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. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.
Fix a category C. Let X1 and X2 be objects of C. A product of X1 and X2 is an object X, typically denoted X1 × X2, equipped with a pair of morphisms π1 : X → X1, π2 : X → X2 satisfying the following universal property:
Whether a product exists may depend on C or on X1 and X2. If it does exist, it is unique up to canonical isomorphism, because of the universal property, so one may speak of the product.
The morphisms π1 and π2 are called the canonical projections or projection morphisms. Given Y and f1, f2, the unique morphism f is called the product of morphisms f1 and f2 and is denoted ⟨f1, f2⟩.
Instead of two objects, we can start with an arbitrary family of objects indexed by a set I.
Given a family (Xi)i∈I of objects, a product of the family is an object X equipped with morphisms πi : X → Xi satisfying the following universal property:
Alternatively, the product may be defined through equations. So, for example, for the binary product:
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 J as the discrete category with two objects, so that CJ is simply the product category C × C. The diagonal functor Δ : C → C × C assigns to each object X the ordered pair (X, X) and to each morphism f the pair (f, f). The product X1 × X2 in C is given by a universal morphism from the functor Δ to the object (X1, X2) in C × C. This universal morphism consists of an object X of C and a morphism (X, X) → (X1, X2) which contains projections.
In the category of sets, the product (in the category theoretic sense) is the Cartesian product. Given a family of sets Xi the product is defined as
Given any set Y with a family of functions fi : Y → Xi, the universal arrow f : Y → Πi∈I Xi is defined by f(y) := (fi(y))i∈I .
An example in which the product does not exist: In the category of fields, the product Q × Fp does not exist, since there is no field with homomorphisms to both Q and Fp.
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").
For any objects X, Y, and Z of a category with finite products and coproducts, there is a canonical morphism X × Y + X × Z → X × (Y + Z), where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct X × Y + X × Z guarantees the existence of unique arrows filling out the following diagram (the induced arrows are dashed):
The universal property of the product X × (Y + Z) then guarantees a unique morphism X × Y + X × Z → X × (Y + Z) 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, one has the canonical isomorphism