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 & Notations
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 .
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 Objects
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 categories
- : 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.