Homotopical algebraic geometry. I. Topos theory.

By Bertrand Toën, G. Vezzosi. 2002

308 citations recorded by the source corpus.

This is the rst of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this rst part we investigate a notion of higher topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of 1-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T , generalizing the model category structure on simplicial presheaves over a Grothendieck site of A. Joyal and R. Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspodence between S-topologies on an S-category T , and certain left exact Bouseld localizations of the model category of pre-stacks on T . Based on the above results, we study the notion of model topos introduced by C. Rezk, and we relate it to our model categories of stacks over S-sites. In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a denition of etale K-theory of ring spectra, extending the etale K-theory of commutative rings.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6364489157d59b268e8c4f536c6cedb2e9daad25

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Citations in the corpus

Non-Archimedean analytic geometry as relative algebraic geometry (2013, background): “ category of (∞,1)-categories for which one could use quasi-categories. ∞-groupoids in this model are Kan simplicial sets. Stacks would be pre-stacks which satisfy descent with respect to hypercovers [14, 33, 34, 63, 64]. Similarly we could deﬁne (pre-)stacks valued in other categories, for instance”
The homotopy theory of dg-categories and derived Morita theory (2004, background): “The model category dg − CatU is furthermore U-cofibrantly generated in the sense of [ HAGI , Appendix].”
Higher and derived stacks: a global overview (2009, background): “e model category of functors SSetsk−CAlg endowed with the levelwise projective model structure, and then deﬁne M as the left Bousﬁeld localization of SSetsk−CAlg along the equivalences in sk−CAlg(see [HAGI] for details). We then have a natural equivalence between dk\−Affand Int(M). The next step is ”
Higher K-theory via universal invariants (2007, background): “Remark 6.9. Notice that lemma 4:2:2 from [ 37 ] implies that for every X inMf, the morphism ( X)”
A model category structure on the category of simplicial categories (2004, background): “In an early online version of the book [2], Dwyer, Hirschhorn, and Kan present a cofibrantly generated model category structure on the category of simplicial categories, but as Toën and Vezzosi point out in their paper [10], this model category structure is incorrect, in that some of the proposed ge”
Loop spaces and connections (2010, methodology): “For more details on derived algebraic geometry, we refer the reader to Toën’s extremely useful survey [31], and to the papers of Toën-Vezzosi [32, 33] and Lurie [21–23].”