Desarrollo normativo de la formación profesional

Some basic notions from category theory will be helpful to understand the constructions we are going to use and their mutual relations. The following account is only aimed at introducing the concepts which are strictly necessary to our purposes.The interested reader can start from [4] to get a gentle introduction to the field, or refer to [24] for a complete overview.

A category is a structure consisting of objects and arrows (or morphisms) between them. Every arrowf has two associated objects, the domaindom(f)and the codomain

their composite) g◦f :A→ C; and each objectA has an identity arrow 1A :A→ A.

For all arrowsf :A→B,g:B→C,h:C →D the following identities hold:

· h◦(g◦f) = (h◦g)◦f

· f ◦1A=f = 1B◦f

Theopposite(ordual) categoryCopof the categoryCis a category with the same objects asC and reversed arrows: for every arrow f :B → A in C we have the corresponding arrowf∗ :A→B in Cop.

An important advantage of the categorical approach is that every statement in the language of category theory has a dual which is equivalent to it. The dual is obtained replacingdomforcod,codfor domand f ◦g forg◦f. If we have a statement holding in a categoryC, the dual statement holds automatically in the category Cop.

Definition 2.1.6 (Mono- and epimorphisms). Given a category C, an arrow f :

A → B is a monomorphism if, given any g, h : C → A, f ◦g = f ◦h implies g = h, whereA, B, C are any objects in C. f :A→B is an epimorphism if giveni, j:B →D,

i◦f =j◦f impliesi=j for any objectA, B, C inC.

Afunctor between two categoriesCand Dis an arrow in the category of all categories, i.e. a mapping associating objects ofD to objects ofC and arrows of Dto arrows ofC, preserving composition and identities.

A contravariant functor from C to D is a functor of the form F : Cop → D. This kind of functor reverses arrows, i.e. it takes f : A → B toF(f) : F(B) → F(A) and

F(g◦f) =F(f)◦F(g).

Boolean algebras form a category together with Boolean homomorphisms, which we will denoteBA. We will soon see that its dual category is a special class of topological spaces: the categoryStone of Stone spaces and continuous maps between them.

Direct and inverse systems

Adirected poset I= (I,⩽) is a set I together with a binary relation ⩽, such that:

· ⩽ is a partial order;

Definition 2.1.7(Direct system). LetC be a category. Adirect system inC consists of an indexed family {Ai|i∈ I} of objects in C whose index set is a directed poset I,

together with a set of morphisms{fij :Ai →Aj} for anyi, j ∈I withi⩽j, satisfying

the following properties:

· fii:Ai→Ai is the identity map;

· For any i, j, k∈I such thati⩽j ⩽kwe have fjk◦fij =fik.

We will use the notation{Ai, fij} to denote a direct system.

The dual structure is then obtained reversing the arrows, as follows:

Definition 2.1.8 (Inverse system). Let C be a category. An inverse system in C consists of an indexed family {Ai|i ∈ I} of objects in C whose index set is a directed

posetI, together with a set of morphisms {hji :Aj → Ai} for any i, j ∈I withi⩽j,

satisfying the following properties:

· hii:Ai →Ai is the identity map;

· For any i, j, k∈I such thati⩽j ⩽kwe have hji◦hkj =hki.

We will use the notation{Ai, hji}to denote such an inverse system.

We now introduce the notion, central to our construction, of inverse limit of an inverse system:

Definition 2.1.9 (Inverse limit). Let {Ai, hji} be an inverse system, indexed by I,

in a categoryC. Theinverse limit of{Ai, hji}consists of an objectLinC together with

morphismspi:L→Ai, for every i∈I, such that the following conditions are satisfied: · hji◦pj =pi, for all i, j∈I with i⩽j;

· If there exists any object X equipped with morphisms gi :X → Ai for anyi∈I

and such that hji◦gj = gi, for all i, j ∈ I with i ⩽ j, then there must exist a

unique morphism f :X→Lsuch that pi◦f =gi for all i∈I.

We will writelim_←−Ai={L, pi}to indicate the limit just defined. The arrowspiare called

projections.

The second property listed in the definition is calleduniversality(oruniversal mapping property, UMP) of the limit and is a fundamental way to characterise objects in category theory as unique up to isomorphism.

A picture will help figure out the situation. The dashed lines represent arrows that are unique.

Figure 2.1: Inverse limit of an inverse system and colimit of a direct system

We are also interested in the dual notion of limit, to represent what happens in the dual category:

Definition 2.1.10 (Colimit of a direct system). Let {Ai, fij} be a direct system,

indexed by I, in a category C. The colimit of {Ai, fij} consists of an object C in C

together with morphismsιi:Ai→C, for everyi∈I, such that the following conditions

are satisfied:

· ιj◦fij =ιi, for all i, j∈I withi⩽j;

· If there exists any object X equipped with morphismsgi :Ai →X, for any i∈I,

such that gj ◦fij =gi, for all i, j ∈I with i⩽j, then there must exist a unique

morphism u:C→X such thatu◦ιi=gi for all i∈I.

We will write lim_−→Ai = {C, ιi} to indicate the colimit just defined. The arrows ιi are

calledimmersions in the colimit.