Moerdijk, Ieke The classifying topos of a continuous groupoid. I. (English) Zbl 0706.18007 Trans. Am. Math. Soc. 310, No. 2, 629-668 (1988). Summary: We investigate some properties of the functor B which associates to any continuous groupoid G its classifying topos BG of equivariant G-sheaves. In particular, it will be shown that the category of toposes can be obtained as a localization of a category of continuous groupoids. Cited in 2 ReviewsCited in 37 Documents MSC: 18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) 18B25 Topoi 18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories) 18D35 Structured objects in a category (MSC2010) 18B30 Categories of topological spaces and continuous mappings (MSC2010) 18B35 Preorders, orders, domains and lattices (viewed as categories) Keywords:classifying topos of equivariant G-sheaves; Grothendieck topos; colimit of toposes; category of fractions; category of generalized spaces; groupoid objects; continuous groupoid; category of toposes; localization PDFBibTeX XMLCite \textit{I. Moerdijk}, Trans. Am. Math. Soc. 310, No. 2, 629--668 (1988; Zbl 0706.18007) Full Text: DOI References: [1] Michael Barr and Radu Diaconescu, Atomic toposes, J. Pure Appl. Algebra 17 (1980), no. 1, 1 – 24. · Zbl 0429.18006 [2] Michael Barr and Robert Paré, Molecular toposes, J. Pure Appl. Algebra 17 (1980), no. 2, 127 – 152. · Zbl 0436.18002 [3] B. J. Day and G. M. Kelly, On topologically quotient maps preserved by pullbacks or products, Proc. Cambridge Philos. Soc. 67 (1970), 553 – 558. · Zbl 0191.20801 [4] P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New York, 1967. · Zbl 0186.56802 [5] Théorie des topos et cohomologie étale des schémas. Tome 2, Lecture Notes in Mathematics, Vol. 270, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963 – 1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. · Zbl 0234.00007 [6] P. T. Johnstone, Topos theory, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1977. London Mathematical Society Monographs, Vol. 10. · Zbl 0368.18001 [7] Peter T. Johnstone, Open maps of toposes, Manuscripta Math. 31 (1980), no. 1-3, 217 – 247. · Zbl 0433.18002 [8] P. T. Johnstone, Factorization theorems for geometric morphisms. I, Cahiers Topologie Géom. Différentielle 22 (1981), no. 1, 3 – 17. Third Colloquium on Categories (Amiens, 1980), Part II. · Zbl 0454.18007 [9] André Joyal and Myles Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71. · Zbl 0541.18002 [10] Gregory Maxwell Kelly, Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series, vol. 64, Cambridge University Press, Cambridge-New York, 1982. · Zbl 0478.18005 [11] Makkai and R. Paré, Accessible categories (to appear). · Zbl 0703.03042 [12] J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. · Zbl 0769.55001 [13] Ieke Moerdijk, An elementary proof of the descent theorem for Grothendieck toposes, J. Pure Appl. Algebra 37 (1985), no. 2, 185 – 191. · Zbl 0559.18001 [14] Ieke Moerdijk, Continuous fibrations and inverse limits of toposes, Compositio Math. 58 (1986), no. 1, 45 – 72. · Zbl 0587.18003 [15] -, Prodiscrete groups (to appear). · Zbl 1228.22001 [16] I. Moerdijk and G. C. Wraith, Connected locally connected toposes are path-connected, Trans. Amer. Math. Soc. 295 (1986), no. 2, 849 – 859. · Zbl 0592.18003 [17] Ross Street, Fibrations in bicategories, Cahiers Topologie Géom. Différentielle 21 (1980), no. 2, 111 – 160. · Zbl 0436.18005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.