Abstract Algebra/Category theory

Category theory is the study of categories, which are collections of objects and morphisms (or arrows), from one object to another. It generalizes many common notions in Algebra, such as different kinds of products, the notion of kernel, etc. See Category Theory for additional information.

Definitions & NotationsEdit

Definition 1: A (locally small) category   consists of

A collection   of objects.
A collection   of morphisms.
For any  ,   is the subcollection of   of morphisms from   to  , where each   is required to be a set (hence the term locally small).

These obey the following axioms:

There is a notion of composition. If  ,   and  , then   and   are called a composable pair. Their composition is a morphism  .
Composition is associative.   whenever the composition is defined.
For any object  , there is an identity morphism   such that if   are objects,   and  , then   and  .

Note that we demand neither   nor   to be sets; if they are both in fact sets, then we call our category small.

Definition 2: A morphism   has associated with it two functions   and   called domain and codomain respectively, such that   if and only if   and  . Thus two morphisms   are composable if and only if  .

Remark 3: Unless confusion is possible, we will usually not specify which Hom-set a given morphism belongs to. Also, unless several categories are in play, we will usually not write  , but just "  is an object". We may write   or   to implicitly indicate the Hom-set   belongs to. We may also omit the composition symbol, writing simply   for  .

Basic PropertiesEdit

Lemma 4: Let   be an object of a category. The identity morphism for   is unique.

Proof: Assume   and   are identity morphisms for  . Then  .

Example 5: We present some of the simplest categories:

i)   is the empty category, with no objects and no morphisms.
ii)   is the category containging only a single object and its identity morphism. This is the trivial category.
iii)   is the category with two objects,   and  , their identity morphisms, and a single morphism  .
iv) We can also have a category like  , but where we have two morphisms   with  . Then   and   are called parallel morphisms.
v)   is the category with three objects  . We have  ,   and  .

Initial and Final ObjectsEdit

Definition An object   in a category is called initial or cofinal, if for any object   there exists a unique morphism  

Lemma If   and   are initial objects, then they are isomorphic.

Proof: Let   and   be the unique morphisms between   and  . Given that both   and   have a unique endomorphism because of their initiality, this morphism must be the identity. Therefore   and   are the respective identity morphisms, making   and   isomorphic.

Definition An object   in a category is called final or coinitial, if for any object   there exists a unique morphism  

Lemma If   and   are final objects, then they are isomorphic.

Proof Pass to isomorphicness of initial objects in the cocategory.

Some examples of categoriesEdit

  •  : the category whose objects are sets and whose morphisms are maps between sets.
  •  : the category whose objects are finite sets and whose morphisms are maps between finite sets.
  • The category whose objects are open subsets of   and whose morphisms are continuous (differentiable, smooth) maps between them.
  • The category whose objects are smooth (differentiable, topological) manifolds and whose morphisms are smooth (differentiable, continuous) maps.
  • Let   be a field. Then we can define  : the category whose objects are vector spaces over   and whose morphisms are linear maps between vector spaces over  .
  •  : the category whose objects are groups and whose morphisms are homomorphisms between groups.

In all the examples given thus far, the objects have been sets with the morphisms given by set maps between them. This is not always the case. There are some categories where this is not possible, and others where the category doesn't naturally appear in this way. For example:

  • Let   be any category. Then its opposite category   is a category with the same objects, and all the arrows reversed. More formally, a morphism in   from an object   to   is a morphism from   to   in  .
  • Let   be any monoid. Then we can define a category with a single object, with morphisms from that object to itself given by elements of   with composition given by multiplication in  .
  • Let   be any group. Then we can define a category with a single object, with morphisms from that object to itself given by elements of   with composition given by multiplication in  .
  • Let   be any small category, and let   be any category. Then we can define a category   whose objects are functors from   to   and whose morphisms are natural transformations between the functors from   to  .
  •  : the category whose objects are small categories and whose morphisms are functors between small categories.