Categories.2C objects.2C and morphisms 2 Category theory

categories

a category c consists of following 3 mathematical entities:

a class ob(c), elements called objects;
a class hom(c), elements called morphisms or maps or arrows. each morphism f has source object , target object b.

the expression f : → b, verbally stated f morphism b .

the expression hom(a, b) – alternatively expressed homc(a, b), mor(a, b), or c(a, b) – denotes hom-class of morphisms b.
a binary operation ∘, called composition of morphisms, such 3 objects a, b, , c, have hom(b, c) × hom(a, b) → hom(a, c). composition of f : → b , g : b → c written g ∘ f or gf, governed 2 axioms:

associativity: if f : → b, g : b → c , h : c → d h ∘ (g ∘ f) = (h ∘ g) ∘ f, and
identity: every object x, there exists morphism 1x : x → x called identity morphism x, such every morphism f : → b, have 1b ∘ f = f = f ∘ 1a.






from axioms, can proved there 1 identity morphism every object. authors deviate definition given identifying each object identity morphism.



morphisms

relations among morphisms (such fg = h) depicted using commutative diagrams, points (corners) representing objects , arrows representing morphisms.

morphisms can have of following properties. morphism f : → b a:

monomorphism (or monic) if f ∘ g1 = f ∘ g2 implies g1 = g2 morphisms g1, g2 : x → a.
epimorphism (or epic) if g1 ∘ f = g2 ∘ f implies g1 = g2 morphisms g1, g2 : b → x.
bimorphism if f both epic , monic.
isomorphism if there exists morphism g : b → such f ∘ g = 1b , g ∘ f = 1a.
endomorphism if = b. end(a) denotes class of endomorphisms of a.
automorphism if f both endomorphism , isomorphism. aut(a) denotes class of automorphisms of a.
retraction if right inverse of f exists, i.e. if there exists morphism g : b → f ∘ g = 1b.
section if left inverse of f exists, i.e. if there exists morphism g : b → g ∘ f = 1a.

every retraction epimorphism, , every section monomorphism. furthermore, following 3 statements equivalent:

f monomorphism , retraction;
f epimorphism , section;
f isomorphism.

cite error: there <ref group=lower-alpha> tags or {{efn}} templates on page, references not show without {{reflist|group=lower-alpha}} template or {{notelist}} template (see page).