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.

Definition

Let C be a category with some objects X1 and X2. A product of X1 and X2 is an object X (often denoted X1 × X2) together with a pair of morphisms π1 : XX1, π2 : XX2 that satisfy the following universal property:

• for every object Y and every pair of morphisms f1 : YX1, f2 : YX2 there exists a unique morphism f : YX1 × X2 such that the following diagram commutes:

The unique morphism f is called the product of morphisms f1 and f2 and is denoted f1, f2. The morphisms π1 and π2 are called the canonical projections or projection morphisms.

Above we defined the binary product. Instead of two objects we can take an arbitrary family of objects indexed by some set I. Then we obtain the definition of a product.

An object X is the product of a family (Xi)iI of objects if there exist morphisms πi : XXi such that for every object Y and every I-indexed family of morphisms fi : YXi, there exists a unique morphism f : YX such that the following diagrams commute for all i in I:

The product is denoted ΠiI Xi. If I = {1, ..., n} then it is denoted X1 × ... × Xn and the product of morphisms is denoted f1, ..., fn.

Equational definition

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

• Existence of f is guaranteed by existence of the operation ⟨ -,- ⟩.
• Commutativity of the diagrams above is guaranteed by the equality f1, ∀f2, ∀i∈{1, 2}, πi of1, f2 = fi.
• Uniqueness of f is guaranteed by the equality g : YX, ⟨ π1 o g, π2 o g = g.

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 J as the discrete category with two objects, so that CJ is simply the product category C × C. The diagonal functor Δ : CC × 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.

Examples

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

ΠiI Xi := { (xi)iI | iI, xiXi }

with the canonical projections

πj : ΠiI XiXj, πj((xi)iI) := xj.

Given any set Y with a family of functions fi : YXi, the universal arrow f : Y → ΠiI Xi is defined by f(y) := (fi(y))iI .

Other examples:

Discussion

The product does not necessarily exist. For example, an empty product (i.e. 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 → 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 CIC. 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 f1 : X1Y1, f2 : X2Y2 we should find a morphism X1 × X2Y1 × Y2. We choose f1 o π1, f2 o π2 . This operation on morphisms is called cartesian product of morphisms. Second, consider the general product functor. For families {X}i,{Y}i, fi : XiYi we should find a morphism ΠiI Xi → ΠiI Yi. We choose the product of morphisms {fi o πi}i.

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 the terminal object of C. We then have natural isomorphisms

$X\times (Y\times Z)\simeq (X\times Y)\times Z\simeq X\times Y\times Z,$ $X\times 1\simeq 1\times X\simeq X,$ $X\times Y\simeq Y\times X.$ These properties are formally similar to those of a commutative monoid; a category with its finite products constitutes a symmetric monoidal category.

Distributivity

For any objects X, Y, and Z of a category with finite products and coproducts, there is a canonical morphism X × Y + X × ZX × (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 × ZX × (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

$X\times (Y+Z)\simeq (X\times Y)+(X\times Z)$ .