Conversely, for any egg (*M*,*e*) of a concrete category, seeing *M*
as a set of function symbols, there is a unique *M*-algebra structure on every object
*X* satisfying both axioms of action and Mor_{M}(*M*,*X*) ⊂
Mor(*M*,*X*).
This structure • on *M* makes (*M*,*e*,•) a monoid acting on all objects
and _{C}*X*,*Y*, Mor(*X*,*Y*) ⊂
Mor_{M}(*X*,*Y*),*X* = *M*.

The hypothesis imply ∀

As (

The last monoid axiom comes as Id

We conclude Mor

In a concrete category with products, the definition of ⋅ can be written

⋅**⃗** ∈ Mor(*M*,*X ^{X}*) ∧
⋅

This monoid (*M*,*e*,•) is essentially the opposite of the monoid End(*M*).
Indeed

∀*x*,*y*∈*M*, •**⃖**_{x} ∈ End(*M*)
∧ •**⃖**_{y} ∈ End(*M*) ∧ •**⃖**_{x} ⚬
•**⃖**_{y} (*e*)
= •**⃖**_{x}(*y*) = *y*•*x*.

For the most general case, consider a concrete category

∀_{C}*E*, ∀*u*∈*E*^{(V)},
∃!*f*∈Mor(*K*,*E*), *f*⚬*b* = *u*

∀_{C}*E*, ∀*u*∈*E*^{(V)},
φ**⃖**_{E}(*u*) = ℩{*f*∈Mor(*K*,*E*) |
*f*⚬*b* = *u*} = *b*_{(E)}^{-1}(*u*)

∴ Mor(*K*,*E*) = {φ**⃖**_{E}(*u*) |
*u*∈*E*^{(V)}}

∀_{C}*E*,*F*, Mor(*E*,*F*) ⊂
Mor_{K}(*E*,*F*)

Elements of Im

As Mor(*K*,*E*) was expressed from φ_{E}, it will more precisely
coincide with Mor_{K}(*K*,*E*), once *K* is given a proper
*K*-structure. A minimal natural candidate is the trajectory of (*b*,*s*),
which depends on End(*K*); but the choice of End(*K*) (fitting axioms of
categories with fixed Mor(*K*,*E*)) will not matter.

Indeed, the smallest End(*K*) =
{Id_{K}} gives to *K* its smallest trajectories-defined *K*-structure
{((*s*,*b*),*s*) | *s*∈*K*}. This gives the greatest candidate
Mor_{K}(*K*,*E*), equal to Mor(*K*,*E*); but the reverse
inclusion Mor(*K*,*E*) ⊂ Mor_{K}(*K*,*E*) is another natural
requirement (preservation of the *K*-structure). So the equality is necessary.

For φ_{E} to be algebraic, we shall assume *V* to be
*structureless*, which means ∀*E*,
*E*^{(V)} = *E ^{V}*. For this to hold with

- The empty structure
**V**= ∅ ensures it in all*U*in any case (then*K*≠ ∅ ⇒ Mor_{K}(*K*,*V*) = ∅). - For example
**V**= {((π_{x},Id_{V}),*x*) |*x*∈*V*} still gives Mor_{K}(*V*,*E*) = Mor(*V*,*E*) for*b*-modules*E*structured by trajectories ; precisely, Mor_{K}(*V*,*E*) =*E*when all π^{V}_{x}work in*E*as projections from*E*to^{V}*E*.

If moreover

For any object *K* in *C* with a chosen basis *b*, and any possible symbol
*s* of (preserved) *V*-ary operation in objects of *C*, the element
*s'* = *s _{K}*(

Any operation *s* with lower arity also has a copy *s'* = *s _{K}*(

(*K*, *j*) : ^{C}∐_{i∈I}
*E _{i}*

∀_{C}*F*, ⊓_{i∈I}
*j*_{i (F)} : *K*_{(F)} ↔
∏_{i∈I} *E*_{i (F)}

While the general definions of product and coproduct only determine their isomorphism classes, we shall usually speak of

The category of sets has coproducts given by the
disjoint union with its
natural injections
(_{i∈I} *F ^{Ei}*
⥬

In categories of all relational systems with fixed language

In other concrete categories, the underlying sets of coproducts may differ from disjoint
unions, more often than those of products differ from the products of sets.
The injectivity of each *j*_{i} often remains, but already fails in
categories of partial algebras (4.2).

(*b* is a basis of *K*^{(M)} in
*C*^{(M)}) ⇔ (*K*, *b*) :
^{C}∐_{x∈V} *M*.

These

Products and coproducts have associativity properties like first seen
with the product of sets : any product of products of objects, is naturally isomorphic to a simple product indexed by
the disjoint union of indexing sets; and the same for coproducts.

In particular, a coproduct
(*K*, *j*) : ^{C}∐_{i∈I} *E _{i}* of
objects

Let us call

- Any
*L*-stable subset of an*L*-system is*b*-stable - Thus, any
*L*-stable subset of a*b*-module is a*b*-module. - For any partial
*L*-algebra*E*, Inj*b*_{(E)}, i.e. ∀*u*∈*E*, !^{V}*g*∈Mor_{L}(*K*,*E*),*g*⚬*b*=*u*.

So, equational systems usually aim to distinguish among algebras (or partial algebras) *E*,
those for which the injection *b*_{(E)} is also surjective. In many cases *L*
has no projection symbol, obliging the chosen *L*-structure of *V* to be empty.

- If all symbols in
*L*are*n*-ary, then*V*↪_{n}*V*⊔_{n}*L*with structure {((*s*, Id_{Vn}),*s*) |*s*∈*L*} is an equational system, whose modules among*L*-systems are the algebraic ones ; - Any monoid
*K*forms a*K*-equational system with*V*= {*e*}; then, the*K*-sets are the partial*K*-algebras which are*b*-modules. Indeed any*b*-module*E*gets a*K*-set structure included in**E**, then equality is deduced from the functionality of**E**. - In the construction of algebraic structures on modules we described
*K*as a*K*-equational system. - The axiom of identity on a system
is expressible as it being a module by an equational system with
*V*a singleton, and*K*a pair (details are left as an exercise to the reader) ; - Similarly for associativity with a 3-elements
*V*and a 6-elements*K*.

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory

- 3.1. Galois
connections

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.

3.12. The category of relations

5. Second-order foundations

6. Foundations of Geometry