dorsal/arxiv
View SchemaFlexible sheaves
| Authors | Carlos Simpson |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9608025 |
| URL | https://arxiv.org/abs/q-alg/9608025 |
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.
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"abstract": "We look at homotopy-coherent diagrams of spaces (after Segal, Leitch, Vogt,\nMather, Cordier) over a Grothendieck site; we call these ``flexible\npresheaves\u0027\u0027. After some preliminary materiel, we define the ``flexible sheaf\u0027\u0027\ncondition. This descent condition (known to Thomason) is the same as what\nJardine called being ``flasque\u0027\u0027 with respect to the presheaves representable\nby objects in the site; and it is more recently known as the condition of being\nan $n$-stack. We construct the flexible sheaf associated to a flexible presheaf\nin the $n$-truncated case, as an application of a certain natural operation\n$n+2$ times. We prove an analogue of Vogt\u0027s theorem for the case where the\nGrothendieck topology is nontrivial, identifying the set of morphisms in\nIllusie\u0027s derived category as the set of homotopy classes of homotopy-coherent\nmorphisms between flexible sheaves. The homotopy-coherent point of view allows\none easily to define the flexible mapping sheaf $Hom (R,T)$ between two\nflexible sheaves.\n This revision fills major gaps in the bibliography. References to the\nadditional items are inserted in the text. A new introduction and abstract are\nadded (the old ones are retained as comments in the source file). A few other\nminor changes in the exposition include arrangement of internal references.",
"arxiv_id": "q-alg/9608025",
"authors": [
"Carlos Simpson"
],
"categories": [
"q-alg",
"alg-geom",
"math.AG",
"math.QA"
],
"title": "Flexible sheaves",
"url": "https://arxiv.org/abs/q-alg/9608025"
},
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