[Paper Review] Flexible sheaves

Carlos Simpson|arXiv (Cornell University)|Aug 29, 1996

Homotopy and Cohomology in Algebraic Topology8 citations

TL;DR

This paper introduces flexible sheaves as homotopy-coherent presheaves on a Grothendieck site satisfying a descent condition equivalent to being flasque or an n-stack. It constructs the flexible sheafification functor via a repeated application of a natural operation n+2 times, and establishes an analogue of Vogt’s theorem, identifying homotopy classes of homotopy-coherent maps between flexible sheaves with morphisms in Illusie’s derived category.

ABSTRACT

We look at homotopy-coherent diagrams of spaces (after Segal, Leitch, Vogt, Mather, Cordier) over a Grothendieck site; we call these ``flexible presheaves''. After some preliminary materiel, we define the ``flexible sheaf'' condition. This descent condition (known to Thomason) is the same as what Jardine called being ``flasque'' with respect to the presheaves representable by objects in the site; and it is more recently known as the condition of being an $n$-stack. We construct the flexible sheaf associated to a flexible presheaf in the $n$-truncated case, as an application of a certain natural operation $n+2$ times. We prove an analogue of Vogt's theorem for the case where the Grothendieck topology is nontrivial, identifying the set of morphisms in Illusie's derived category as the set of homotopy classes of homotopy-coherent morphisms between flexible sheaves. The homotopy-coherent point of view allows one easily to define the flexible mapping sheaf $Hom (R,T)$ between two flexible sheaves. This revision fills major gaps in the bibliography. References to the additional items are inserted in the text. A new introduction and abstract are added (the old ones are retained as comments in the source file). A few other minor changes in the exposition include arrangement of internal references.

Motivation & Objective

• To formalize flexible presheaves as homotopy-coherent diagrams of spaces over a Grothendieck site.
• To define and analyze the flexible sheaf condition, showing it coincides with known notions like flasqueness and n-stacks.
• To construct the flexible sheaf associated to a flexible presheaf in the n-truncated case using a repeated natural operation.
• To establish a homotopy-coherent analogue of Vogt’s theorem, identifying derived category morphisms with homotopy classes of coherent maps.
• To provide a coherent framework for mapping sheaves and correct gaps in the literature.

Proposed method

• Model flexible presheaves as homotopy-coherent diagrams over a Grothendieck site, following Segal, Vogt, and Cordier.
• Define the flexible sheaf condition as a descent condition equivalent to flasqueness and n-stacks.
• Construct the flexible sheafification functor by applying a natural operation n+2 times in the n-truncated setting.
• Use the homotopy-coherent point of view to define the flexible mapping sheaf $\mathrm{Hom}(R,T)$ between two flexible sheaves.
• Prove an analogue of Vogt’s theorem by identifying $\mathrm{Hom}_{\text{ho}}(R,T)$ with morphisms in Illusie’s derived category.
• Revise and expand the bibliography, integrating key references and improving exposition.

Experimental results

Research questions

• RQ1 How can flexible presheaves be systematically defined and characterized on a Grothendieck site using homotopy-coherent methods?
• RQ2 What is the precise relationship between the flexible sheaf condition and established concepts like flasqueness and n-stacks?
• RQ3 Can a flexible sheafification functor be constructed in the n-truncated case, and if so, via what mechanism?
• RQ4 To what extent does Vogt’s theorem on homotopy classes of maps extend to the setting of flexible sheaves and derived categories?
• RQ5 How can the flexible mapping sheaf $\mathrm{Hom}(R,T)$ be naturally defined and used in this framework?

Key findings

• The flexible sheaf condition is equivalent to the classical notions of flasqueness and being an n-stack.
• The flexible sheaf associated to a flexible presheaf in the n-truncated case is constructed via n+2 applications of a natural operation.
• The homotopy classes of homotopy-coherent morphisms between flexible sheaves are canonically identified with morphisms in Illusie’s derived category.
• The flexible mapping sheaf $\mathrm{Hom}(R,T)$ is well-defined and naturally arises from the homotopy-coherent structure.
• The revised framework corrects major omissions in the bibliography and improves the clarity and coherence of foundational results in the area.