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 or 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.

DefinitionEdit

Product of two objectsEdit

Fix a category   Let   and   be objects of   A product of   and   is an object   typically denoted   equipped with a pair of morphisms     satisfying the following universal property:

  • For every object   and every pair of morphisms     there exists a unique morphism   such that the following diagram commutes:

Whether a product exists may depend on   or on   and   If it does exist, it is unique up to canonical isomorphism, because of the universal property, so one may speak of the product. This has the following meaning: let   be another cartesian product, there exists a unique isomorphism   such that   and  .

The morphisms   and   are called the canonical projections or projection morphisms. Given   and     the unique morphism   is called the product of morphisms   and   and is denoted  

Product of an arbitrary familyEdit

Instead of two objects, we can start with an arbitrary family of objects indexed by a set  

Given a family   of objects, a product of the family is an object   equipped with morphisms   satisfying the following universal property:

  • For every object   and every  -indexed family of morphisms   there exists a unique morphism   such that the following diagrams commute for all  

The product is denoted   If   then it is denoted   and the product of morphisms is denoted  

Equational definitionEdit

Alternatively, the product may be defined through equations. So, for example, for the binary product:

  • Existence of   is guaranteed by existence of the operation  
  • Commutativity of the diagrams above is guaranteed by the equality: for all   and all    
  • Uniqueness of   is guaranteed by the equality: for all    [1]

As a limitEdit

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).

Universal propertyEdit

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.

ExamplesEdit

In the category of sets, the product (in the category theoretic sense) is the Cartesian product. Given a family of sets   the product is defined as

 
with the canonical projections
 
Given any set   with a family of functions   the universal arrow   is defined by  

Other examples:

DiscussionEdit

An example in which the product does not exist: In the category of fields, the product   does not exist, since there is no field with homomorphisms to both   and  

Another example: An empty product (that is,   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 one can treat each product as a functor  [3] How this functor maps objects is obvious. Mapping of morphisms is subtle, because the product of morphisms defined above does not fit. First, consider the 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.[4] Second, consider the general 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[4] (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 a terminal object of   We then have natural isomorphisms

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

DistributivityEdit

For any objects   of a category with finite products and coproducts, there is a canonical morphism   where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct   guarantees the existence of unique arrows filling out the following diagram (the induced arrows are dashed):

The universal property of the product   then guarantees a unique morphism   induced by the dashed arrows in the above diagram. A distributive category is one in which this morphism is actually an isomorphism. Thus in a distributive category, there is the canonical isomorphism

 

See alsoEdit

ReferencesEdit

  1. ^ Lambek J., Scott P. J. (1988). Introduction to Higher-Order Categorical Logic. Cambridge University Press. p. 304.
  2. ^ Qiaochu Yuan (June 23, 2012). "Banach spaces (and Lawvere metrics, and closed categories)". Annoying Precision.
  3. ^ Lane, S. Mac (1988). Categories for the working mathematician (1st ed.). New York: Springer-Verlag. p. 37. ISBN 0-387-90035-7.
  4. ^ a b Michael Barr, Charles Wells (1999). Category Theory – Lecture Notes for ESSLLI. p. 62. Archived from the original on 2011-04-13.

External linksEdit