---
tag: 6837RS7N82
kind: concept
name: Topos
msc: [18B25, 03G30]
sources:
  - url: https://en.wikipedia.org/wiki/Topos
same_as:
  wikidata: Q2143621
  wikipedia: Topos
---

## Summary

An (elementary) topos is a [[category|A2DE66937D]] with finite limits, [[exponential objects|RHWD8DY41R]] and a [[subobject classifier|NKP5TPQ79X]].
The category of sets is a topos, and so is the category of [[sheaves|RT8A8FJ566]] on any [[site|NYE0Z94MME]].
A topos behaves like a universe of sets in which one can do much of mathematics, but its internal logic is in general [[intuitionistic|Q11N8X99FK]].
Toposes connect geometry, where they arise as generalized spaces, with logic, where they serve as models of higher-order theories.

## Explanations

### L1 human

The category of sets has products, function sets and a set $\set{0, 1}$ of truth values that classifies subsets.
A topos is any category with these three features.
Sets on which a fixed group acts form a topos, as do sets that vary continuously over a space (sheaves).
In such a world most constructions with sets still work.
But the law of excluded middle may fail, because a statement can be true on part of a space and false elsewhere.