Classifying topos
{{technical|date=November 2020}}
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
- {{citation|authorlink1=Olivia Caramello|last=Caramello|first=Olivia|title=Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic 'bridges'|publisher= Oxford University Press|year= 2017|isbn= 9780198758914|doi=10.1093/oso/9780198758914.001.0001}}
- {{citation|mr=1300636|last1=Mac Lane|first1= Saunders|last2=Moerdijk|first2= Ieke
|title=Sheaves in geometry and logic. A first introduction to topos theory|series= Universitext|publisher= Springer-Verlag|place= New York|year=1992|isbn= 0-387-97710-4 }}
- {{citation|mr=1440857|last=Moerdijk|first= I.|title=Classifying spaces and classifying topoi|series=Lecture Notes in Mathematics|volume= 1616|publisher= Springer-Verlag|place= Berlin|year= 1995|isbn= 3-540-60319-0|doi=10.1007/BFb0094441}}