Product (category theory)
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.
Let be a category with some objects and . An object is a product of and , denoted , iff it satisfies this universal property:
- there exist morphisms such that for every object and pair of morphisms there exists a unique morphism such that the following diagram commutes:
The unique morphism is called the product of morphisms and and is denoted . The morphisms and are called the canonical projections or projection morphisms.
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, the product may be defined through equations. So, for example, for the binary product:
- Existence of is guaranteed by the operation .
- Commutativity of the diagrams above is guaranteed by the equality .
- Uniqueness of is guaranteed by the equality .
As a limit
The product is a special case of a limit. This may be seen by using a discrete category (a family of objects without any morphisms, other than their identity morphisms) as the diagram required for the definition of the limit. The discrete objects will serve as the index of the components and projections. If we regard this diagram as a functor, it is a functor from the index set considered as a discrete category. The definition of the product then coincides with the definition of the limit, being a cone and projections being the limit (limiting cone).
Just as the limit is a special case of the universal construction, so is the product. Starting with the definition given for the universal property of limits, take as the discrete category with two objects, so that is simply the product category . The 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
- 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).
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. 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 (although some authors use this phrase to mean "a category with all finite limits").
The product is associative. Suppose is a cartesian category, product functors have been chosen as above, and denotes the terminal object of . We then have natural isomorphisms
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
- Coproduct – the dual of the product
- Diagonal functor – the left adjoint of the product functor.
- Limit and colimits
- Inverse limit
- Cartesian closed category
- Categorical pullback
- Lambek J., Scott P. J. (1988). Introduction to Higher-Order Categorical Logic. Cambridge University Press. p. 304.
- Michael Barr, Charles Wells (1999). Category Theory - Lecture Notes for ESSLLI. p. 62.
- Adámek, Jiří; Horst Herrlich, and George E. Strecker (1990). Abstract and Concrete Categories. John Wiley & Sons. ISBN 0-471-60922-6.
- Barr, Michael; Charles Wells (1999). Category Theory for Computing Science. Les Publications CRM Montreal (publication PM023). Chapter 5.
- Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed. ed.). Springer. ISBN 0-387-98403-8.