A morphism *f*∈Mor(*E*,*F*) is called *monic*, or a
*monomorphism*, if all *f*^{(X)} are injective:

∀_{C}*X*, ∀*g*,*h*∈Mor(*X*,*E*),
*f*∘*g* = *f*∘*h* ⇒ *g* = *h*.

Dually, a morphism *f*∈Mor(*E*,*F*) is called *epic*, or an *epimorphism*,
if all *f*_{(X)} are injective :

∀_{C}*X*, ∀*g*,*h*∈Mor(*F*,*X*),
*g*∘*f* = *h*∘*f* ⇒ *g* = *h*

∀_{C}*X*, *f*_{(X)} :
*F*_{(X)} ↪ *E*_{(X)}

*f*_{(C)} : *C*^{(F)} ↪ *C*^{(E)}.

In the category of sets, epimorphisms are the surjections.

Then a

- ∃
*f*∈Mor(*E*,*F*),*f*;*g*= 1_{E} - For any action α of
*C*,*g*^{α}is surjective (Im*g*^{α}=*E*^{α}). *g*^{(E)}is surjective, i.e. Im*g*^{(E)}= End(*E*).

1. ⇒ 2. : ∀

2. ⇒ 3. obvious;

3. ⇒ 1. as 1

Dually, *f*∈Mor(*E*,*F*) is a section
(or *section in F* if the category is concrete), if, equivalently,

- ∃
*g*∈Mor(*F*,*E*),*f*;*g*= 1_{E}. - For any co-action β of
*C*,*f*_{β}is surjective (Im*f*_{β}=*E*_{β}). *f*_{(E)}is surjective, i.e. Im*f*_{(E)}= End(*E*).

- All actions of sections are injective; thus, all sections are monic;
- All co-actions of retractions are injective; thus, all retractions are epic.

Section ⇒ Injective morphism
⇒ Monomorphism

Retraction ⇒ Surjective morphism ⇒ Epimorphism

If *b* is an isomorphism then all objects are *b*-modules, i.e. *b* is an epic section, and

∀_{C}*X*, (*b*_{(M)}^{-1}
= *b*^{-1}_{(M)}) ∧ (*b*^{(M) -1} =
*b*^{-1 (M)}).

Proof : ∃

Thus :

- If both
*X*,*Y*are*b*-modules then all other objects are also*b*-modules ; - Epic section ⇔ Isomorphism ⇔ Monic retraction ;
- If an element of a monoid is both left invertible and right cancellative, then it is invertible.

An important kind of examples is when *b* is bijective (and thus epic), in the category
of relational systems for a given language *L*. To simplify, let *b* be the identity
Id_{X} into *Y* with larger structure **Y** = **X**
∪ *Z*.

For example, given a binary symbol *r*∈*L*, the properties of reflexivity, symmetry and
transitivity of the interpretation of *r* in a system *M*, are respectively expressible
as *M* being a Re-module, a Sy-module and a Tr-module, where the morphisms Re, Sy and Tr are
respectively defined by

- (Re) :
*X*= {0},**X**= ∅,*Z*= {(*r*,(0,0))} - (Sy) :
*X*= {0,1},**X**= {(*r*,(0,1))},*Z*= {(*r*,(1,0))} - (Tr) :
*X*= {0,1,2},**X**= {(*r*,(0,1)), (*r*,(1,2))},*Z*= {(*r*,(0,2))}

So formalized, this general case of a bijective *b* can be thought of as giving
*Z* the role of a set of *L*-typed **X**-ary algebraic symbols, which for
every set *M* gives to * ^{L}M* the

(*M*, **M**) is a *b*-module
⇔ **M** ∈ Sub_{Z} * ^{L}M*.

- The anti-preservation of ⚬ by
*C*_{(X)}is written Hom_{F}(*f*,*X*) ⚬ Hom_{G}(*g*,*X*) = Hom_{G}(*g*∘*f*,*X*). *f*∈Mor(*E*,*F*) is*F-epic*, or an*F-epimorphism*, if all Hom_{F}(*f*,*X*) are injective.

Let us analyze the concept of an object *S* being included in an object *E* of a
concrete category, to re-express it as a separate object *X* with an isomorphism to *S*
(by which references to target sets of morphisms could be omitted). This concept has 2 variants.

The "weak" version is the concept of subobject
(by the standard terminology), applicable to any abstract category. It amounts to only requiring
the inclusion morphism of the subobject *S* in *E* (usually Id_{S} :
*S* ↪ *E* in concrete categories) to be monic, and not even necessarily
injective.

Namely, a subobject of *E* is formalized by a *presentation* in the form
(*X*,*u*) where *u* ∈ Mor(*X*,*E*) is monic (indirect description of
Im *u* seen as isomorphic to *X*, thus defining the morphisms to and from Im
*u* as copied from those *X*, but this direct meaning is lost in abstract categories).

The class of presentations (*X*,*u*) of subobjects of an object *E*, is
meta-preordered by

(*X*, *u*) ⊆ (*Y*, *v*) ⇔ ∃ϕ∈Mor(*X*,*Y*),
*u* = *v*∘ϕ

Then, ϕ is also monic because

∀*g*,*h*∈Mor(*Z*,*X*),
ϕ∘*g* = ϕ∘*h* ⇒ *u*∘*g* = *v*∘ϕ∘*g* = *v*∘ϕ∘*h* = *u*∘*h*
⇒ *g* = *h*.

(*X*, *u*) ≡ (*Y*, *v*) ⇔ ((*X*, *u*) ⊆
(*Y*, *v*) ∧ (*Y*, *v*) ⊆ (*X*, *u*)) ⇔ (∃ϕ∈Iso(*X*,*Y*) | *u* = *v*∘ϕ).

The "strong" version requires *u* to be an embedding. This concept of embedding,
first introduced for relational systems in 3.4, will be generalized to any concrete category in 3.9
(while expressible classes of monomorphisms in abstract categories which may play a similar role are
not equivalent).

**Theorem.** Any small category is isomorphic to one made of a family of typed algebras
with all morphisms between them.

The proof essentially repeats the formulas on acts as algebraic structures, transposed.

From the given small category, a family of typed algebras is formed as follows.

- 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 ∐_{t∈T}*E*^{(t)}. *L*= ∐_{t,t'∈T}Mor(*t'*,*t*) seeing Mor(*t'*,*t*) as a set of functions symbols from*t*to*t'*, co-acting on each*E*.

Let us prove that ψ : Mor(

Im ψ ⊂ Mor

The existence of an isomorphism

∀

In particular,

- any monoid is isomorphic to the monoid of endomorphisms of an algebra on a language of function symbols;
- any group is isomorphic to an automorphism group of such an algebra.

Notice the symmetry of roles (called duality) between sides, which not only switches the orientation of morphisms between two objects, but also lets a category be somehow reworded as a special kind of mathematical theory (so viewing category theory as a weak version of one-theory theory, despite contextual differences):

- "Objects" in categories become "types",
- "Morphisms" become "invariant functions", implicitly generated by a language of function symbols only.

- An epimorphism
*b*∈ Mor(*X*,*Y*) gives to*Y*a role of sub-type of*X*, so that a*b*-module is an object whose component of type*X*is filled by*Y*. - To link both concepts of invariance : for any monoid
*M*(resp. small category) generated by a set thought of as a language of (typed) function symbols (so*M*is made of functions expressible as terms in this language), there exists a (set*I*of) algebraic interpretation(s), i.e. actions, where*M*coincides with the category*M'*of all functions (resp. interpreted function symbols) which are invariant by all endomorphisms (resp. by all morphisms between interpretations in*I*). Anyway in any interpretation, the concrete image of*M*is included in the corresponding*M'*, because the set*M'*of invariant functions by any given set of functions is (Id, ⚬)-stable (this can be dually rephrased by qualifying*M'*as the set of morphisms there).

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.

3.9. Initial and final objects

3.10. Products of systems

3.11. Basis

3.12. Composition of relations

5. Second-order foundations

6. Foundations of Geometry

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