- A class of "objects" of that category, regarded as pure elements (ignoring any inclusion order); the category is called small if this class is a set;
- Between any two objects
*A*,*B*is given a set Mor(*A*,*B*); these are regarded as pairwise disjoint sets of pure elements; - To any object
*A*is given an element 1_{A}∈Mor(*A*,*A*) - To any 3 objects
*A*,*B*,*C*is given a composition operation we shall abusively denote by the same symbol • : Mor(*B*,*C*)×Mor(*A*,*B*)→Mor(*A*,*C*)

satisfying the axioms

- For any objects
*A*,*B*, ∀*x*∈Mor(*A*,*B*),*x*•1_{A}=*x*= 1_{B}•*x* - For any objects
*A,B,C,D,*∀*x*∈Mor(*A*,*B*),∀*y*∈Mor(*B*,*C*),∀*z*∈Mor(*C*,*D*), (*z*•*y*)•*x*=*z*•(*y*•*x*)

Let us fix the set of types as a copy of the set of objects : from each objectXwe make a typeX'(not giving to this bijective correspondence any special status).

Each objectMis interpreted as a system where each typeX'is interpreted as the set Mor(X',M).

As a language, let us take all morphisms between types: the set of function symbols from typeX'to typeY'is defined as Mor(Y',X') (with reverse order, as symbols act on the right).

The proof goes on just like with monoids.∎

- 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*). Simplified as Hom(*f*,*X*) in abstract categories where*f*determines*F*.

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.

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

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

- A morphism
*f*∈Mor(*E*,*F*) is 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*). -
A morphism
*g*∈Mor(*F*,*E*) is 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*).

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

If *f* is an isomorphism then Hom(*X*,*f*) and Hom(*X*,*g*)
are bijections, inverse of each other, between Mor(*X*,*E*) and Mor(*X*,*F*).

Similarly, any monic retraction is an isomorphism.

These dependencies between qualities of morphisms, can be mapped as follows:

Isomorphism ⇔ (Retraction ∧ Monomorphism) ⇔ (Section ∧ Epimorphism)Retraction ⇒ Surjective morphism ⇒ Epimorphism

Section ⇒ Embedding ⇒ Injective morphism ⇒ Monomorphism

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:

- Singletons are final objects (where all relations are constantly true); and also in any category of algebras with a fixed language where they are admitted as objects. In the case of multi-type systems, final objects are made of one singleton per type.
- 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 ?

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. Algebras

3.3. Special morphisms

3.4. Monoids

3.5. Actions of monoids

3.6.Categories

3.7. Algebraic terms and term algebras

3.8. Integers and recursion

3.9. Arithmetic with addition