- A class of "objects" of that category (which need not be sets);
the category is
*small*if this class is a set; - to any objects
*A*,*B*is given a set Mor(*A*,*B*) of «morphisms from*A*to*B*»; these are usually regarded as pairwise disjoint; - to any object
*A*is given 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*) ;

- 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*)

Again, an *automorphism* of an object *E*, is an isomorphism from *E* to itself.
Their set Aut(*E*) is the group of invertible elements of the monoid
End(*E*)=Mor(*E*,*E*).

This gives back the concept of concrete category as a category with a chosen effective action (where effectiveness is defined by the injectivity of this function from Mor(

Mor(* ^{k}X*,

Any object *X* in a category, naturally defines an action of this category by giving
to any object *E* the set * ^{X}E* = Mor(

Hom(*X*, *f*) = (Mor(*X*,*E*)∋*g*↦
*f*•*g*), with target Mor(*X*,*F*) for any target *F* of *f*.

∀*f*,*g*∈Mor(*X*,*Y*), Hom(*M*,*f*) =
Hom(*M*,*g*) ⇔ ∀*h*∈* ^{M}X*,

Hom(*f*,*X*) =
(Mor(*F*,*X*)∋*g*↦ *g*•*f*)

The composition axioms satisfied by these actions and co-actions, are written: for any 4 objects

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

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

**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*:

∀*g*,*h*∈Mor(*X*,*E*),
*f*•*g* = *f*•*h* ⇒ *g* = *h*.

∀*g*,*h*∈Mor(*F*,*X*),
*g*•*f* = *h*•*f* ⇒ *g* = *h*.

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.

**Sections, retractions.** When *g*•*f* = 1_{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 1_{E}∈Im(Hom_{F}(*f*,*E*)), i.e. ∃*g*∈Mor(*F*,*E*),*g*•*f*=1_{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 1_{E}∈Im(Hom(*E*,*g*)), i.e. ∃*f*∈Mor(*E*,*F*),*g*•*f*=1_{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*•*g*•*f*= Hom_{F}(*f*,*X*)(*h*•*g*).

A morphism *f* is an isomorphism if and only if Hom(*X*,*f*) :
Mor(*X*,*E*) ↔ Mor(*X*,*F*); its inverse is then
Hom(*X*, *f*^{ -1}).

In concrete categories, Section ⇒ Injective morphism ⇒ Monomorphism

In categories of relational systems, Retraction ⇒ Quotient ⇒ Surjective morphism ⇒ Epimorphism

**Theorem.** Any small category is isomorphic to that of all morphisms in a family of typed algebras.

- As a set
*T*of types, we can take a copy of the set of objects, but one type per isomorphism class suffices. - Each object
*E*is interpreted as a typed set ∐_{i∈T}where^{i}E= Mor(^{i}E*i*,*E*). *L*= ∐_{i,j∈T}Mor(*j*,*i*) seeing Mor(*j*,*i*) as a set of function symbols from*i*to*j*.

Im ψ ⊂ Mor

The existence of an isomorphism

∀

In particular for any monoid *M* there is a language
*L* of function symbols and an *L*-algebra *X*
such that End_{L} *X* is
isomorphic to *M*.

Any group is isomorphic to a permutation group, namely the group of automorphisms
of an algebra.

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory (continued)

3.1. Morphisms of relational systems and concrete categories4. Arithmetic and first-order foundations

3.2. Algebras

3.3. Special morphisms

3.4. Monoids

3.5. Actions of monoids

3.6. Invertibility and groups

3.7.Categories

3.8. Initial and final objects

3.9. Eggs, basis, clones and varieties

5. Second-order foundations

6. Foundations of Geometry