×

Sheaves and local equivalence relations. (English) Zbl 0559.18006

A local equivalence relation r on a topological space X is a global section of the sheaf of germs of equivalence relations on X. One can construct a category sh(X;r) of r-sheaves, which under certain conditions on r is a topos and in fact an étendue. Thus, it can be described as a topos of \({\mathfrak G}\)-sheaves, where \({\mathfrak G}\) is an étale topological groupoid. The groupoid \({\mathfrak G}\) can be explicitly described for a large class of r by utilizing the construction of the holonomy groupoid of a differentiable piece of a groupoid due to J. Pradines.

MSC:

18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
54B40 Presheaves and sheaves in general topology
18B10 Categories of spans/cospans, relations, or partial maps
18F10 Grothendieck topologies and Grothendieck topoi
18B25 Topoi
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] 1 N. Bourbaki , Variétés différentielles et analytiques (Fasc. de Rés.) , Hermann , Paris 1967 . MR 219078 | Zbl 0171.22004 · Zbl 0171.22004
[2] 2 R. Brown , Holonomy and monodromy groupoids , Univ. College of North Wales , Pure Math., Preprint n^\circ 82-2, 1982 . · Zbl 1051.18008
[3] 3 R. Brown , G. Danesh-Naruie & J.P.L. Hardy , Topological groupoids II: Covering morphisms and G-spaces , Math. Nachr. 74 ( 1976 ), 143 - 156 . MR 442937 | Zbl 0281.22002 · Zbl 0281.22002 · doi:10.1002/mana.3210740110
[4] 4 C. Ehresmann , Structures locales , Ann. di Mate. Pura e appl. 36 ( 1954 ) ; republished in ” Charles Ehresmann: Oeuvres complètes et commentées ”, Partie I, Amiens , 1984 . MR 65922 | Zbl 0055.42002 · Zbl 0055.42002 · doi:10.1007/BF02412835
[5] 5 C. Ehresmann , Les connexions infinitésimales dans un espace fibré différentiable , Coll. Top. Bruxelles 1951 , in ” Charles Ehresmann: Oeuvres complètes et commentées ”, Partie I, Amiens , 1984 . MR 42768 | Zbl 0054.07201 · Zbl 0054.07201
[6] 6 C. Ehresmann , Structures feuilletées , Proc. 5th Canad. Math. Cong. Montréal 1961; in ” Charles Ehresmann: Oeuvres complètes et commentées ”, Partie II-2, Amiens 1982 . Zbl 0146.19501 · Zbl 0146.19501
[7] 7 C. Ehresmann , Catégories topologiques, catégories différentiables , Coll. Géom. Diff. Glob. Bruxelles , CBRM 1958 ; in ” Charles Ehresmann: Oeuvres complètes et commentées ”, Partie I, Amiens 1984 . · Zbl 0205.28202
[8] 8 C. Ehresmann , Catégories topologiques III , Indig. Math. 28 - 1 , 1966 ; in ” Charles Ehresmann: Oeuvres complètes et commentées ”, Partie II-2, Amiens 1982 . MR 215766 | Zbl 0163.26802 · Zbl 0163.26802
[9] 9 C. Ehresmann , Prolongements des catégories différentiables , Cahiers Top. et Géom. Diff. VI , 1964 ; in ” Charles Ehresmann: Oeuvres complètes et commentées ” Partie I, Amiens 1984 . Numdam | MR 179225 | Zbl 0192.10102 · Zbl 0192.10102
[10] 10 C. Ehresmann , Propriétés infinitésimales des catégories différentiables , Cahiers Top. et Géom. Diff. IX - 1 , 1967 ; in ” Charles Ehresmann: Oeuvres complètes et commentées ”, Partie I, Amiens , 1984 . Numdam | MR 244336 | Zbl 0192.10103 · Zbl 0192.10103
[11] 11 A. Grothendieck & J.L. Verdier , Théorie des topos (SGA 4, Vol. 1 ), Lecture Notes in Math. 269 , Springer ( 1972 ). MR 354653 | Zbl 0256.18008 · Zbl 0256.18008
[12] 12 P.T. Johnstone , Topos Theory , Academic Press , 1977 . MR 470019 | Zbl 0368.18001 · Zbl 0368.18001
[13] 13 J. Pradines , Théorie de Lie pour les groupoides différentiables. Relations entre propriétés locales et globales , C.R.A.S. A 26 ( 1966 ), 907 - 910 . MR 214103 | Zbl 0147.41102 · Zbl 0147.41102
[14] 14 G. Reeb , Sur la théorie générale des systèmes dynamiques , Ann. Inst. Fourier VI , Grenoble ( 1955 ), 89 - 116 . Numdam | MR 81464 | Zbl 0071.11001 · Zbl 0071.11001 · doi:10.5802/aif.61
[15] 15 K. Rosenthal , Aspects of Etendues , Ph.D. Thesis, State Univ. of N.Y. at Buffalo , 1979 .
[16] 16 K. Rosenthal , Etendues and categories with monic maps , J. Pure and Appl. Algebra 22 ( 1981 ), 193 - 212 . MR 624572 | Zbl 0473.18004 · Zbl 0473.18004 · doi:10.1016/0022-4049(81)90059-1
[17] 17 K. Rosenthal , Local equivalence relations , Top. and its Appl. 13 ( 1982 ), 167 - 176 . MR 644111 | Zbl 0486.18012 · Zbl 0486.18012 · doi:10.1016/0166-8641(82)90018-9
[18] 18 K. Rosenthal , Quotient systems in Grothendieck topoi , Caniers Top. et Géom. Diff. XXIII- 4 ( 1982 ), 425 - 438 . Numdam | MR 693508 | Zbl 0521.18005 · Zbl 0521.18005
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.