Automorphism ⇔ (Endomorphism ∧ Isomorphism)

Isomorphism ⇔ (Retraction ∧ Monomorphism) ⇔ (Section ∧ Epimorphism)

Retraction ⇒ Surjective morphism ⇒ Epimorphism

Section ⇒ Embedding ⇒ Injective morphism ⇒ Monomorphism

Isomorphism ⇒ Elementary embedding ⇒ Embedding

- Hom(
*X*,*f*) = (Mor(*X*, Dom*f*)∋*g*↦*f*০*g*), with target Mor(*X*,*F*) for any target*F*of*f*. - Hom
_{F}(*f*,*X*) = (Mor(*F*,*X*)∋*g*↦*g*০*f*), with target Mor(*E*,*X*). In abstract categories where*f*determines*F*, the notation simplifies as Hom(*f*,*X*).

Hom(*X*,*g*) ০ Hom(*X*,*f*) =
Hom(*X*,*g*০*f*)

Hom_{F}(*f*,*X*) ০
Hom_{G}(*g*,*X*) =
Hom_{G}(*g*০*f*,*X*)

Hom

**Monomorphism**. In a category, a morphism
*f*∈Mor(*E*,*F*)
is called *monic*, or a *monomorphism*, if Hom(*X*,*f*)
is injective for all objects *X*.

In any concrete category, all injective morphisms are monic, and
any morphism with image *F* is *F-epic*.
However, the converses may not hold, and exceptions may be uneasy
to classify, especially as the condition depends on the whole
category and not just the morphism at hand.

The following 2 concepts may be considered cleaner as they admit a local characterization:

**Sections.** A morphism *f*∈Mor(*E*,*F*) is
called a *section* (or *section in F* if the category
is concrete), if Id_{E}∈Im(Hom_{F}(*f*,*E*)),
i.e. ∃*g*∈Mor(*F*,*E*),*g*০*f*=Id_{E}.
Then *f* is monic and for all objects *X* we have
Im(Hom_{F}(*f*,*X*)) = Mor(*E*,*X*).

**Retraction.** A morphism *g*∈Mor(*F*,*E*)
is called a *retraction* (or *retraction on E* if the
category is concrete), if Id_{E}∈Im(Hom(*E*,*g*)),
i.e. ∃*f*∈Mor(*E*,*F*),*g*০*f*=Id_{E}.
Then *g* is epic and for all objects *X* we have
Im(Hom(*X*,*g*)) = Mor(*X*,*F*).

When *g*০*f*=Id_{E} we also say that *f*
is a section of *g*, and that *g* is a retraction of *f*.

- Hom
_{E}(*g*,*X*) is injective (*g*is epic) - Im(Hom
_{F}(*f*,*X*)) = Mor(*E*,*X*). Namely, ∀*h*∈Mor(*E*,*X*),*h*=*h*০*g*০*f*= Hom_{F}(*f*,*X*)(*h*০*g*).

- Hom(
*X*,*f*) is injective (*f*is monic) - Im(Hom(
*X*,*g*)) = Mor(*X*,*F*).∎

**Isomorphism**. In a concrete category, an *isomorphism*
between objects *E*,*F* , is a bijection *f*
: *E* ↔ *F* such that *f* ∈Mor(*E*,*F*)
and *f*^{ -1}∈Mor(*F*,*E*).

In any category, an isomorphism is an *f* ∈Mor(*E*,*F*) such
that ∃*g*∈Mor(*F*,*E*),
*g*০*f*= Id_{E}
∧ *f*০*g*= Id_{F} (this inverse *g* of *f*
is unique).

Any epic section

Similarly, any monic retraction is an isomorphism.

Two objects

**Endomorphisms.** An *endomorphism* of an object *E*
in a category, is a morphism from *E* to itself. Their set
is written End(*E*)=Mor(*E*,*E*).

**Automorphisms.** An *automorphism* of an object *E* is an isomorphism
of *E* to itself.

Embeddings will usually be supposed injective, as it means strongly preserving the equality
relation. Things can come down to this case by replacing equality in the concept of injectivity by a
properly defined equivalence relation, or replacing systems by their quotient by this relation,
where the canonical surjections would be non-injective embeddings.

Injective embeddings are isomorphisms to their images.

Embeddings still strongly preserve structures defined using the
symbols in *L* and the logical symbols ∧,∨,0,1,¬, and also = in
the case of injective embeddings.

Thus, they also preserve invariant structures defined
using symbols of *L* and ∧,∨,¬,0,1,∃ where any
occurrence of ¬ comes after (inside) any occurrence of ∃.

**Elementary embedding.** An *f* ∈
Mor_{L}(*E*,*F*) is called an *elementary embedding*
(or *elementary* *L*-*embedding*) if it (strongly) preserves all
invariant structures (defined by first-order formulas with language *L*).

Every isomorphism is an elementary embedding.

If *f*∈ End(*E*) is an invariant elementary embedding then it is an automorphism:

ImThe existence of an elementary embeddingfis also invariant (as a unary relation ∃y,f(y)=x)

∴ ∀x∈E,x∈Imf⇔f(x)∈Imf

∴ Imf=E. ∎

**Elementary equivalence.** 2 systems are said to be
elementarily equivalent, if every ground first-order formula true
in the one is true in the other.

Such objects have this remarkable property: when they exist, all such objects are isomorphic, by a unique isomorphism between any two of them.

Proof: For any initial objects *X* and *Y*, ∃*f*∈Mor(*X*,*Y*),
∃*g*∈Mor(*Y*,*X*), *g*০*f* ∈Mor(*X*,*X*)
∧ *f*০*g* ∈Mor(*Y*,*Y*).

But Id_{X} ∈ Mor(*X*,*X*) which is a
singleton, thus *g*০*f*= Id_{X}.
Similarly, *f*০*g*=Id_{Y}.

Thus *f* is an isomorphism, unique because Mor(*X*,*Y*)
is a singleton.∎

Such objects exist in many categories, but are not always interesting. For example, in any category of relational systems containing representatives (copies) of all possible ones with a given language:

- Final objects are the singletons (where all relations are
constantly true),

- The only initial object is the empty set (where any nullary relation, i.e. boolean constant, is false).

Does it have an initial object ? a final object ?

More texts on algebra

Back to homepage : Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory (continued)

3.1. Morphisms of relational systems and concrete categories4. Model Theory

3.2.Special morphisms

3.3. Algebras

3.4. Algebraic terms and term algebras

3.5. Integers and recursion

3.6. Arithmetic with addition

4.1. Finiteness and countability

4.2. The Completeness Theorem

4.3. Infinity and the axiom of choice

4.4. Non-standard models of Arithmetic

4.5. How theories develop

4.6. The Incompleteness Theorem