Product (category theory)

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.

satisfying the following universal property: Whether a product exists may depend on

If it does exist, it is unique up to canonical isomorphism, because of the universal property, so one may speak of the product.

is another product, there exists a unique isomorphism

satisfying the following universal property: The product is denoted

Alternatively, the product may be defined through equations.

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.

considered as a discrete category.

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

This universal morphism consists of an object

In the category of sets, the product (in the category theoretic sense) is the Cartesian product.

Other examples: An example in which the product does not exist: In the category of fields, the product

is the empty set) is the same as a terminal object, and some categories, such as the category of infinite groups, do not have a terminal object: given any infinite group

is a set such that all products for families indexed with

exist, then one can treat each product as a functor

[3] How this functor maps objects is obvious.

First, consider the binary product functor, which is a bifunctor.

[4] Second, consider the general product functor.

A category where every finite set of objects has a product is sometimes called a Cartesian category[4] (although some authors use this phrase to mean "a category with all finite limits").

is a Cartesian category, product functors have been chosen as above, and

denotes a terminal object of

These properties are formally similar to those of a commutative monoid; a Cartesian category with its finite products is an example of a symmetric monoidal category.

of a category with finite products and coproducts, there is a canonical morphism

To see this, note that the universal property of the coproduct

guarantees the existence of unique arrows filling out the following diagram (the induced arrows are dashed): The universal property of the product

induced by the dashed arrows in the above diagram.

A distributive category is one in which this morphism is actually an isomorphism.

Thus in a distributive category, there is the canonical isomorphism