Six-Functor Formalisms III: The construction and extension of 6FFs

TL;DR

This paper completes a series on six-functor formalisms by proving a key theorem on partial adjoints, utilizing simplicial techniques and $irc$-categorical methods to construct and extend these formalisms in geometric contexts.

Contribution

It introduces the theorem of partial adjoints and combines it with $irc$-categorical compactification to construct and extend six-functor formalisms in geometric settings.

Findings

Proved the theorem of partial adjoints using simplicial techniques.

Constructed six-functor formalisms in geometric contexts.

Extended formalisms from smaller to larger setups using simplified descent.

Abstract

This article is the last of the series of articles where we reprove the foundational ideas of abstract six-functor formalisms developed by Liu-Zheng. We prove the theorem of partial adjoints, which is a simplicial technique of encoding various functors altogether by taking adjoints along specific directions. Combined with the $\infty$-categorical compactification theorem from the previous article, we can construct abstract six-functor formalisms in reasonable geometric setups of our interest. We also reprove the simplified versions of the DESCENT program due to Liu-Zheng, which allows us to extend such formalisms from smaller to larger geometric setups.