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.
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Let be a category with some objects and . An object is the product of and , denoted , iff it satisfies this universal property:
The unique morphism is called the product of morphisms and and is denoted .
Above we defined the binary product. Instead of two objects we can take an arbitrary family of objects indexed by some set . Then we obtain the definition of a product.
An object is the product of a family of objects iff there exist morphisms , such that for every object and a -indexed family of morphisms there exists a unique morphism such that the following diagrams commute for all :
The product is denoted ; if , then denoted and the product of morphisms is denoted .
Alternatively, product may be defined totally by equations, here is an example for binary product:
Also product may be derived from limit. A family of objects is a diagram without morphisms. If we regard our diagram as a functor, it is a functor from considered as a discrete category. Then the definition of product coincides with the definition of limit, being a cone and projections being the limit (limiting cone).
As well as limit, product may be defined via universal property. For comparison see Limit#Universal property. Lets unfold that definition for binary product. In our case is a discrete category with two objects, is simply the product category , diagonal functor assigns to each object the ordered pair and to each morphism the pair . The product in is given by a universal morphism from the functor to the object in . This universal morphism consists of an object of and a morphism which contains projections.
with the canonical projections
Given any set Y with a family of functions
the universal arrow f is defined as
The product does not necessarily exist. For example, an empty product (i.e. 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 there are infinitely many morphisms , so cannot be terminal.
If is a set such that all products for families indexed with exist, then it is possible to choose the products in a compatible fashion so that the product turns into a functor . 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 we should find a morphism . We choose . This operation on morphisms is called cartesian product of morphisms.[2] Second, consider product functor. For families we should find a morphism . We choose the product of morphisms .
A category where every finite set of objects has a product is sometimes called a cartesian category[2] (although some authors use this phrase to mean "a category with all finite limits").
Suppose is a cartesian category, product functors have been chosen as above, and denotes the terminal object of . We then have natural isomorphisms
These properties are formally similar to those of a commutative monoid; a category with its finite products constitutes a symmetric monoidal category.
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 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