TL;DR
This paper provides an overview of recent advances in six-functor formalisms, including their abstract theory, related 2-categories, duality results, and various examples, highlighting their significance in modern algebraic geometry.
Contribution
It introduces the latest developments in six-functor formalisms, emphasizing their theoretical foundations, categorical structures, and diverse applications.
Findings
Simplifications in Poincaré--Verdier duality proofs
Connections between 6-functor formalisms and geometric rings
Compilation of numerous examples of 6-functor formalisms
Abstract
These are lecture notes for a course in Winter 2022/23, updated and completed in October 2025. The goal of the lectures is to present some recent developments around six-functor formalisms, in particular: the abstract theory of 6-functor formalisms; the 2-category of cohomological correspondences, and resulting simplifications in the proofs of Poincar\'e--Verdier duality results; the relation between 6-functor formalisms and ``geometric rings''; many examples of 6-functor formalisms, both old and new.
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