---
tag: 6ZS77J1873
kind: definition
name: Product
msc: [18A30]
defines: S5744QZV4N
uses: [A2DE66937D, 859F0TFVA3, CHS5AFJGWE, 09CEDJYBKP, 00HF4RJ5QN]
sources:
  - url: https://en.wikipedia.org/wiki/Product_(category_theory)
---

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $(X_i)_{i \in I}$ be a family of objects of $\mathcal{C}$ indexed by a set $I$.
A *product* of the family is an object $P$ with [[morphisms|859F0TFVA3]] $\pi_i \colon P \to X_i$ for $i \in I$, called *projections*, with the following property.
For every object $Y$ and every family of morphisms $f_i \colon Y \to X_i$, there is exactly one morphism $f \colon Y \to P$ with $\pi_i \circ f = f_i$ for all $i$.
The object $P$ is written $\prod_{i \in I} X_i$, or $X_1 \times X_2$ for two objects, and $f$ is written $(f_i)_{i \in I}$ or $\langle f_1, f_2 \rangle$.

## Remarks

A product is the [[limit|CHS5AFJGWE]] of the diagram whose shape is the set $I$ seen as a category with only identity morphisms.
So it is unique up to unique isomorphism, and the product of the empty family is a terminal object.

Example: in the category of sets, the [[Cartesian product|09CEDJYBKP]] $X_1 \times X_2$ with the coordinate maps is a product.
Given $f_1 \colon Y \to X_1$ and $f_2 \colon Y \to X_2$, the map $f(y) = (f_1(y), f_2(y))$ satisfies $\pi_i \circ f = f_i$, and any such $f$ must have these coordinates.
In the category of groups the product is the [[direct product|00HF4RJ5QN]], and in the category of topological spaces it is the Cartesian product with the product topology.

Non-example: in the category of fields, $\mathbb{F}_2$ and $\mathbb{F}_3$ have no product.
Field homomorphisms preserve the characteristic, so no field $P$ has morphisms to both $\mathbb{F}_2$ and $\mathbb{F}_3$.