---
tag: Q05K04QJ71
kind: concept
name: Pullback
msc: [18A30]
specializes: [CHS5AFJGWE]
sources:
  - url: https://en.wikipedia.org/wiki/Pullback_(category_theory)
same_as:
  wikidata: Q1397439
  wikipedia: Pullback (category theory)
---

## Summary

The pullback, or fibre product, of two morphisms $f \colon X \to Z$ and $g \colon Y \to Z$ with a common target is a universal commutative square over them.
It is the [[limit|CHS5AFJGWE]] of the diagram $X \to Z \leftarrow Y$.
In sets it is the set of pairs $(x, y)$ with $f(x) = g(y)$, and it includes preimages and intersections as special cases.
Pullbacks are used to restrict, intersect and change the base of objects throughout geometry and topology.

## Explanations

### L1 human

Two maps land in the same place $Z$.
The pullback collects the pairs of points, one from $X$ and one from $Y$, that land on the same point of $Z$.
If $Y$ is a subset of $Z$ and $g$ is the inclusion, the pairs are just the points $x$ with $f(x) \in Y$, so the pullback is the preimage $f^{-1}(Y)$.
If both $X$ and $Y$ are subsets of $Z$ with their inclusions, the pullback is the intersection $X \cap Y$.