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. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

Definition

Product of two objects

Fix a category C. Let X_1 and X_2 be objects of C. A product of X_1 and X_2 is an object X, typically denoted equipped with a pair of morphisms satisfying the following universal property: Whether a product exists may depend on C or on X_1 and 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: if is another product, there exists a unique isomorphism such that and. The morphisms \pi_1 and \pi_2 are called the canonical projections or projection morphisms; the letter \pi alliterates with projection. Given Y and f_1, f_2, the unique morphism f is called the product of morphisms f_1 and f_2 and is denoted

Product of an arbitrary family

Instead of two objects, we can start with an arbitrary family of objects indexed by a set I. Given a family of objects, a product of the family is an object X equipped with morphisms satisfying the following universal property: The product is denoted If then it is denoted and the product of morphisms is denoted

Equational definition

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

As a limit

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 I considered as a discrete category. The definition of the product then coincides with the definition of the limit, {f}_i being a cone and projections being the limit (limiting cone).

Universal property

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 \mathbf{J} as the discrete category with two objects, so that is simply the product category The diagonal functor assigns to each object X the ordered pair (X, X) and to each morphism f the pair (f, f). The product in C is given by a universal morphism from the functor \Delta to the object in This universal morphism consists of an object X of C and a morphism which contains projections.

Examples

In the category of sets, the product (in the category theoretic sense) is the Cartesian product. Given a family of sets X_i the product is defined as with the canonical projections Given any set Y with a family of functions the universal arrow is defined by Other examples: l∞ ]] norm.

Discussion

An example in which the product does not exist: In the category of fields, the product does not exist, since there is no field with homomorphisms to both \Q and F_p. Another example: An empty product (that is, I 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 G there are infinitely many morphisms \Z \to G, so G cannot be terminal. If I is a set such that all products for families indexed with I exist, then one can treat each product as a functor How this functor maps objects is obvious. Mapping of morphisms is subtle, because the product of morphisms defined above does not fit. First, consider the binary product functor, which is a bifunctor. For we should find a morphism We choose This operation on morphisms is called Cartesian product of morphisms. Second, consider the general product functor. For families we should find a morphism We choose the product of morphisms A category where every finite set of objects has a product is sometimes called a Cartesian category (although some authors use this phrase to mean "a category with all finite limits"). The product is associative. Suppose C is a Cartesian category, product functors have been chosen as above, and 1 denotes a terminal object of C. We then have natural isomorphisms 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.

Distributivity

For any objects of a category with finite products and coproducts, there is a canonical morphism where the plus sign here denotes the coproduct. 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 then guarantees a unique morphism 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