[Paper Review] Topological exodromy with coefficients

Mauro Porta, Jean-Baptiste Teyssier|arXiv (Cornell University)|Nov 9, 2022

Algebraic structures and combinatorial models4 citations

TL;DR

This paper generalizes the exodromy equivalence to stratified spaces with locally weakly contractible strata, removes noetherianity and ascending chain condition assumptions, and extends coefficients to stable or compactly generated $∞$-categories. It establishes a functorial exodromy equivalence and constructs a derived moduli stack of hyperconstructible hypersheaves, leading to a new cohomological Hall algebra generalizing Schiffmann-Vasserot and Davison-Mistry's constructions.

ABSTRACT

We improve the exodromy equivalence of MacPherson, Treumann and Lurie in several ways: first, we allow stratified spaces that have locally weakly contractible strata, rather than being locally of singular shape, we remove all noetherianity assumptions and we consider more general coefficients (e.g. compactly assembled or stable presentable $\infty$-categories). Furthermore, our approach shows that the exodromy equivalence is functorial for every morphism of stratified spaces. As an application, we construct, under suitable finiteness assumptions, a higher Artin derived stack of hyperconstructible hypersheaves and prove that every perversity function gives rise to an open substack of perverse hypersheaves. Using the derived structure, we provide the construction of a new cohomological Hall algebra, generalizing the ones of character varieties studied in the past by Schiffmann-Vasserot, Davison and Mistry.

Motivation & Objective

• To extend the exodromy equivalence beyond the classical setting of locally singular-shaped spaces to those with locally weakly contractible strata.
• To remove the noetherianity and ascending chain condition assumptions on the poset of strata.
• To generalize the coefficients of the exodromy equivalence to include stable or compactly generated $∞$-categories.
• To provide a functorial formulation of the exodromy equivalence for all morphisms of stratified spaces.
• To construct a derived moduli stack of hyperconstructible hypersheaves and relate it to perverse sheaves via perversity functions.

Proposed method

• Replacing the $∞$-category of constructible sheaves with that of hyperconstructible hypersheaves to relax the singular shape assumption.
• Using the $∞$-category of exit paths $Π_{∞}^{Σ}(X,P)$ as the domain of functors for the exodromy correspondence.
• Introducing a functorial framework for the exodromy equivalence that works for all morphisms of stratified spaces.
• Constructing the derived moduli stack ${}^{π}\mathbf{Perv}_{P}(X)$ of hyperconstructible hypersheaves using derived algebraic geometry.
• Proving that every perversity function defines an open substack of perverse hypersheaves within the moduli stack.
• Defining a new cohomological Hall algebra via the derived structure on the moduli stack of perverse hypersheaves.

Experimental results

Research questions

• RQ1 Can the exodromy equivalence be extended to stratified spaces with locally weakly contractible strata, without requiring the singular shape condition?
• RQ2 Does the exodromy equivalence remain valid when removing the ascending chain condition on the poset of strata?
• RQ3 Can the exodromy correspondence be generalized to coefficients in stable or compactly generated $∞$-categories?
• RQ4 Is there a functorial formulation of the exodromy equivalence that holds for all morphisms of stratified spaces?
• RQ5 Can a derived moduli stack of hyperconstructible hypersheaves be constructed, and does it admit a natural structure of a cohomological Hall algebra?

Key findings

• The exodromy equivalence is extended to stratified spaces with locally weakly contractible strata, replacing the singular shape assumption.
• The exodromy equivalence is now valid without any noetherianity or ascending chain condition assumptions on the poset $P$.
• The coefficients of the exodromy correspondence are generalized to include all stable or compactly generated $∞$-categories.
• An explicit formula for the inverse of the exodromy equivalence is provided, enhancing its functoriality.
• A derived moduli stack of hyperconstructible hypersheaves is constructed, and every perversity function gives rise to an open substack of perverse hypersheaves.
• A new cohomological Hall algebra is defined on the derived moduli stack of perverse hypersheaves, generalizing earlier constructions by Schiffmann-Vasserot and Davison-Mistry.