Monoids were introduced as an abstraction of transformation monoids, which
are concrete categories with one object. Generalizing this to pluralities of
objects, a category (also called abstract category for insistence)
differs from a concrete category, by forgetting that objects
are sets and that morphisms are functions. It is made of:
satisfying both axioms, of identity and associativity:
- 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 1A ∈ 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) ;
Generalizing isomorphisms in
concrete categories, an isomorphism f
between objects E and F in an abstract category, is an
f∈Mor(E,F) such that ∃g∈Mor(F,E),
f•g=1F. This g is unique, called the inverse of
f and written f -1.
- For any objects A,B, ∀x∈Mor(A,B),
x•1A = x = 1B•x
- For any objects A,B,C,D, ∀x∈Mor(A,B),
(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
The concept of opposite category is defined similarly to opposite monoids,
as re-reading the category with sides reversed, exchanging the roles of
Functions defined by composition
In any category, any f ∈ Mor(E,F) defines functions
by currying composition with other morphisms to or from another object X:
let us denote (almost following wikipedia but adapted to our concept of concrete category)
The former respects composition, while the latter reverses
it: for any 4 objects E,F,G,X , ∀f
- Hom(X, f) = (Mor(X,E)∋g↦
f•g), with target Mor(X,F) for any target F of f.
- HomF(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) =
The concepts of cancellativity and invertibility are generalized to categories as follows.
HomF(f, X) ০
HomG(g, X) =
Monomorphism. In a category, a morphism
is called monic, or a monomorphism, if Hom(X,f)
is injective for all objects X:
f•g = f•h ⇒ g = h.
Epimorphism. In an abstract category, a morphism f∈Mor(E,F)
is called epic, or an epimorphism, if Hom(f,X)
is injective for all objects X:
g•f = h•f ⇒ g = h.
In our concept of concrete category, we must specify F, saying
f∈Mor(E,F) is F-epic, or an F-epimorphism,
if all HomF(f,X) are injective.
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
Sections, retractions. When g•f = 1E
we say that f is a section of g, and that g is a retraction of f.
Proof: if g•f=1E then for all objects X,
HomF(f,X) ০ HomE(g,X)
= IdMor(E,X), thus
- A morphism f∈Mor(E,F) is a section
(or section in F if the category is concrete), if
i.e. ∃g∈Mor(F,E), g•f=1E.
Then f is monic and for all objects X we have
Im(HomF(f,X)) = Mor(E,X).
A morphism g∈Mor(F,E) is a retraction (or retraction on E
if the category is concrete), if 1E∈Im(Hom(E,g)),
i.e. ∃f∈Mor(E,F), g•f=1E.
Then g is epic and for all objects X we have
Im(Hom(X, g)) = Mor(X,F).
Similarly, Hom(X,g) ০ Hom(X,f) =
Hom(X,1E) = IdMor(X,E) thus
f is monic and Im(Hom(X,g)) = Mor(X,F).∎
- HomE(g,X) is injective (g is epic)
- ∀h∈Mor(E,X), h =
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 any category, Isomorphism ⇔ (Retraction ∧ Monomorphism) ⇔
(Section ∧ Epimorphism)
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.
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.
Each u ∈ Mor(E,F) acts on each type t by Hom(t,u) : tE → tF. These for all u and t
define f : Mor(E,F) →
FE, so that the system of these for all E, F
respects identities and compositions (they form an action in a sense generalized from monoids
to small categories).
Let us prove that f : Mor(E,F) ↔ MorL(E,F).
- As a set T of types, we can take a copy of the set of objects, but one type per isomorphism class suffices.
- L = ∐t,t'∈T Mor(t',t) seeing
Mor(t',t) as a set of function symbols from
t to t'.
- Each object E is interpreted as a typed set ∐t∈T
tE where tE =
Im f ⊂ MorL(E,F) by associativity
(f commutes with the action of L).
The existence of an isomorphism k ∈ tE ensures that
f is injective (as k is epic) and
MorL(E,F) ⊂ Im f:
∀g∈MorL(E,F), u =
g(k)•k-1 ∈ Mor(E,F)
⇒ (∀x∈E, ∃s∈L, k•s = x ∴ g(x) = g(k•s) =
g(k)•s = u•k•s = u•x)
⇒ g = f(u) ∎
In particular for any monoid M there is a language
L of function symbols and an L-algebra X
such that EndL 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
1. First foundations of
2. Set theory (continued)
3. Algebra 1
of relational systems and concrete categories
4. Model Theory
3.3. Special morphisms
3.5. Actions of monoids
3.6. Invertibility and groups
3.8. Initial and final objects
3.9. Algebraic terms
3.10. Term algebras
3.11. Integers and recursion
3.12. Presburger Arithmetic