Classifying topos

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
  • 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.
  • The classifying space of topological groups in homotopy theory.

References
  • 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, 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, 1616, Berlin: Springer-Verlag, doi:10.1007/BFb0094441, ISBN 3-540-60319-0, MR 1440857
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