---
tag: 5H16H7FY5D
kind: concept
name: Extensive category
msc: [18B50]
sources:
  - url: https://en.wikipedia.org/wiki/Extensive_category
same_as:
  wikidata: Q55611299
  wikipedia: Extensive category
---

## Summary

An extensive category is a category with finite [[coproducts|4FJ0S71P58]] that behave like disjoint unions of sets.
Coproducts are disjoint and stable under pullback.
So an object over $A + B$ splits uniquely into a part over $A$ and a part over $B$.
Sets, topological spaces, small categories and every topos are extensive, while categories of modules are not.
Extensivity is the categorical form of the idea that a space can be cut into disjoint pieces.

## Explanations

### L1 human

If a set $X$ maps into a disjoint union $A \sqcup B$, then $X$ breaks into the part that lands in $A$ and the part that lands in $B$.
These two parts do not overlap, and their union is $X$.
An extensive category is one where every object over a sum breaks up in this way.
In vector spaces this fails: the diagonal line in $\R \oplus \R$ meets both axes only at zero, yet it is not the sum of those two meetings.