Morphisms

Definition Isomorphisms

NOTE

A morphism $f\in Ho{m}_C(A,B)$ is an isomorphism if it has an inverse under composition: There exists a morphism $g\in Ho{m}_C(B,A)$ so that: $gf=1_A,fg=1_B$

Note that in $gf=1_A$ f is calculated first. so we have $A\to B\to A$

If $f$ is an isomorphism, then $f^{-1}$ is an isomorphism and $(f^{-1}{)}^{-1}=f$

Examples

the isomorphisms in the category Set are precisely the bijections see Ways to prove Isomorphism in group theory which is the isomorphism in the category Grp Grp has the morphism: Homomorphism (see Section 13 Homomorphisms)

Definition Endomorphisms

A morphism of an object A of a category C to itself is called an endomorphism In Grp It is defined as: Rings of Endomorphisms

Definition Groupoid

NOTE

A category, in which every morphism is an isomorphism, is called groupoid.

• What is a Category
• Difference between Morphism and Function
• Definition Isomorphisms

Definition: Automorphism

An Automorphism of an object A of category C is an isomorphism from A to itself.

Algebra Chapter 0, p.51

If a morphism is a endomorphism and an isomorphism at the same time, it is a automorphism

The set of automorphisms of A is denoted $Au{t}_C(A)$

• Every element $f\in Au{t}_C(A)$ has an inverse $f^{-1}\in Au{t}_C(A)$
• $Au{t}_C(A)$ contains the element $1_A$
• composition is associative
• the composition of two elements $f,g$ in this set is again an element $gf\in Au{t}_C(A)$

Tip

This Definition seems familiar, doesn’t it? See Section 4 Groups

See how Automorphism is defined in Group Theory: 14.15 Definition: Automorphism

Difference between Automorphism and endomorphism

Property	Endomorphism	Automorphism
Definition	A morphism $f:A\to A$ in a category.	An invertible endomorphism $f:A\to A$.
Hom-set	$f\in \mathrm{Hom}(A,A)$.	$f\in \mathrm{Hom}(A,A)$ and $f^{-1}$ exists.
Invertibility	Not required.	Required: there exists gg with $gf=1_A$ and $fg=1_A$.
Algebraic structure	Monoids under composition.	Groups under composition.
Role	“Self-map” preserving structure.	“Symmetry” or “automagic move” of A.
Examples	Any linear map $V\to V$ (e.g. projection).	Any invertible linear map (e.g. rotation).
Note that $Au{t}_C(A)\subseteq En{d}_C(A)$
Automorphism is a stricter morphism.

Definition Monomorphism

NOTE

Let C be a category. A morphism $f\in Ho{m}_C(A,B)$ is a monomorphism if:

for all objects Z of C and all morphisms ${\alpha }^{\prime }$ ${\alpha }^{\prime \prime }$ $\in Ho{m}_C(Z,A)$ $f\circ {\alpha }^{\prime }=f\circ {\alpha }^{\prime \prime }⟹{\alpha }^{\prime }={\alpha }^{\prime \prime }$

Or we can write it in this way: $f({\alpha }^{\prime })=f({\alpha }^{\prime \prime })⟹{\alpha }^{\prime }={\alpha }^{\prime \prime }$ Similar to the definition of injective functions in the Set Theory

In the category Set the monomorphisms are precisely the injective functions (one-to-one)

Definition Epimorphism

NOTE

Let C be a category. A morphism $f\in Ho{m}_C(A,B)$ is a Epimorphism if:

for all objects Z of C and all morphisms ${\alpha }^{\prime }$ ${\alpha }^{\prime \prime }$ $\in Ho{m}_C(A,Z)$ ${\alpha }^{\prime }\circ f={\alpha }^{\prime \prime }\circ f⟹{\alpha }^{\prime }={\alpha }^{\prime \prime }$

In the category Set the monomorphisms are precisely the surjective functions (onto)

Difference between monomorphism and Epimorphism is just left composition and right composition.