- 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*).

**Representation of small categories.** Any small category has an interpretation
as a family of typed algebras, with all
morphisms between them.

See each object

Take the language of function symbols where the set of those from type

The proof goes on just like with monoids.∎

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

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*).

For any initial objectsBy this unique isomorphism,X,Y, ∃f∈Mor(X,Y), ∃g∈Mor(Y,X),g•f∈ Mor(X,X) ∧ 1_{X}∈ Mor(X,X) ∴g•f= 1_{X}.

Similarly,f•g= 1_{Y}. Thusfis an isomorphism, unique because Mor(X,Y) is a singleton.∎

Similarly, an object

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 where relations are constantly true, are final objects in categories of one-type systems or algebras; for multi-type systems, final objects are made of one singleton per type.
- The empty set where Boolean constants are false is the initial object.

- Objects are all (
*X*,φ) where*X*is a set and φ:*X*×*K*→*B*; - Mor((
*X*,φ),(*Y*,φ') = {*f*∈*Y*| ∀^{X}*a*∈*X*,∀*k*∈*K*, φ(*a*,*k*) = φ'(*f*(*a*),*k*)}.

- Id
_{A}∈Mor(*A*,*E*) - For any object
*N*, Mor(*N*,*A*) = {*h*∈Mor(*N*,*E*) | Im*h*⊂*A*} - An
*f*∈Mor(*N*,*E*) is an isomorphism to*A*when it is an embedding to*E*and Im*f*=*A*.

namely copying that of

Such disagreements could already occur in categories of relational systems. Then let an injective

{*f*੦*g* | *g*∈Mor(*N*,*K*)} = Mor(*N*,*A*).

From this, consider the category where

- Objects are all morphisms of
*C*into*A*, we may conceive as (*F*,*f*) where*F*is an object of*C*,*f*∈Mor(*F*,*E*) and Im*f*⊂*A*; - Mor((
*F*,*f*),(*G*,*g*)) = {*h*∈Mor(*F*,*G*) |*f*=*g*০*h*}

Dependencies between qualities of morphisms, can be mapped as follows:

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

Section ⇒ Embedding ⇒ Injective morphism ⇒ Monomorphism

- Objects are all (
*X*,*x*) where*X*is an object of*C*and*x*∈*X* - Mor((
*X*,*x*),(*Y*,*y*)) = {*f*∈Mor(*X*,*Y*) |*f*(*x*)=*y*}.

- If
*C*is the category of*M*-sets for a monoid (*M*,*e*, •) then, seeing*M*as an*M*-set interpreting • as left action, (*M*,*e*) is an initial object of*C'*; initial objects are the (*X*,*x*) where*x*is a free and generating element of*X*. - Conversely, for any initial object (
*M*,*e*) of*C'*(if that exists), there is a unique monoid structure (*M*,*e*,•) with an action on every other object*X*of*C*(beyond • on*M*itself), such that for all objects*X*,*Y*of*C*we have Mor(*X*,*Y*) ⊂ Mor_{M}(*X*,*Y*) and Mor(*M*,*X*) = Mor_{M}(*M*,*X*).

2. Defining ∀

The composition in

The last axiom of monoid,

∀

This monoid (

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. Invertibility and groups

3.7.Categories

3.8. Algebraic terms

3.9. Term algebras

3.10. Integers and recursion

3.11. Presburger Arithmetic