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Math and science::Algebra::Aluffi

Category. Definition.

Category

A category C consists of:

  • a class of objects, denoted as Obj(C)
  • a set, denoted as HomC(A,B), for any objects A and B of C. The elements are called morphisms.

The set of morphisms must have the following properties:

Identity
For each object AObj(C), there exists (at least) one morphism 1AHomC(A,A), called the identity on A.
Composition
Morphisms can be composed: any two morphisms fHomC(A,B) and gHomC(B,C) determine the existance of another morphism gfHomC(A,C).
Associativity of composition
For any fHomC(A,B), gHomC(B,C) and hHomC(C,D), we have:
(hg)f=h(gf).
Identity law
The identity morphisms are identities with respect to composition. For any fHomC(A,B), we have:
f1A=f,1Bf=f
Morphism sets are disjoint
For any A,B,C,DObj(C), then HomC(A,B) and HomC(C,D) are disjoint unless A=C and B=D.

Composition as the existance of a function

The statement of morphism composition above can be stated alternatively as the existance of a function.

For any A,B,CObj(C), there exists a function HomC(A,B)×HomC(B,C)HomC(A,C) which is denoted as fg.

The existance of a function imposes the requirement that for any element of the domain HomC(A,B)×HomC(B,C) there must be an element in the co-domain HomC(A,C). The function existance requirement allows for an empty domain if one or both of HomC(A,B) or HomC(B,C) are empty.

Using functions to imply non-empty sets

I find the following bi-implication interesting:

Let Sa and Sb be sets. Then

(SaSb)f:SaSb

This bi-implication allows function existance to be used instead of the LHS. How does the nature of morphism composition change when the RHS is replaced with the LHS of the bi-implication above? Maybe this has an effect on what we view as the datum of a category.

Example

Category of sets

There is a category whose objects are sets and whose morphisms are all the functions between sets. This set often is given special syntax, such as Set or Set.

Category from relation

Let S be a set and be a relation on S which is both reflexive and transitive. Then we can form a category like so:

  • The elements of S are the objects of the category.
  • For some a,bS, let Hom(a,b)=(a,b) iff ab, else let Hom(a,b)=.

It's best to keep in mind the relation as being represented by the graph ΓS×S, where (a,b)Γ iff ab. The case where is the equivalence relation '=' and thus all morphisms are identity morphisms, produces a category which is said to be discrete.

This example demonstrates the flexibility of the definition of morphisms: the morphisms do not need to be functions, they can be elements of a set (or any other type of object) as long as the requirements of the morphisms in the above definition are satisfied.


Source

Aluffi, p19
Leinster, p11