---
tag: S5744QZV4N
kind: concept
name: Product
msc: [18A30]
specializes: [CHS5AFJGWE]
sources:
  - url: https://en.wikipedia.org/wiki/Product_(category_theory)
same_as:
  wikidata: Q919107
  wikipedia: Product (category theory)
---

## Summary

The product of a family of objects in a [[category|A2DE66937D]] is an object with a projection to each member of the family.
Any family of morphisms into the members factors uniquely through it.
It is the [[limit|CHS5AFJGWE]] of a diagram with no non-identity arrows.
It generalises the [[Cartesian product|09CEDJYBKP]] of sets, the [[direct product of groups|00HF4RJ5QN]] and the product of topological spaces.

## Explanations

### L1 human

A map into a product is the same thing as a list of maps, one into each factor.
For sets, a function $f \colon Z \to X \times Y$ is the same as the pair of functions $z \mapsto$ first coordinate and $z \mapsto$ second coordinate of $f(z)$.
The categorical product takes this property as the definition and forgets how the product is built.
In a partially ordered set seen as a category, the product of two elements is their greatest lower bound.