S.Albayrak Anasayfa Kategoriler

3.3 Categories

Definition: A category C consists of

  1. at least 1 morphism, identity morphism 1A, in Hom(A,A)
  2. one can compose morphisms: Hom(A,B) × Hom(B,C) Hom(A,C)
  3. compositions are associative
  4. identity morphisms are identities with respect to composition, i.e. for f Hom(A,B), f1A = 1Bf = f
  5. Hom(A,B) and Hom(C,D) are distinct unless A = C and B = D.

Definition: if domain and codomain of a morphism is same, then it is called endomorphism: End(A) := Hom(A,A).

Definition: a morphism f Hom(A,B) is an isomorphism if it has a two sided inverse, i.e. g s.t. gf = 1A and fg = 1B.

If all morphisms in a category are isomorphisms, then that category is called groupoids.

If an endomorphism is also an isomorpihm, it is called automorphism. The set of automorphisms is denoted as Aut(A), which is a subset of End(A).

Aut(A) is a group for all objects A of all categories.

Definitions: A morphism f Hom(A,B) is a monomorphism if for all objects Z and all morphisms α,α Hom(Z,A) of the category, we have

αα f αf α (3.3)

In words monomorphisms cannot morph different morphisms into the same one. Similarly, epimorphisms cannot be morphed into the same morphism by different morphisms, with the definition: A morphism f Hom(A,B) is a epimorphism if for all objects Z and all morphisms β,β Hom(B,Z) of the category, we have

ββ β fβ f (3.4)

A morphism being both a monomorphism and epimorphism does not necessarily guarantee that it is also an isomorphism. This is only true in some categories such as Set.