Automorphism ⇔ (Endomorphism ∧ Isomorphism)
Isomorphism ⇔ (Retraction ∧ Monomorphism) ⇔ (Section ∧ Epimorphism)
Retraction ⇒ Surjective morphism ⇒ Epimorphism
Section ⇒ Embedding ⇒ Injective morphism ⇒ Monomorphism
Isomorphism ⇒ Elementary embedding ⇒ Embedding
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.Epimorphism. In an abstract category, a morphism f∈Mor(E,F) is called epic, or an epimorphism, if Hom(f,X) 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 IdE∈Im(HomF(f,E)),
Then f is monic and for all objects X we have
Im(HomF(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 IdE∈Im(Hom(E,g)),
Then g is epic and for all objects X we have
Im(Hom(X,g)) = Mor(X,F).
When g০f=IdE we also say that f is a section of g, and that g is a retraction of 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), g০f= IdE ∧ f০g= IdF (this inverse g of f is unique).
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.Thus it is both an endomorphism and an isomorphism. However an endomorphism of E which is an isomorphism to a strict subset of E, is not an automorphism. Strong preservation. A function f ∈ FE is said to strongly preserve a relation symbol or formula r interpreted in each of E and F, if it preserves both r and ¬r :
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 ∈ MorL(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:
Im f is also invariant (as a unary relation ∃y, f(y)=x)The existence of an elementary embedding f ∈MorL(E,F) implies that E and F are elementarily equivalent:
∴ ∀x∈ E, x∈Im f ⇔ f(x)∈Im f
∴ Im f = 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.The most usual practice of mathematics focuses on systems where all elementarily equivalent ones are connected by isomorphisms, which are the only elementary embeddings. However, non-surjective elementary embeddings exist and play a special role in foundational issues, such as Skolem's paradox and non-standard models of arithmetic.
Proof: For any initial objects X and Y, ∃f∈Mor(X,Y),
∃g∈Mor(Y,X), g০f ∈Mor(X,X)
∧ f০g ∈Mor(Y,Y).
But IdX ∈ Mor(X,X) which is a singleton, thus g০f= IdX. Similarly, f০g=IdY.
Thus f is an isomorphism, unique because Mor(X,Y) is a singleton.∎
3.1. Morphisms of relational systems and concrete categories4. Model Theory
3.2. Special morphisms
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