In mathematics, a classifying topos for some sort of structure is a toposT such that there is a natural equivalence between geometric morphisms from a cocomplete topos E to T and the category of models for the structure in E.
Examplesedit
The classifying topos for objects of a topos is the topos of presheaves over the opposite of the category of finite sets.
The classifying topos for rings of a topos is the topos of presheaves over the opposite of the category of finitely presented rings.
The classifying topos for local rings of a topos is the topos of sheaves over the opposite of the category of finitely presented rings with the Zariski topology.
The classifying topos for linear orders with distinct largest and smallest elements of a topos is the topos of simplicial sets.
If G is a discrete group, the classifying topos for G-torsors over a topos is the topos BG of G-sets.
Caramello, Olivia (2017), Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic 'bridges', Oxford University Press, doi:10.1093/oso/9780198758914.001.0001, ISBN9780198758914
Mac Lane, Saunders; Moerdijk, Ieke (1992), Sheaves in geometry and logic. A first introduction to topos theory, Universitext, New York: Springer-Verlag, ISBN0-387-97710-4, MR 1300636
Moerdijk, I. (1995), Classifying spaces and classifying topoi, Lecture Notes in Mathematics, vol. 1616, Berlin: Springer-Verlag, doi:10.1007/BFb0094441, ISBN3-540-60319-0, MR 1440857
classifying, topos, this, article, technical, most, readers, understand, please, help, improve, make, understandable, experts, without, removing, technical, details, november, 2020, learn, when, remove, this, message, mathematics, classifying, topos, some, sor. This article may be too technical for most readers to understand Please help improve it to make it understandable to non experts without removing the technical details November 2020 Learn how and when to remove this message In mathematics a classifying topos for some sort of structure is a topos T such that there is a natural equivalence between geometric morphisms from a cocomplete topos E to T and the category of models for the structure in E Examples editThe classifying topos for objects of a topos is the topos of presheaves over the opposite of the category of finite sets The classifying topos for rings of a topos is the topos of presheaves over the opposite of the category of finitely presented rings The classifying topos for local rings of a topos is the topos of sheaves over the opposite of the category of finitely presented rings with the Zariski topology The classifying topos for linear orders with distinct largest and smallest elements of a topos is the topos of simplicial sets If G is a discrete group the classifying topos for G torsors over a topos is the topos BG of G sets The classifying space of topological groups in homotopy theory References editCaramello Olivia 2017 Theories Sites Toposes Relating and studying mathematical theories through topos theoretic bridges Oxford University Press doi 10 1093 oso 9780198758914 001 0001 ISBN 9780198758914 Mac Lane Saunders Moerdijk Ieke 1992 Sheaves in geometry and logic A first introduction to topos theory Universitext New York Springer Verlag ISBN 0 387 97710 4 MR 1300636 Moerdijk I 1995 Classifying spaces and classifying topoi Lecture Notes in Mathematics vol 1616 Berlin Springer Verlag doi 10 1007 BFb0094441 ISBN 3 540 60319 0 MR 1440857External links editClassifying topos at the nLab nbsp This category theory related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Classifying topos amp oldid 1026573566, wikipedia, wiki, book, books, library,
