Teorías del aprendizaje

(ii) LetC _{be a small groupoid. Define the star ofx, denotedSt(x) orStC(x),}

to be the set of objects of x\C_{, that is, the set of morphisms of}C _{with sourcex.}

WriteC_{(x, x) =π(}C_{, x) for the group of automorphisms of the objectx.}

(iii) LetE _andB _{be small connected groupoids. A coveringp:}E _−→B_{is a}

functor that is surjective on objects and restricts to a bijection

p:St(e)−→St(p(e))

for each object eofE_{. For an objectb of}B_{, letFb denote the set of objects of}E

such thatp(e) =b. Thenp−1(St(b)) is the disjoint union overe∈Fb ofSt(e). Parts (i) and (ii) of the unique path lifting theorem can be restated as follows. Proposition. _{If p:E−→B is a covering of spaces, then the induced functor} Π(p) : Π(E)−→Π(B)is a covering of groupoids.

Parts (iii), (iv), and (v) of the unique path lifting theorem are categorical consequences that apply to any covering of groupoids, where they read as follows.

Proposition. _{Let p:}E _−→B _{be a covering of groupoids, let b be an object}

of B_{, and leteande}′ _{be objects of F}

b.

(i) p:π(E_{, e)−→π(}B_{, b)is a monomorphism.}

(ii) p(π(E_{, e}′_{))is conjugate to p(π(E, e)).}

(iii) As e′ _{runs through Fb, the groups p(π(E, e}′_{)) run through all conjugates}

of p(π(E_{, e))in π(}B_{, b).}

Proof. _{For (i), ifg, g}′_∈π(E_{, e) andp(g) =p(g}′_{), theng=g}′_{by the injectivity} ofponSt(e). For (ii), there is a mapg:e−→e′_{sinceE is connected. Conjugation} byg gives a homomorphismπ(E_{, e)−→π(}E_{, e}′_{) that maps underpto conjugation} of π(B_{, b) by its elementp(g). For (iii), the surjectivity ofpon St(e) gives that}

anyf ∈π(B_{, b) is of the formp(g) for someg∈St(e). Ife}′ _{is the target ofg, then}

p(π(E_{, e}′_{)) is the conjugate ofp(π(E, e)) by f.} The fibersFb of a covering of groupoids are related by translation functions. Definition. _{Let p :} E _−→ B _{be a covering of groupoids. Define the fiber}

translation functorT =T(p) :B_−→S _{as follows. For an objectbof}B_{,T(b) =Fb.}

For a morphism f :b −→b′ _{of B,T(f) :F}

b −→Fb′ is specified by T(f)(e) = e′,

wheree′ _{is the target of the unique gin St(e) such thatp(g) =f.}

It is an exercise from the definition of a covering of a groupoid to verify thatT

is a well defined functor. For a covering spacep:E−→B and a pathf :b−→b′_,

T(f) :Fb −→Fb′ is given byT(f)(e) =g(1) whereg is the path in E that starts

ateand coversf.

Proposition. _{Any two fibers Fb and Fb}′ of a covering of groupoids have the

same cardinality. Therefore any two fibers of a covering of spaces have the same cardinality.

Proof. _{Forf :b−→b}′_{, T(f) :F}

b −→Fb′ is a bijection with inverseT(f−1).

24 COVERING SPACES

4. Group actions and orbit categories

The classification of coverings is best expressed in categorical language that involves actions of groups and groupoids on sets.

A (left) action of a group Gon a set S is a function G×S −→ S such that

es = s (where e is the identity element) and (g′_{g)s =g}′_{(gs) for all s ∈ S. The}

isotropy groupGsof a pointsis the subgroup{g|gs=s}ofG. An action isfreeif

gs=simpliesg=e, that is, ifGs=efor everys∈S.

The orbit generated by a point sis {gs|g ∈G}. An action is transitiveif for every pair s, s′ _{of elements of S, there is an element g of G such that gs = s}′_. Equivalently, S consists of a single orbit. If H is a subgroup of G, the set G/H

of cosetsgH is a transitive G-set. When Gacts transitively on a setS, we obtain an isomorphism of G-sets between S and the G-set G/Gs for any fixed s∈S by sendinggsto the cosetgGs.

The following lemma describes the group of automorphisms of a transitive

G-setS. For a subgroupH ofG, letN Hdenote the normalizer ofH inGand define

W H=N H/H. Such quotient groupsW H are sometimes called Weyl groups. Lemma. _{Let G act transitively on a set S, choose s ∈ S, and let H = Gs.}

Then W H is isomorphic to the group AutG(S)of automorphisms of theG-setS.

Proof. _{Forn ∈ N H with image ¯n ∈W H, define an automorphism φ(¯n) of}

S byφ(¯n)(gs) =gns. For an automorphismφ of S, we haveφ(s) =ns for some

n ∈ G. For h ∈ H, hns = φ(hs) = φ(s) = ns, hence n−1_{hn ∈ G}

s = H and

n∈N H. Clearlyφ=φ(¯n), and it is easy to check that this bijection betweenW H

and AutG(S) is an isomorphism of groups. We shall also need to considerG-maps between differentG-setsG/H.

Lemma. _{A G-mapα:G/H −→G/K has the form α(gH) =gγK, where the}

elementγ∈Gsatisfiesγ−1_{hγ∈K for allh∈H.}

Proof. _{Ifα(eH) =γK, then the relation}

γK=α(eH) =α(hH) =hα(eH) =hγK

implies thatγ−1_{hγ∈K forh∈H.}

Definition. _{The category} O_{(G) of canonical orbits has objects the G-sets} G/H and morphisms theG-maps ofG-sets.

The previous lemmas give some feeling for the structure of O_{(G) and lead to}

the following alternative description.

Lemma. _{The category}O_{(G)is isomorphic to the category}G _{whose objects are}

the subgroups of G and whose morphisms are the distinct subconjugacy relations

γ−1_{Hγ⊂K for γ∈G.}

If we regardGas a category with a single object, then a (left) action ofGon a setSis the same thing as a covariant functorG−→S_{. (A right action is the same}

thing as a contravariant functor.) If B_{is a small groupoid, it is therefore natural}

to think of a covariant functorT :B_−→S _{as a generalization of a group action.}

For each object b of B_{, T restricts to an action of π(}B_{, b) onT(b). We say that}

the functorT is transitiveif this group action is transitive for each object b. If B