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Descent theory and mapping spaces. (English) Zbl 1462.14003

Summary: The purpose of this paper is to develop a theory of \((\infty,1)\)-stacks, in the sense of Hirschowitz-Simpson’s ‘Descent Pour Les \(n\)-Champs’, using the language of quasi-category theory and the author’s local Joyal model structure. The main result is a characterization of \((\infty,1)\)-stacks in terms of mapping space presheaves. An important special case of this theorem gives a sufficient condition for the presheaf of quasi-categories associated to a presheaf of model categories to be a higher stack. In the final section, we apply this result to construct the higher stack of unbounded complexes associated to a ringed site.

MSC:

14A20 Generalizations (algebraic spaces, stacks)
18N40 Homotopical algebra, Quillen model categories, derivators
14A30 Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.)
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
55P42 Stable homotopy theory, spectra
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[1] Grothendieck, A., Sur quelques points dalgbre homologique, Tohoku Math. J., 9, 2, 119-221 (1957) · Zbl 0118.26104
[2] Artin, M., Grothendieck, A., Verdier, J.: Thorie des topos et cohomologie tale des schmas, Tome 1: Thorie des topos, Sminaire de Gomtrie Algbrique du Bois-Marie 19631964 (SGA 4). Lecture Notes in Mathematics, vol. 269. Springer, Berlin (1972)
[3] Bergner, J., Complete segal spaces arising from simplicial categories, Trans. AMS, 361, 3, 323-347 (2007)
[4] Bergner, J., A model category structure on the category of simplicial categories, Trans. AMS, 359, 2048-2063 (2007) · Zbl 1114.18006
[5] Bergner, J., Homotopy limits of model categories and more general homotopy theories, Bull. LMS, 44, 2, 311-322 (2012) · Zbl 1242.55006
[6] De Jong, J.: Stacks project (2019). https://stacks.math.columbia.edu/(Online Book)
[7] Dugger, D.; Spivak, DI, Mapping spaces in quasi-categories, Algebraic. Geom. Topol., 11, 1, 263-325 (2011) · Zbl 1214.55013
[8] Dugger, D.; Hollander, S.; Isaksen, D., Hypercovers and simplicial presheaves, Math Proc. Cam. Phil. Soc., 136, 1, 9-51 (2004) · Zbl 1045.55007
[9] Dwyer, W.; Kan, D., Calculating simplicial localizations, JPAA, 18, 1, 17-35 (1980) · Zbl 0485.18013
[10] Dwyer, W.; Kan, D., Function complexes in homotopical algebra, Topology, 19, 4, 427-440 (1980) · Zbl 0438.55011
[11] Giraud, J., Cohomologie Non-Abelienne. Grundlehren der Mathematischen Wissenschaften (1971), Berlin: Springer, Berlin · Zbl 0135.02401
[12] Goerss, P., Jardine, J.F.: Simplicial homotopy theory. Modern Birkhäuser Classics. Birkhäuser, Basel (2009). doi:10.1007/978-3-0346-0189-4. doi:10.1007/978-3-0346-0189-4(reprint of the 1999 edition) · Zbl 1195.55001
[13] Hirschowitz, A., Simpson, C.: Descent pour les n-champs (2001). arXiv:math/9807049(preprint)
[14] Jardine, JF, Boolean localization, in practice, Doc. Math., 13, 690-711 (1996) · Zbl 0858.55018
[15] Jardine, JF, Presheaves of chain complexes, K Theory, 30, 365-420 (2003) · Zbl 1047.18011
[16] Jardine, JF, Categorical homotopy theory, Homol. Homotopy Appl., 8, 1, 71-144 (2006) · Zbl 1087.18009
[17] Jardine, JF, Local homotopy theory. Springer monographs in mathematics (2015), New York: Springer, New York · Zbl 1320.18001
[18] Joyal, A., Quasi-categories and Kan complexes, J. Pure Appl. Algebra, 175, 207-222 (2002) · Zbl 1015.18008
[19] Joyal, A.: The theory of quasi-categories and its applications (2008). http://mat.uab.cat/ kock/crm/hocat/advanced-course/Quadern45-2.pdf(preprint)
[20] Joyal, A.; Tierney, M., Quasi-categories vs. Segal spaces, Categories in algebra, geometry and mathematical physics, contemporary mathematics, 277-326 (2007), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1138.55016
[21] Kapulkin, K.; Szumilo, K., Quasicategories of frames of cofibration categories, Appl. Categorical Struct., 25, 3, 323-347 (2017) · Zbl 1372.55014
[22] Lurie, J., Higher Topos Theory. Annals of Mathematics Studies (2009), Princeton, Oxford: Princeton University Press, Princeton, Oxford · Zbl 1175.18001
[23] Mac Lane, S., Categories for the working mathematician, no. 5 in Graduate Texts in Mathematics (1978), Berlin: Springer, Berlin · Zbl 0232.18001
[24] Mac Lane, S.; Moerdijk, I., Sheaves in geometry and logic. Universitext (1994), New York: Springer, New York
[25] Meadows, N., The local Joyal model structure, Theory Appl. Categorical, 31, 24, 690-711 (2016) · Zbl 1352.18008
[26] Meadows, N.: Local complete Segal spaces. Appl. Categorical Struct. (2018). doi:10.1007/s10485-018-9535-1 · Zbl 1409.18017
[27] Rezk, C.: A model for the homotopy theory of homotopy theories. Trans. AMS 353 (2001) · Zbl 0961.18008
[28] Simpson, C.; Alexandre, G., Descent, A mathematical portrait, 83-141 (2014), Somerville: Int. Press, Somerville · Zbl 1303.14013
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