1996/08/29 by Carlos Simpson, Simpson, Carlos
Engineering · #Algebraic Geometry (math.AG) #FOS: Mathematics #Quantum Algebra (math.QA) #Slime Mold and Myxomycetes Research
paper · pdf · doi:10.48550/arxiv.q-alg/9608025
openalex publication_date 1996/08/29 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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.