*E*=^{n}*E*×...×*E*=*E*, set of all^{Vn}*n*-tuples of elements of*E*(*n*-ary product or exponentiation).- Rel
_{E}^{(n)}= ℘(*E*), set of all^{n}*n*-ary relations in*E* - Op
_{E}^{(n)}=*E*, set of all^{En}*n*-ary operations in*E*.

^{L}*E* = ∐_{s∈L}
*E ^{ns}*

A *relational language* is a language *L* of *relation symbols*,
where each *s*∈*L* aims to be interpreted in any *L*-system *E* as
an *n _{s}*-ary relation. These form a family called an

∏_{s∈L} ℘(*E ^{ns}*)
⥬ ℘(

^{L}Id_{E} = Id_{LE}

∀_{fnc}*f*,*g*, Im *f*
⊂ Dom *g* ⇒ ^{L}(*g*⚬*f*) =
* ^{L}g* ⚬

Inj

Im

We shall omit parenthesis in notations of direct and inverse images in the way

∀*A*⊂*E*, * ^{L}f*[

∀

The case of an *algebraic language*, whose symbols aim to represent operations,
will be studied in 3.3.

*s _{E}* = {

*f* ∈ Mor_{L}(*E*,*F*) ⇔
∀(*s*,*x*)∈**E**, (*s*,*f*⚬*x*)∈**F**

⇔ (∀*s*∈*L*,∀*x*∈*E*^{ns},
*s*(*x*) ⇒ *s*(*f*⚬*x*))

⇔
* ^{L}f*[

- a class of sets called
*objects*; - a class of functions called
*morphisms*; then for any objects*E*,*F*, we define

Mor(*E*,*F*) = {*f*∈*F*|^{E}*f*is a morphism}

- Every morphism belongs to some Mor(
*E*,*F*), i.e. its domain is an object and its image is included in an object (in practice, images of morphisms will be objects too); - For any object
*E*, Id_{E}∈ Mor(*E*,*E*) ; - Any composite of morphisms is a morphism: for any 3 objects
*E*,*F*,*G*, ∀*f*∈ Mor(*E*,*F*), ∀*g*∈Mor(*F*,*G*),*g*⚬*f*∈ Mor(*E*,*G*).

The simplest example is the

A category is *small* if its class of objects is a set.

For example, we shall speak of the category of all

In any given category of

Proof. For any

- Substituting arguments of a
*s*∈*L*by a map σ to*n*' other variables (∀*E*,∀*x*∈*E*^{n'},*s'*(*x*) ⇔*s*(*x*⚬σ)), works :*s'*(*x*) ⇒*s*(*x*⚬σ) ⇒*s*(*f*⚬*x*⚬σ) ⇒*s'*(*f*⚬*x*). - ∀
*s*,*s*'∈*L*,*n*=_{s}*n*⇒ ∀_{s'}*x*∈*E*^{ns}, (*s*(*x*) ∧*s'*(*x*)) ⇒ (*s*(*f*⚬*x*) ∧*s'*(*f*⚬*x*)) - ∀
*s*,*s*'∈*L*,*n*=_{s}*n*⇒ ∀_{s'}*x*∈*E*^{ns}, (*s*(*x*) ∨*s'*(*x*)) ⇒ (*s*(*f*⚬*x*) ∨*s'*(*f*⚬*x*)) - For 0 and 1 it is trivial

- ∀
*x*,*y*∈*E*,*x*=*y*⇒*f*(*x*) =*f*(*y*) - ∀
*x*∈*E*^{ns},(∃*y*∈*E*,*s*(*x*,*y*)) ⇒ (∃*z*=*f*(*y*)∈*F*,*s*(*f*⚬*x*,*z*)) ∎

- In the nullary case: for any
*f*∈ Mor_{L}(*E*,*F*), if a ground formula with language*L*using only these logical symbols is true in*E*, then it is also true in*F*. - The graph of any operation defined by a term is expressible in this way, and thus also preserved. But the proofs given by the above means are only a schema of one proof for each "small" (concretely written) term. A single proof for the full range of all terms will be given in 4.1.

The above cases of 0, 1, ∨ and ∧ are mere particular cases (the nullary and binary cases) of the following:

- Any union of a family of preserved structures in a concrete category is a preserved structure.
- Any intersection of a family of preserved structures is also a preserved structure.

Then the concept of morphism between systems

- As a tuple or family of functions (
*f*)_{i}_{i∈τ}, where ∀*i*∈τ,*f*:_{i}*E*→_{i}*F*;_{i} - As a function
*f*:*E*→*F*that is a τ-morphism seeing τ as a list of unary relation symbols (like for the use of classes as notions in set theory), or equivalently such that*t*⚬_{F}*f*=*t*._{E}

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory

3.1. Galois connections4. Arithmetic and first-order foundations

3.2.Relational systems and concrete categories

3.3. Algebras

3.4. Special morphisms

3.5. Monoids and categories

3.6. Actions of monoids and categories

3.7. Invertibility and groups

3.8. Properties in categories

3.9. Initial and final objects

3.10. Products of systems

3.11. Basis

3.12. The category of relations

5. Second-order foundations

6. Foundations of Geometry

Other languages:

FR : Systèmes relationnels et catégories concrètes