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Topological Groupoids

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Topological Groupoids Alberto Santini Topological groupoids Kandidatproject Institut for Matematiske Fag, Københavns Universitet. Lektor: Jesper Michael Møller 2 Contents 1Basicconcepts 2 1.1 Introduction to categories ......................... 2 1.2 Groupoids and the fundamental groupoid ................. 8 1.3 Functors ................................... 13 1.4 Pushouts ................................... 18 1.5 Further properties of the fundamental groupoid ............. 22 1.6 The fundamental group of the circle .................... 27 1.7 Fibrations of groupoids and cofibrations of spaces ............ 29 2Productsandquotientsofgroupoids 34 2.1 Universal morphisms and universal groupoids .............. 34 2.2 Free groupoids and free product of groupoids ............... 36 2.3 Quotient groupoids ............................. 40 2.4 Adjunction spaces .............................. 41 2.5 Van Kampen theorem for the fundamental groupoid ........... 43 Index 47 Bibliography 49 1 CHAPTER ONE BASIC CONCEPTS 1.1 Introduction to categories There are basically two ways of approaching groupoids. The first one is algebraically, considering them as a particular generalization of the algebraic structure of group. The second one, that is the one we’ll use and need to prove our results is the category theoretical approach. To do this, we’ll introduce some basic aspects of category theory, with in mind the fact that we’ll always use it as tool for our topological aim. Categories have been introduced by Saunders Mac Lane ([3]) and Samuel Eilenberg as a way to generalize particular dual properties of groups and lattices, but then they have proven useful as a generalization of almost all algebraic structures mathematicians usually deal with. Their basic constituents are objects and morphisms (sometimes called arrows) that satisfy certain axioms, as explained in what follows. Definition 1.1.1. A category is a quadruple A =(Obj(A), Hom, Id, ) where ◦ Obj(A) is a class, whose members are called the objects of A. • For each A, B Obj(A), HomA(A, B) is a set whose elements are called mor- • ∈ f phisms and are denoted as A B;wealsocallA the domain and B the −→ codomain of f. IdA,A For each A Obj(A), IdA,A is a morphism A A,calledtheidentity on A. • ∈ −→ 2 CHAPTER 1. BASIC CONCEPTS f g For each A, B, C, Obj(A) and each A B,B C, g f is a morphism • g f ∈ −→ −→ ◦ A ◦ C; is called a composition law. −→ ◦ such that the following holds: f g h C1 The composition is associative: given A B, B C, C D we have ◦ −→ −→ −→ (f g) h = f (g h). ◦ ◦ ◦ ◦ f C2 The identities IdA,A actually act as identities for the composition: given A B −→ we have that f IdA,A = f and IdA,B f = f. ◦ ◦ C3 The various sets HomA(A, B) are pairwise disjoint. We will sometimes refer to HomA := HomA(A, B) (1.1.1) A,B Obj(A) ∈ HomA(A, ) := HomA(A, B) (1.1.2) − B Obj(A) ∈ HomA( ,B) := HomA(A, B) (1.1.3) − A Obj(A) ∈ (1.1.4) Example 1.1.2. Examples of categories are: Set where the objects are the sets and the morphisms are functions between • them. Pos where the objects are partially ordered sets and the morphisms are the • isomorphisms between posets (i.e. order-preserving maps). Lin where the objects are finite-dimensional K-vector spaces and the morphisms • are the linear maps between them. Grp where the objects are groups and the morphisms are group homomorphisms. • Every monoid M is also a category with just an object, M itself and whose • morphisms are the elements of M (so that the identity is the identity of M and the composition is just the multiplication on M). 3 CHAPTER 1. BASIC CONCEPTS Top where the objects are topological spaces and the morphisms are continuous • functions between them. Given a topological space X,wedefinethecategoryP X;theobjectsarethe • points of X and the morphisms are the paths between them (a path from x X ∈ to y X is a continuous function px,y :[0, 1] X such that px,y(0) = x and ∈ → px,y(1) = y). The identity for x X is the null-path i.e. the constant function ∈ px,x(t)=x t [0, 1] and the composition is the concatenation of paths. ∀ ∈ Definition 1.1.3. Given two categories A and B we say that B is a subcategory of A if Obj(B) Obj(A). • ⊆ Given two objects A, B Obj(B) we have that HomB(A, B) HomA(A, B). • ∈ ⊆ Composition in B is the same as the one in A. • For each A Obj(B) we have IdB,A =IdA,A. • ∈ B is said to be wide if it has exactly the same objects as A, while it is said to be full if each couple of objects of B has the same morphisms both in A and B. f g Definition 1.1.4. Given two morphisms A B and B A,wesaythatthey −→ −→ are respectively a co-retraction and a retraction if g f =IdA,A.Sometimeswe ◦ say that g is a left-inverse of f and f is a right-inverse of g. The following theorem will show that, fixed a morphism that has both a left and right inverse, it is legitimate to simply refer to its inverse without specifying if left or right. We will call such a morphism invertible or an isomorphism. f g h Theorem 1.1.5. Given A B and B A, B A,ifg f =IdA,B and −→ −→ −→ ◦ f h =IdA,A,theng = h. ◦ Proof. g = g IdA,A = g (f h)=(g f) h =IdA,B h = h (1.1.5) ◦ ◦ ◦ ◦ ◦ ◦ 4 CHAPTER 1. BASIC CONCEPTS Definition 1.1.6. Two objects A, B Obj(A) are called isomorphic if there exists f ∈ an isomorphism A B.It’strivialtoprovethattherelation“is isomorphic to”is −→ an equivalence relation on Obj(A),ifthisisaset. Exercise 1.1.7. An object Oi Obj(C) is said to be initial if HomC(Oi,A) =1for ∈ | | all A Obj(C).AnalogouslyOf Obj(C) is called final if HomC(A, Of ) =1for ∈ ∈ | | all A Obj(C).Anobjectthatisbothinitialandfinaliscalledazero object. ∈ Prove that all initial objects are isomorphic as are all final objects. • Prove that Set and Top have initial and final objects but no zero object. • Prove that Grp has a zero object. • Proof. Let A, B Obj(C) be two initial objects. By hypothesis there are unique mor- • ∈ f A g f phisms A B and B ,sothereisamorphismA ◦ A.ButsinceA is −→ −→ −→ initial HomC(A, A) has only one element, so we must have that g f =IdC,A ◦ and analogously f g =IdC,B,soA and B are isomorphic. An entirely similar ◦ reasoning may be applied to final objects. Since functions can have empty domains, but can’t have empty codomains it’s • clear that the only initial object in Set is .Ontheotherhand,theonlyobject ∅ that allow only one map to be constructed such that it has the object as codomain, must have only one element and so the final objects are the singletons. Hence it’s clear then there is no zero object in Set.Reasoningintheexactsamewayone can find that the empty space is the only initial object of Top and that one-point space are its only final objects. Consider any group G = eG .Bytheverydefinition,thereisjustonehomomor- • { } phism from G to any other group H (and it is eG eH )andonehomomorphism → from H to G (and it is h eG for all h H). → ∈ f Exercise 1.1.8. AmorphismA B of C is called a monomorphism if −→ C Obj(C) g, h HomC(C, A) f g = f h g = h (1.1.6) ∀ ∈ ∀ ∈ ◦ ◦ ⇒ 5 CHAPTER 1. BASIC CONCEPTS Analogously it is called an epimorphism if C Obj(C) g, h HomC(B,C) g f = h f g = h (1.1.7) ∀ ∈ ∀ ∈ ◦ ◦ ⇒ f f We write monomorphisms as A B and epimorphisms as A B. Prove that a morphism of Set is a monomorphism if and only if it is injective. • Prove that a morphism of Set is an epimorphism if and only if it is surjective. • Prove that every isomorphism is both a monomorphism and an epimorphism. • Give an example of a category that has a morphism that is both an epimorphism • and a monomorphism, but not an isomorphism. Proof. f Given A B and a, a A with a = a,considerasingleton s Obj(Set) • ∈ { } ∈ and the functions fa,fa : s A : fa(x)=a fa (x)=a (1.1.8) { } → ∧ Since f is a monomorphism, we have that a = a f fa = f fa and so ⇒ ◦ ◦ f(a)=f(fa(x)) = f(fa (x)) = f(a) (1.1.9) On the other hand, if f is injective, given g, h : C A with g = h,letc C → ∈ be a point such that g(c) = h(c). Then by injectivity f(g(c)) = f(h(c)) and so f g = f h. ◦ ◦ f Given A B, consider a set with two elements s, t Obj(Set) and the • { } ∈ functions g,h B s, t (1.1.10) −→ { } g(x)=s x B (1.1.11) ∀ ∈ s if x f(A) h(x)= ∈ (1.1.12) t if x f(A) ∈ 6 CHAPTER 1. BASIC CONCEPTS Now notice that a A we have that g(f(a)) = s and h(f(a)) = s,sog f = h f ∀ ∈ ◦ ◦ and since f is an epimorphism this implies that g = h,thatistosaythat x Bx f(A) and so f is injective. On the other hand, given the injection ∀ ∈f ∈ g,h A B,considerC Obj C and the morphisms B C such that g f = h f. −→ ∈ −→ ◦ ◦ By surjectivity, each b B can be written as f(a) for some a A.Sowehave ∈ ∈ that g(b)=g(f(a)) = (g f)(a)=(h f)(a)=h(f(a)) = h(b) b B (1.1.13) ◦ ◦ ∀ ∈ ans so g = h,thatistosayf is an epimorphism.
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