3.6. Categories

Categories (also called abstract categories for insistence) differ from concrete categories, by forgetting that objects are sets (ordered by inclusion) and that morphisms are functions. The concept of monoid was the particular case of an abstract category with only one object. The general case of categories is similarly formalized as the data of:

satisfying the axioms

Representation theorem. Any small category can be interpreted as that of all morphisms in some given list of typed algebras.
Let us fix the set of types as a copy of the set of objects : from each object X we make a type X' (not giving to this bijective correspondence any special status).
Each object M is interpreted as a system where each type X' is interpreted as the set Mor(X',M).
As a language, let us take all morphisms between types: the set of function symbols from type X' to type Y' is defined as Mor(Y',X') (with reverse order, as symbols act on the right).
The proof goes on just like with monoids.∎

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 ∈Mor(E, F), ∀g∈Mor(F,G),
Hom(X, g) ০ Hom(X, f) = Hom(X, gf)
HomF(f, X) ০ HomG(g, X) = HomG(gf, 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.

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,h∈Mor(F,X), gf=hfg=h.
In our concept of concrete category, we must specify F: we say that 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 category.

The following 2 concepts may be considered cleaner as they admit a local characterization:

Sections, retractions. When gf=IdE we say that f is a section of g, and that g is a retraction of f.

Proof: if gf=IdE then for all objects X, HomF(f,X) ০ HomE(g,X) = HomE(IdE,X) = IdMor(E,X), thus Similarly, Hom(X,g) ০ Hom(X,f) = Hom(X,IdE) = IdMor(X,E) thus f is monic and Im(Hom(X,g)) = Mor(X,F).∎

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).

Any epic section f∈Mor(E,F) is an isomorphism : gf=IdEfgf = IdFffg=IdF
Similarly, any monic retraction is an isomorphism.

These dependencies between qualities of morphisms, can be mapped as follows:

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

Initial and final objects

In any category, an object X is called an initial object (resp. a final object) if for all objects Y the set Mor(X,Y) (resp. Mor(Y,X)) is a singleton.
Such objects have this remarkable property: when they exist, all such objects are isomorphic, by a unique isomorphism between any two of them.

Proof: For any initial objects X and Y, ∃f∈Mor(X,Y), ∃g∈Mor(Y,X), gf ∈Mor(X,X) ∧ fg ∈Mor(Y,Y).
But IdX ∈ Mor(X,X) which is a singleton, thus gf= IdX. Similarly, fg=IdY.
Thus f is an isomorphism, unique because Mor(X,Y) is a singleton.∎

By this unique isomorphism, X and Y may be treated as identical to each other. We say that an initial object is essentially unique.
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: Exercise. Given two fixed sets X and Y, consider the category whose objects are all (E,φ) where E is a set and φ: E×XY, and the morphisms from (E,φ) to (E',φ') are all f : EE' such that ∀aE,∀xX, φ(a,x) = φ'(f(a),x).
Does it have an initial object ? a final object ?


Set theory and foundations of mathematics
1. First foundations of mathematics
2. Set theory (continued)
3. Algebra 1
3.1. Morphisms of relational systems and concrete categories
3.2. Algebras
3.3. Special morphisms
3.4. Monoids
3.5. Actions of monoids
3.6. Categories
3.7. Algebraic terms and term algebras
3.8. Integers and recursion
3.9. Arithmetic with addition
4. Model Theory