he 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.

Let $C$ be a category with some objects $X\_1$ and $X\_2$. An object $X$ is the product of $X\_1$ and $X\_2$, denoted $X\_1\; \backslash times\; X\_2$, iff it satisfies this universal property:

there exist morphisms $\backslash pi\_1$

X \to X_1, \pi_2

X \to X_2, called the canonical projections or projection morphisms, such that for every object $Y$ and pair of morphisms $f\_1$

Y \to X_1, f_2

Y \to X_2 there exists a unique morphism $f$

Y \to X such that the following diagram commutes:

X \to X_i, such that for every object $Y$ and a $I$-indexed family of morphisms $f\_i$

Y \to X_i there exists a unique morphism $f$

Y \to X such that the following diagrams commute for all $i\; \backslash in\; I$:

Existence of $f$ is guaranteed by the operation $\backslash lang\; -,-\; \backslash rang$.

Commutativity of diagrams above are guaranteed by the equality $\backslash forall\; f\_1,\; \backslash forall\; f\_2,\; \backslash forall\; i\; \backslash in\; \backslash ,\backslash \; \backslash pi\_i\; \backslash circ\; \backslash langle\; f\_1,\; f\_2\backslash rangle\; =\; f\_i$.

Uniqueness of $f$ is guaranteed by the equality $\backslash forall\; f,\backslash \; \backslash langle\; \backslash pi\_1\; \backslash circ\; f,\backslash pi\_2\; \backslash circ\; f\; \backslash rangle\; =\; f$.

C \to C \times C assigns to each object $X$ the ordered pair $(X,X)$ and to each morphism $f$ the pair $(f,f)$. The product $X\_1\; \backslash times\; X\_2$ in $C$ is given by a universal morphism from the functor $\backslash Delta$ to the object $(X\_1,X\_2)$ in $C\; \backslash times\; C$. This universal morphism consists of an object $X$ of <
Examples

In 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

$\backslash prod\_\; X\_i$

= \

with the canonical projections

$\backslash pi\_j$

\prod_ X_i \to X_j \mathrm \quad \pi_j((x_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

$f:Y\; \backslash to\; \backslash prod\_\; X\_i\; \backslash mathrm\; \backslash quad\; f(y)$

= (f_i(y))_

Other examples:

In the category of topological spaces, the product is the space whose underlying set is the cartesian product and which carries the product topology. The product topology is the coarsest topology for which all the projections are continuous.

In the category of modules over some ring R, the product is the cartesian product with addition defined componentwise and distributive multiplication.

In the category of groups, the product is the direct product of groups given by the cartesian product with multiplication defined componentwise.

In the category of relations (Rel), the product is given by the disjoint union. (This may come as a bit of a surprise given that the category of sets (Set) is a subcategory of Rel.)

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).

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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 $\backslash mathbb\; \backslash to\; G$, so $G$ cannot be terminal. 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 $C^I\; \backslash to\; C$. How this functor maps objects is obvious. Mapping of morphisms is subtle, because product of morphisms defined above does not fit. First, consider binary product functor, which is a bifunctor. For $f\_1:X\_1\backslash to\; Y\_1,\; f\_2:X\_2\backslash to\; Y\_2$ we should find a morphism $X\_1\backslash times\; X\_2\; \backslash to\; Y\_1\backslash times\; Y\_2$. We choose $\backslash langle\; f\_1\; \backslash circ\; \backslash pi\_1,\; f\_2\; \backslash circ\; \backslash pi\_2\; \backslash rangle$. This operation on morphisms is called cartesian product of morphisms.(although some authors use this phrase to mean "a category with all finite limits").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

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

$X\backslash times\; 1\; \backslash simeq\; 1\backslash times\; X\; \backslash simeq\; X$

$X\backslash times\; Y\; \backslash simeq\; Y\backslash times\; X$

These properties are formally similar to those of a commutative monoid; a category with its finite products constitutes a symmetric monoidal category.

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Distributivity

In 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 we have various canonical projections and injections which fill out the diagramcenterThe 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

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

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See also

Coproduct – the dual of the product

Limit and colimits

Equalizer

Inverse limit

Cartesian closed category

Categorical pullback

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