Applied Topology - Lectures NINE Semester II,...

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Applied Topology

Lectures NINE

Semester II, 2009-10

Graham EllisNUI Galway, Ireland

Section 9

Categories and functors

The language of categories and functors is needed for concisestatements in topology.

A category C consists of:

The axioms for a category are:

◮ a collection Ob(C) of things called objects,

◮ for each pair of objects A,B ∈ Ob(C) a (possibly empty)collection MorC(A,B) of things called morphisms,

◮ a function

MorC(B ,C ) × MorC(A,B) −→ MorC(A,C ),

(f , g) 7→ f ◦ g

called composition which is defined for each triple of objectsA,B ,C ∈ Ob(C).

1. Associativity: The equation

(f ◦ g) ◦ h = f ◦ (g ◦ h)

holds for all morphisms f , g , h for which (f ◦ g) ◦ h is defined.

1. Associativity: The equation

(f ◦ g) ◦ h = f ◦ (g ◦ h)

holds for all morphisms f , g , h for which (f ◦ g) ◦ h is defined.

2. Identity: For every object B ∈ Ob(C) there is a morphism1B ∈ MorC(B ,B) such that 1B ◦ f = f and g ◦ 1B = g for anymorphisms f ∈ MorC(A,B), g ∈ MorC(B ,C ).

EXAMPLE:We have the category VecR whose objects are the vectors spacesover R, and whose morphisms are the R-linear homomorphisms.

EXAMPLE:We have the category ChnR whose objects are the chain complexesover R, and whose morphisms are the chain maps.

EXAMPLE:We have the category VecR whose objects are the vectors spacesover R, and whose morphisms are the R-linear homomorphisms.

EXAMPLE:We have the category ChnR whose objects are the chain complexesover R, and whose morphisms are the chain maps.

EXAMPLE:Any group G can be considered as a category G with one object ∗and with MorG(∗, ∗) = G .

A Functor F : C → D between two categories C,D consists offunctions

F : Ob(C) → Ob(D),

F : MorC(A,B) → MorD(F(A),F(B))

satisfyingF(f ◦ g) = F(f ) ◦ F(g),

F(1A) = 1F(A)

for all composable morphisms f , g and all identity morphisms 1A.

EXAMPLE/EXERCISEFor each n ≥ 0, homology provides a functor

Hn : ChnR → VecR

from the category of chain complexes to the category of vectorspaces.

Hn : ChnR → VecR

EXAMPLEAny group homomorphism φ : G → G ′ can be regarded as afunctor Φ: G → G′ between one-object categories.

Hn : ChnR → VecR

EXAMPLEAny group homomorphism φ : G → G ′ can be regarded as afunctor Φ: G → G′ between one-object categories.

EXAMPLEWe have the category whose objects are (small) categories andwhose morphisms are functors.

Some basic functors in topology

over K

Chain Complexes

Simplicial

Cubical

Haussdorfftopologicalspaces complexes

complexes

Vector Spaces

Abeliangroups

if K isa field

if K is the integers

CW spaces

cell spaces)(also called

over K

Precise descriptions need to be given. Categories and functorswithin the box can be represented on a computer and the efficiencyof their representations needs to be considered.

Some basic functors in topology

over K

Chain Complexes

Simplicial

Cubical

Haussdorfftopologicalspaces complexes

complexes

Vector Spaces

Abeliangroups

if K isa field

if K is the integers

CW spaces

cell spaces)(also called

over K

Precise descriptions need to be given. Categories and functorswithin the box can be represented on a computer, and theefficiency their representations needs to be considered.

The functor

C∗ : Simp → Chn

Simplicial complexes

A simplicial complex consists of an ordered set V of vertices and aset K of finite nonempty subsets of V called simplexes such that:

1. {v} ∈ K for any v ∈ V .

2. Any nonempty subset of a simplex is a simplex.

Simplicial complexes

A simplicial complex consists of an ordered set V of vertices and aset K of finite nonempty subsets of V called simplexes such that:

1. {v} ∈ K for any v ∈ V .

2. Any nonempty subset of a simplex is a simplex.

A simplex σ containing exactly k + 1 vertices will be called ann-simplex. If σ′ ⊆ σ then σ′ will be called a face of σ.

EXAMPLEFor any set V the set of all finite nonempty subsets of V is asimplicial complex.

EXAMPLEFor any set A and any collection V = {vλ ⊂ A} of subsets of A,the set N(V ) of all subsets of V with non-empty intersection is asimplicial complex called the nerve of V .

EXAMPLEFor any set V the set of all finite nonempty subsets of V is asimplicial complex.

EXAMPLEFor any set A and any collection V = {vλ ⊂ A} of subsets of A,the set N(V ) of all subsets of V with non-empty intersection is asimplicial complex called the nerve of V .

ILLUSTRATION: Suppose that A is a circle S1 and thatV = {R ,S ,T} is a covering of S1 by three open subsets R ,S ,T

which intersect pairwise but for which R ∩ S ∩ T = ∅.

{S} {T}

{R,S} {R,T}

Then N(V ) consists of three 0-simplexes and three 1-simplexes.

EXAMPLEThe Rips complex R(S , t) is a simplicial complex arising from ann × n symmetric matrix S = (sij) and t ≥ 0. The vertex set isV = {1, . . . , n} and there is one simplex for each subset σ ⊂ V

satisfying sij ≤ t for all distinct i , j ∈ σ.

EXAMPLEThe Rips complex R(S , t) is a simplicial complex arising from ann × n symmetric matrix S = (sij) and t ≥ 0. The vertex set isV = {1, . . . , n} and there is one simplex for each subset σ ⊂ V

satisfying sij ≤ t for all distinct i , j ∈ σ.

Recall from Lecture 6 that simplicial complexes can be investigatedusing gap.

EXAMPLEFor any set A and any collection S = {sλ ⊆ A} of subsets of A, wecan consider the set V = {vλµ : sλ ( sµ} of proper inclusions withsλ, sµ ∈ S .

A chain in V is a finite subset

{sλ1⊂ sλ2

, sλ2⊂ sλ3

. . . , sλn⊂ sλn+1

of V consisting of composable inclusions.

{sλ1⊂ sλ2

, sλ2⊂ sλ3

. . . , sλn⊂ sλn+1

The collection of all chains in V is a simplicial complex called theorder complex of S .

{sλ1⊂ sλ2

, sλ2⊂ sλ3

. . . , sλn⊂ sλn+1

The collection of all chains in V is a simplicial complex called theorder complex of S .

(This order complex is particularly interesting when A is a finitegroup and S is the collection of elementary abelian p-subgroups forsome prime p. There is an unsolved conjecture about thissimplicial complex due to Daniel Quillen.)

gap commands for Quillen’s complex

We can comstruct the order complex on the set S of elementaryabelian 2-groups of S5 as follows. (The simplicial complex canviewed as a graph with 45 vertices and 60 edges.)

gap> G:=SymmetricGroup(5);;

gap> K:=QuillenComplex(G,2);

Simplicial complex of dimension 1.

gap> K!.nrSimplices(0);

Simplicial maps

A simplicial map f : K → K ′ from a simplicial complex K to asimplicial complex K ′ is a function f : V → V ′ of the vertex setssuch that,

1. for each simplex σ in K , the set

f (σ) = {f (v) : v ∈ σ}

is a simplex in K ′;

2. for each simplex σ in K , and each v , v ′ ∈ σ, if v ≤ v ′ thenf (v) ≤ f (v ′).

Simplicial maps

A simplicial map f : K → K ′ from a simplicial complex K to asimplicial complex K ′ is a function f : V → V ′ of the vertex setssuch that,

1. for each simplex σ in K , the set

f (σ) = {f (v) : v ∈ σ}

is a simplex in K ′;

2. for each simplex σ in K , and each v , v ′ ∈ σ, if v ≤ v ′ thenf (v) ≤ f (v ′).

.We let Simp denote the category whose objects are simplicialcomplexes and whose morphisms are simplicial maps.

EXAMPLEConsider the simplicial complex K with vertices VK ={0, 1, 2, 3, 4, 5} and six maximal simplices

{1, 2}, {2, 3}, {3, 0}, {0, 4}, {4, 5}, {1, 5}.

Consider also the simplicial complex L with vertices VL = {a, b, c}and three maximal simplices

{a, b}, {b, c}, {a, c}.

EXAMPLEConsider the simplicial complex K with vertices VK ={0, 1, 2, 3, 4, 5} and six maximal simplices

{1, 2}, {2, 3}, {3, 0}, {0, 4}, {4, 5}, {1, 5}.

Consider also the simplicial complex L with vertices VL = {a, b, c}and three maximal simplices

{a, b}, {b, c}, {a, c}.

{b}{c}

{0,3}{0,4}

In the picture the ordering of the vertices is represented usingarrow heads.

There is a simplicial map f : K → L between the two illustratedsimplicial complexes K ,L defined on vertices by

f (1) = a, f (2) = b, f (3) = c , f (0) = a, f (4) = b, f (5) = c .

In the picture the ordering of the vertices is represented usingarrow heads.

There is a simplicial map f : K → L between the two illustratedsimplicial complexes K ,L defined on vertices by

f (1) = a, f (2) = b, f (3) = c , f (0) = a, f (4) = b, f (5) = c .

REMARKIn other examples the ordering of vertices of 2-simplices could berepresented by circular arrows.

{y}{z}

{y} {z}

The chain complex of a simplicial complex

For a simplicial complex K with ordered vertex set V we let Cn(K )denote the free abelian group with one basis element σ for eachn-simplex σ ∈ K .

For each n-simplex σ = {v1 < v2 < · · · < vn+1} ⊂ V ,n ≥ 1, wedefine σi to be the (n − 1)-simplex obtained by deleting vi from σ.

For n ≥ 1 we define

∂n : Cn(K ) −→ Cn−1(K ) ,

to be the homomorphism which sends the basis element σ to

∂n(σ) =∑

1≤i≤n+1

(−1)iσi .

EXERCISE: ∂n(∂n+1(σ)) = 0 for all basis elements σ.

We denote by C∗(K ) the chain complex

→ Cn(K )∂n→ Cn−1(K ) → . . . → C0(K ) .

Given a simplicial map f : K → K ′ and an n-simplex σ ∈ K , wedefine f (σ) = {f (v) : v ∈ σ}. If f (σ) is an n-simplex in K ′ thenwe set

Cn(f )(σ) = f (σ) ∈ Cn(K′);

otherwise we setCn(f )(σ) = 0 ∈ Cn(K

Given a simplicial map f : K → K ′ and an n-simplex σ ∈ K , wedefine f (σ) = {f (v) : v ∈ σ}. If f (σ) is an n-simplex in K ′ thenwe set

Cn(f )(σ) = f (σ) ∈ Cn(K′);

otherwise we setCn(f )(σ) = 0 ∈ Cn(K

We define the chain map

C∗(f ) : C∗(K ) → C∗(K′)

to be that consisting of the abelian group homomorphisms

Cn(f ) : Cn(K ) → Cn(K′)

defined on generators as above.

It is an exercise to see that C∗ is a chain map.

It is a further exercise to see that we have defined a functor

C∗ : Simp → Chn

from the category of simplicial complexes to the category Chn

whose objects are chain complexes of free abelian groups andwhose morphisms are chain maps.

It is an exercise to see that C∗ is a chain map.

It is a further exercise to see that we have defined a functor

C∗ : Simp → Chn

from the category of simplicial complexes to the category Chn

whose objects are chain complexes of free abelian groups andwhose morphisms are chain maps.

NOTATION: We write Hn(K ) for Hn(C∗(K )) when K is asimplicial complex.

gap commands for constructing the homology of a simplicial map

gap> K_faces:=[[1,2],[2,3],[3,0],[0,4],[4,5],[5,1]];;

gap> K:=MaximalSimplicesToSimplicialComplex(K_faces);;

gap> L_faces:=[[’a’,’b’],[’b’,’c’],[’c’,’a’]];;

gap> L:=MaximalSimplicesToSimplicialComplex(L_faces);;

gap> g:=function(x);

> if x=1 or x=0 then return ’a’;fi;

> if x=2 or x=4 then return ’b’;fi;

> if x=3 or x=5 then return ’c’;fi;

> end;;

gap> Simp_g:=SimplicialMap(K,L,g);;

gap> Chain_g:=ChainMapOfSimplicialMap(Simp_g);;

gap> Homology(Chain_g,1);

[ f1 ] -> [ f1^2 ]

gap commands for constructing the homology of a simplicial map

gap> K_faces:=[[1,2],[2,3],[3,0],[0,4],[4,5],[5,1]];;

gap> K:=MaximalSimplicesToSimplicialComplex(K_faces);;

gap> L_faces:=[[’a’,’b’],[’b’,’c’],[’c’,’a’]];;

gap> L:=MaximalSimplicesToSimplicialComplex(L_faces);;

gap> g:=function(x);

> if x=1 or x=0 then return ’a’;fi;

> if x=2 or x=4 then return ’b’;fi;

> if x=3 or x=5 then return ’c’;fi;

> end;;

gap> Simp_g:=SimplicialMap(K,L,g);;

gap> Chain_g:=ChainMapOfSimplicialMap(Simp_g);;

gap> Homology(Chain_g,1);

[ f1 ] -> [ f1^2 ]

This ilustrates that the above simplicial map induces

H1(K )×2→ H1(L).

Applied Topology - Lectures NINE Semester II,...

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