---
tag: 85ZZ921DWH
kind: concept
name: Classifying topos
msc: [18F10, 03G30, 18B25]
specializes: [S3YVE24AQK]
sources:
  - url: https://en.wikipedia.org/wiki/Classifying_topos
same_as:
  wikidata: Q17006942
  wikipedia: Classifying topos
---

## Summary

A classifying topos of a geometric theory $\mathbb{T}$ is a [[Grothendieck topos|S3YVE24AQK]] that contains a generic model of $\mathbb{T}$.
Models of $\mathbb{T}$ in any Grothendieck topos $\mathcal{E}$ correspond to [[geometric morphisms|N5R1WAEPFB]] from $\mathcal{E}$ to the classifying topos.
Every geometric theory has a classifying topos, unique up to equivalence.
This turns questions about theories into questions about toposes, and lets one read a topos as a theory.

## Explanations

### L1 human

A classifying topos is a universe that contains one generic model of a theory.
Every model of the theory in another topos is obtained by pulling back this generic model along a map of toposes.
It plays the role that a classifying space plays for vector bundles in topology.
For example, the topos of functors from finite sets to sets classifies "objects": a map into it from a topos $\mathcal{E}$ is the same as one object of $\mathcal{E}$.