Morphisms

Monic and Epic

Monomorphism

In any category , an morphism is called a monomorphism (monic) if given any , implies . We write if is a monomorphism.

e.g. A function is monic if and only if it is injective.

Epimorphisim

In any category , an morphism is called an epimorphism (epic) if given any , implies . Write if is a monomorphism.

e.g.

• A function is epic if and only if it is surjective.
• In a fixed poset category , every morphism is both monic and epic. Because for any , we have unique and . Thus monic and epic.

Proposition

The composition of monomorphisms is a monomorphism.

Proof Suppose and . For all with , we have , it follows that , thus , therefore is also monic.

Proposition

Given any monoid homomorphisms , if the restrictions to are equal, , then .

Proof Note that and similarly for . Indeed, Hence .

e.g. In the category of monoids and monoid homomorphisms, there is a monic homomorphism where is the additive monoid of natural numbers, and is the additive monoid of integers. This map, given by the inclusion of sets, is also epic in .

Isomorphisms

Isomorphism

In any category , an arrow is called an isomorphism if there is an arrow in such thatSince inverses are unique, we write . We say that is isomorphic to , written , if there exists an isomorphism between them.

We shall check the inverse is unique. Suppose and are both inverse of . Then we have:

Proposition

Every isomorphism is both monic and epic. The converse is not true in general.

Proof Suppose is an isomorphism, thenThus is monic. Similarly, it is also epic.

Endomorphism & Automorphism

An endomorphism is a morphism whose domain equals its codomain. An endomorphism that is an isomorphism is called an automorphism.

Groupoid

A groupoid is a category in which every morphism is an isomorphism.

Proposition

A group is a groupoid with one object .

Proof Clearly that singleton is a category . For all arrows , we have $$f_{x}\circ f_{x^{-1}}= 1_{G}$$$\square$

Lemma Any category contains a maximal groupoid, the subcategory containing all of the objects and only those morphisms that are isomorphisms.

Cayley’s Theorem

Cayley’s theorem in group theory says that any abstract group can be represented as a “concrete” one. The theorem can in fact be generalized to show that any category that is not “too big” can be represented as one that is “concrete”, i.e. a category of sets and functions.

Theorem

Every category with a set of arrows is isomorphic to one in which the objects are sets and the arrows are functions.

Proof Define the Cayley representation of to be the following concrete category with objects for all . Morphisms are functions for any in , defined by . Indeed, they are isomorphic as functor is invertible.

Sections and Retractions

Proposition

Prop If an morphism has a left inverse such that then then must be monic and epic. Proof Suppose , then is monic as any such that follows that , implies that . Similarly is epic.

Def Split Monomorphism A split monomorphism (epimorphism) is an arrow with a left (right) inverse. e.g. In , every monomorphism splits except those with domain .

Def Section and Retraction Given arrows and such that , then is called a section or splitting of , and e is called a retraction of . The object is called a retract of .

Prop Functors also preserve split epimorphisms and split monomorphisms. Proof

Def Projective Objects An object is called projective if for any epimorphism and morphism there is some morphism such that :

Prop In any category, any retract of a projective object is also projective. Proof Suppose arrows and such that with projective . For all epimorphism and morphism , let . Since is projective, there exists such that . Let . Then Therefore is also projective.

Hom-Set

Hom Set

In any category , we call the following set of morphisms as hom-set:And any any morphism in induces a function :

Def Representable Functor The (covariant) representable functor of isIt is indeed a functor as and .

Proposition

Prop A diagram of the form with object : is a product for and iff for every object , the canonical function is an isomorphism. Proof by definition is equivalent to say that for all , there is unique such that , that is is bijective.

Preserve Binary Products

Let , be categories with binary products. A functor is said to preserve binary products if That is is an isomorphism.

e.g. The forgetful functor preserves binary products.

Corollary For any object in a category with products, the (covariant) representable functor preserves products. Proof For any , the proposition says that there is a canonical isomorphism :

Def Discrete Category A discrete category is a category whose only morphisms are the identity morphisms: for all be -objects.