebra and sketch the intuition behind the resulting definitions of kernel and cokernel. Unfortunately, the resulting category does not satisfy the axioms required of homological categories in Grandis's sense. Our main result solves this problem by showing that an arrow category is a homological category already under very weak assumptions. It follows that derived functors and long exact sequences can be constructed for arrow categories quite generally, and for quantitative homological algebra in particular. Talk given at the Prague-Hradec Králové seminar on Cohomology in algebra, geometry, physics and statistics: https://www.math.cas.cz/index.php/eve... https://researchseminars.org/talk/PHK...