Unusual functorialities for weakly constructible sheaves

TL;DR

This paper investigates the behavior of the six Grothendieck operations on weakly constructible sheaves, establishing conditions under which certain morphisms are isomorphisms and exploring properties of these sheaves.

Contribution

It demonstrates that various morphisms related to the six Grothendieck operations become isomorphisms for weakly constructible sheaves and studies their properties in this context.

Findings

Certain morphisms are isomorphisms for weakly constructible sheaves.

Compatibility of the six operations with weakly $R$-constructible sheaves.

Properties of weakly cohomologically constructible sheaves.

Abstract

We prove that various morphisms related to the six Grothendieck operations on sheaves become isomorphisms when restricted to (weakly) constructible sheaves. To this end, we first study some properties of weakly cohomologically constructible sheaves. We then deduce several compatibilities of the six operations in the context of (weakly) $\mathbb{R}$-constructible sheaves.