Product (category theory)

In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of 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.

Suppose C is a category, I is a set, and for each i in I, an object X_i in C is given. An object X, together with morphisms p_i : X → X_i for each i in I is called a product of the family (X_i) if, whenever Y is an object of C and q_i : Y → X_i are given morphisms, then there exists precisely one morphism r : Y → X such that q_i = p_ir.

The above definition is an example of a universal property; in fact, it is a special limit. Not every family (X_i) needs to have a product, but if it does, then the product is unique in a strong sense: if p_i : X → X_i and p'_i : X ' → X_i are two products of the family (X_i), then there exists a unique isomorphism r : X → X ' such that p'_ir = p_i for each i in I.

An empty product (i.e. I is the empty set) is the same as a terminal object in C.

If I is a set such that all products for families indexed with I exist, then it is possible to choose the products in a compatible fashion so that the product turns into a functor C^I → C. The product of the family (X_i) is then often denoted by ΠX_i, and the maps p_i are known as the natural projections. We have a natural isomorphism[?]

<math>\operatorname{Mor}_C\left(Y,\prod_{i\in I}X_i\right) \simeq \prod_{i\in I}\operatorname{Mor}_C(Y,X_i)</math>

(where Mor_C(U,V) denotes the set of all morphisms from U to V in C, the left product is the one in C and the right is the cartesian product of sets).

If I is a finite set, say I = {1,...,n}, then the product of objects X₁,...,X_n is often denoted by X₁×...×X_n. Suppose all finite products exist in C, product functors have been chosen as above, and 1 denotes the terminal object of C corresponding to the empty product. We then have natural isomorphisms[?]

<math>X\times (Y \times Z)\simeq (X\times Y)\times Z\simeq X\times Y\times Z</math>

<math>X\times 1 \simeq 1\times X \simeq X</math>

<math>X\times Y \simeq Y\times X</math>

These properties are formally similar to those of a commutative monoid.