# 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

- 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

## External links

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