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Categories: Category theory

Product (category theory)

Contents

Motivation

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.

Definition

Let C be a category and let {X_i | i ∈ I} be an indexed family of objects in C. The product of the set {X_i} is an object X together with a collection of morphisms π_i : X → X_i (called projections) which satisfy a universal property: for any object Y and any collection of morphisms f_i : Y → X_i, there exists a unique morphism f : Y → X such that for all i ∈ I it is the case that f_i = π_i f. That is, the following diagram commutes (for all i):

If the family of objects consists of only two members X, Y, the product is usually written X×Y, and the diagram takes a form along the lines of:

The unique arrow h making this diagram commute is notated <f,g>.

Discussion

The product construction given above is actually a special case of a limit in category theory. The product can be defined as the limit of any discrete subcategory in C. Not every family {X_i} needs to have a product, but if it does, then the product is unique in a strong sense: if π_i : X → X_i and π'_i : X' → X_i are two products of the family {X_i}, then (by the definition of products) there exists a unique isomorphism f : X → X' such that π_i = π'_i f 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 ∏_i X_i, and the maps π_i are known as the natural projections. We have a natural isomorphism

$\operatorname{Mor}_C\left(Y,\prod_{i\in I}X_i\right) \simeq \prod_{i\in I}\operatorname{Mor}_C(Y,X_i)$

(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

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

$X\times 1 \simeq 1\times X \simeq X$

$X\times Y \simeq Y\times X$

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

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