nLab
product

Definition

In a category, a product (also called a cartesian product) of two objects $X$ and $Y$ is an object $X\times Y$ equipped with morphisms $p:X\times Y\to X$ and $q:X\times Y\to Y$, called projections, such that for any object $Z$ equipped with maps $f:Z\to X$ and $g:Z\to Y$ there exists a unique $h:Z\to X\times Y$ (called the pairing of $f$ and $g$) such that $p\circ h=f$ and $q\circ h=g$.

Such a categorical product is equivalently a limit over a diagram of the shape of the category $\{a,b\}$ with two objects and no non-trivial morphisms.

More generally, for $S\in$ Set $\hookrightarrow$ Cat any discrete category, an $S$-indexed limit is a product of $\mid S\mid$ factors. If $S$ is a finite set we speak of a finite product. Notice that this includes the case of the empty set, the product over which is, if it exists, the terminal object of the category in question.

• A product created by the forgetful functor of a concrete category, especially in algebra, is often called a direct product.

Properties

• When they exist, products are unique up to unique canonical isomorphism, so we often say “the product.”

• One can define in a similar way a product of any family of objects. A product of the empty family is a terminal object.

• A product is a special case of a limit in which the domain category is discrete.

• A product in $C$ is the same as a coproduct in its opposite category $C^{\mathrm{op}}$.

Examples

• This interactive demonstration in FinSet lets you type two sets and see a product, showing the unique map to it from a randomly chosen $Z$ equipped with $f$ and $g$. It also generates a second product, showing the unique canonical isomorphism between this and the first product.

• product of simplices

Related concepts

• product type