An axiomatization of six-functor formalisms

TL;DR

This paper axiomatizes six-functor formalisms in $alculus$, introducing Nagata and local variants, and characterizes their structures using adjoint triples, recollements, and sheaf conditions within $alculus$-categories.

Contribution

It provides a new axiomatic framework for six-functor formalisms, including Nagata and local types, with characterizations based on adjoint triples, recollements, and sheaf conditions.

Findings

Nagata six-functor formalism can be specified by adjoint triples satisfying compatibilities.

Recollements are nearly equivalent to a hypersheaf condition for a Grothendieck topology.

The category of local six-functor formalisms embeds into lax symmetric monoidal functors.

Abstract

In this paper, we consider some variations on Mann's definition $\infty$-categorical definition of abstract six-functor formalisms. We consider Nagata six-functor formalisms, that have the additional requirement of having Grothendieck and Wirthm\"uller contexts. We also consider local six-functor formalisms, which in addition to this, take values in presentable stable $\infty$-categories, and have recollements. Using Nagata's compactification theorem, we show that Nagata six-functor formalism on varieties can be given by just specifying adjoint triples for open immersions and for proper morphisms, satisfying certain compatibilities. The existence of recollements is (almost) equivalent to a hypersheaf condition for a Grothendieck topology on the category of ``varieties and spans consisting of an open immersion and a proper map''. Using this characterisation, we show that the category of local six-functor formalisms embeds faithfully into the category of lax symmetric monoidal functors from the category of smooth and complete varieties to the category of presentable stable $\infty$-categories and adjoint triples. We characterise which lax symmetric monoidal functors on complete varieties, taking values in the category of presentable stable $\infty$-categories and adjoint triples, extend to local six-functor formalisms.