Product (category theory)

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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, the direct 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.

1 Definition
2 Examples
3 Discussion
4 Distributivity
5 See also

[edit] Definition

Let C be a category and let {X_i | i ∈ I} be an indexed family of (not necessarily distinct) 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 the canonical projections or projection morphisms, which are often, but not always, epimorphisms) 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 the product is usually written X₁×X₂, and the diagram takes the form:

The unique arrow f making this diagram commute is sometimes denoted 〈f₁,f₂〉.

[edit] Examples

In the category Set (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

$\prod_{i \in I} X_i := \{(x_i)_{i \in I} | x_i \in X_i \, \forall i \in I\}$

with the canonical projections

$\pi_j : \prod_{i \in I} X_i \to X_j \mathrm{ , } \quad \pi_j((x_i)_{i \in I}) := x_j$

Given any set Y with a family of functions

$f_i : Y \to X_i$

the universal arrow f is defined as

$f:Y \to \prod_{i \in I} X_i \mathrm{ , } \quad f(y) := (f_i(y))_{i \in I}$

In the category of algebraic varieties, the categorical product is given by the Segre embedding.

In the category of semi-abelian monoids, the categorical product is given by the history monoid.

A partially ordered set can be treated as a category, using the order relation as the morphisms. In this case the products and coproducts correspond to greatest lower bounds (meets) and least upper bounds (joins).

[edit] 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 functor from a discrete category to 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.

As with any universal property, the product can be understood as a universal morphism. Let Δ: C → C×C be the diagonal functor which assigns to each object X the ordered pair (X,X) and to each morphism f:X → Y the pair (f,f). Then the product X×Y in C is given by a universal morphism from the functor Δ to the object (X,Y) in C×C.

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

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

(where Hom_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). Thus the covariant hom-functor takes products to products. This is a consequence of the fact that the hom-functor is continuous.

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; a category with its finite products and terminal object constitutes a symmetric monoidal category.

[edit] Distributivity

In general, 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 we have various canonical projections and injections which fill out the diagram

The universal property for X×(Y+Z) then guarantees a unique morphism X×Y+X×Z → X×(Y+Z). A distributive category is one in which this morphism is actually an isomorphism. Thus in a distributive category, one has the canonical isomorphism