A Classification of Six Functor Formalisms via Structured Spaces

TL;DR

This paper provides an infinity categorical framework for understanding six functor formalisms in noncommutative algebraic geometry, unifying various approaches through structured spaces and animated S-stacks.

Contribution

It introduces a new infinity categorical interpretation of reconstruction theorems and establishes conditions for factorization of six functor formalisms via animated S-stacks.

Findings

Unified six functor formalism through structured spaces

Conditions for factorization via animated S-stacks

Resolution of space versus quantity tension in formalism

Abstract

We lay out an infinity categorical interpretation of reconstruction theorems which are germane to the symmetric monoidal perspective of noncommutative algebraic geometry, present sufficient conditions which allow for the factorization of certain six functor formalisms through animated S-stacks, and give a six functor formalism through which the aforementioned six functor formalisms factor through. Furthermore, and what is arguably the main feat of this article, these achievements, though in appearance arising from disparate concerns, are realized in the dissipation of a familiar thematic tension: that between space and quantity.