Product (category theory)

Fix a category ${\displaystyle C.}$ Let ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ be objects of ${\displaystyle C.}$ A product of ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ is an object ${\displaystyle X,}$ typically denoted ${\displaystyle X_{1}\times X_{2},}$ equipped with a pair of morphisms ${\displaystyle \pi _{1}:X\to X_{1},}$ ${\displaystyle \pi _{2}:X\to X_{2}}$ satisfying the following universal property:

• For every object ${\displaystyle Y}$ and every pair of morphisms ${\displaystyle f_{1}:Y\to X_{1},}$ ${\displaystyle f_{2}:Y\to X_{2},}$ there exists a unique morphism ${\displaystyle f:Y\to X_{1}\times X_{2}}$ such that the following diagram commutes:

  

  Universal property of the product

Whether a product exists may depend on ${\displaystyle C}$ or on ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}.}$ If it does exist, it is unique up to canonical isomorphism, because of the universal property, so one may speak of the product. This has the following meaning: let ${\displaystyle X',\pi _{1}',\pi _{2}'}$ be another cartesian product, there exists a unique isomorphism ${\displaystyle h:X'\to X_{1}\times X_{2}}$ such that ${\displaystyle \pi _{1}'=\pi _{1}\circ h}$ and ${\displaystyle \pi _{2}'=\pi _{2}\circ h}$.

The morphisms ${\displaystyle \pi _{1}}$ and ${\displaystyle \pi _{2}}$ are called the canonical projections or projection morphisms; the letter ${\displaystyle \pi }$ (pronounced pi) alliterates with projection. Given ${\displaystyle Y}$ and ${\displaystyle f_{1},}$ ${\displaystyle f_{2},}$ the unique morphism ${\displaystyle f}$ is called the product of morphisms ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ and is denoted ${\displaystyle \langle f_{1},f_{2}\rangle .}$

Instead of two objects, we can start with an arbitrary family of objects indexed by a set ${\displaystyle I.}$

Given a family ${\displaystyle \left(X_{i}\right)_{i\in I}}$ of objects, a product of the family is an object ${\displaystyle X}$ equipped with morphisms ${\displaystyle \pi _{i}:X\to X_{i},}$ satisfying the following universal property:

• For every object ${\displaystyle Y}$ and every ${\displaystyle I}$-indexed family of morphisms ${\displaystyle f_{i}:Y\to X_{i},}$ there exists a unique morphism ${\displaystyle f:Y\to X}$ such that the following diagrams commute for all ${\displaystyle i\in I:}$

  

  Universal product of the product

The product is denoted ${\displaystyle \prod _{i\in I}X_{i}.}$ If ${\displaystyle I=\{1,\ldots ,n\},}$ then it is denoted ${\displaystyle X_{1}\times \cdots \times X_{n}}$ and the product of morphisms is denoted ${\displaystyle \langle f_{1},\ldots ,f_{n}\rangle .}$

Alternatively, the product may be defined through equations. So, for example, for the binary product:

• Existence of ${\displaystyle f}$ is guaranteed by existence of the operation ${\displaystyle \langle \cdot ,\cdot \rangle .}$
• Commutativity of the diagrams above is guaranteed by the equality: for all ${\displaystyle f_{1},f_{2}}$ and all ${\displaystyle i\in \{1,2\},}$ ${\displaystyle \pi _{i}\circ \left\langle f_{1},f_{2}\right\rangle =f_{i}}$
• Uniqueness of ${\displaystyle f}$ is guaranteed by the equality: for all ${\displaystyle g:Y\to X_{1}\times X_{2},}$ ${\displaystyle \left\langle \pi _{1}\circ g,\pi _{2}\circ g\right\rangle =g.}$[1]

The product is a special case of a limit. This may be seen by using a discrete category (a family of objects without any morphisms, other than their identity morphisms) as the diagram required for the definition of the limit. The discrete objects will serve as the index of the components and projections. If we regard this diagram as a functor, it is a functor from the index set ${\displaystyle I}$ considered as a discrete category. The definition of the product then coincides with the definition of the limit, ${\displaystyle \{f\}_{i}}$ being a cone and projections being the limit (limiting cone).

Just as the limit is a special case of the universal construction, so is the product. Starting with the definition given for the universal property of limits, take ${\displaystyle \mathbf {J} }$ as the discrete category with two objects, so that ${\displaystyle \mathbf {C} ^{\mathbf {J} }}$ is simply the product category ${\displaystyle \mathbf {C} \times \mathbf {C} .}$ The diagonal functor ${\displaystyle \Delta$ :\mathbf {C} \to \mathbf {C} \times \mathbf {C} } assigns to each object ${\displaystyle X}$ the ordered pair ${\displaystyle (X,X)}$ and to each morphism ${\displaystyle f}$ the pair ${\displaystyle (f,f).}$ The product ${\displaystyle X_{1}\times X_{2}}$ in ${\displaystyle C}$ is given by a universal morphism from the functor ${\displaystyle \Delta }$ to the object ${\displaystyle \left(X_{1},X_{2}\right)}$ in ${\displaystyle \mathbf {C} \times \mathbf {C} .}$ This universal morphism consists of an object ${\displaystyle X}$ of ${\displaystyle C}$ and a morphism ${\displaystyle (X,X)\to \left(X_{1},X_{2}\right)}$ which contains projections.