Unusual functorialities for weakly constructible sheaves
Andreas Hohl, Pierre Schapira
TL;DR
This paper addresses when the natural morphisms among the six Grothendieck functors become isomorphisms on (weakly) constructible derived categories, focusing on real analytic and b-analytic settings. It introduces the notions of weakly cohomologically constructible and weakly ${\mathbb{R}}$-constructible objects, proves the relevant categories are triangulated, and establishes key local isomorphisms that reduce global six-functor questions to stalkwise data. Using these structural tools, the authors derive eight core compatibilities for pullbacks, tensor products, and Hom, as well as for pushforwards, including extensions to open embeddings and to infinity via b-analytic manifolds. The results generalize and organize known functorialities for constructible sheaves to weaker constructibility hypotheses, enabling compatibility with scalar extension and duality in broader analytic contexts. The work provides a robust framework for applying the six-functor formalism to weakly constructible sheaves, with potential implications for microlocal analysis and applications where constructibility is limited.
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.