Theory and Applications of Categories Vol No pp

FUNCTORIAL AND ALGEBRAIC PROPERTIES OF BROWNS P

FUNCTOR

LUISJAVIER HERNANDEZPARICIO

Transmitted by Ronald Brown

ABSTRACT In E M Brown constructed a functor P which carries the tower

of fundamental groups of the end of a nice space to the BrownGrossman fundamental

In this work we study this functor and its extensions and analogues dened

for prosets prop ointed sets progroups and proab elian groups The new versions of

the P functor are provided with more Examples given in the pap er

prove that in general the P functors are not faithful however one of our main results

establishes that the restrictions of the corresp onding P functors to the full sub categories

of towers are faithful Wealsoprove that the restrictions of the P functors to the

corresp onding full sub categories of nitely generated towers are also full Consequently

in these cases the towers of ob jects in the categories of sets p ointed sets groups and

ab elian groups can b e replaced by adequate algebraic mo dels M sets M p ointed sets

nearmo dules and mo dules The article also contains the construction of left adjoints

for the P functors

Intro duction

This article contains a detailed study of the prop erties of the P functor An interesting

consequence is that in many cases we can replace an inverse system of sets or groups

by sp ecial algebraic mo dels that contain all the information of the corresp onding inverse

systems Firstly we recall the context in whichLRTaylor and EM Brown dened the

homotopy groups and the P functor

In LR Taylor Tay dened the fundamental group of a space by taking a

of base p oints suchthatany innite pathcomp onent of the complement of a compact

subspace contains base p oints He used the fundamental groups of these pathcomp onents

based at these base p oints to dene the fundamental group of a space

In EM Brown Br dened the prop er fundamental group of a space with a

B

base rayby using the string of spheres S which is dened byattaching one

sphere S at each p ositiveinteger of the half line Given a space X with a

B B

base ray he considered the prop er fundamental group X S X as the set

B

of germs at innity of prop er maps from S to X mo dulo germs at innity of prop er

homotopies Let X be a well rayed space the inclusion of the base ray is a cobration

and supp ose that K K K is a conal sequence of compact subsets

Denote by X cl X K ray and by X fX g the asso ciated end tower of X The

i i i

Received by the editors March

Published on April

AMS sub ject classication BE A Y N N Q

Key words and phrases Category of fractions Procategory Monoid M set Browns P functor

Tower Proob ject Nearring Nearmo dule Generator Proset Progroup Proab elian group

c

LuisJavier HernandezParicio Permission to copy for private use granted

Theory and Applications of Categories Vol No

fundamental progroup X of X is isomorphic to the tower f X g Brown dened a

i

B

X If one takes a one ended space functor P tow Gps Gps satisfying P X

having a countable base of neighbourhoods at innity a base ray gives a set of base p oints

and it can b e checked that the fundamental group of this space agrees with the global

fundamental Brown group which is dened by considering global prop er maps and global

prop er homotopies instead of prop er germs see HP

Later Grossman Gr Gr Gr develop ed a homotopy theory for towers of sim

plicial sets and dened the analogues of Browns groups for towers of simplicial sets

Using a well known exact sequence he proved that his notion of fundamental group

G

Z Y was isomorphic to tow GpscZ f Y g where cZ is dened bycZ

j i i

j i

where Z Z and denotes the copro duct in the category of groups Applying the

j

B G

EdwardsHastings emb edding theorem EH wehavethat X X Therefore

the EdwardsHastings emb edding relates the Brown denition of the P functor and the

functor tow GpscZ It is not hard to see that P and tow GpscZ are isomorphic

for anytower of groups and then the P functor can b e seen as a representable homgroup

functor on the category tow Gps

Wehave considered this last formulation of the P functor in order to dene our more

algebraically structured version of the P functor Next wegive some notation and our

denition of the P functor and afterwards we establish some of the main results of this

work

Given an ob ject H of a category C we can consider the of ob jects generated from

it by taking arbitrary sums of copies of H and eective epimorphisms When this class

contains all the ob jects of the category it is said that H is a generator for the category C

Notice that the onep oint set is a generator for the category Set of sets the twop oint

set S is a generator for the category Set of p ointed sets the innite cyclic group Z is a

generator for the category Gr p of groups and the innite cyclic ab elian group denoted in

this pap er by Z is a generator for the category Ab of ab elian groups We will denote by

a

C one of the categories Set Set GrpAb The generator of C will b e denoted by G

Asso ciated with the category C one has the category tow C oftowers in C and the

category pr oC of proob jects in C The ob ject G induces a proob ject cG N C

dened by

cG G i N

i

j i

Given a proob ject X in pr oC one has the canonical action

pr oC cG X pr oC cG cG pr oC cG X f f

For the dierent cases C Set Set GrpAbwe note that pr oC cG X is a set a

p ointed set a group and an ab elian group resp ectively On the other hand P cG

pr oC cG cG has resp ectively the structure of a monoid a monoid see section

a nearring and a ring As a consequence of this fact we will use dierent algebraic

categories but b ecause they havemany common functorial prop erties we will use the

following unied notation

Theory and Applications of Categories Vol No

If C Set and G C will b e the category of P cGsets A P cGset consists

P cG

of a set X together with an action of the monoid P cG pr oC cG cG

If C Set and G S C will b e the category of P cGp ointed sets In this

P cG

case the structure is given by an action of a monoid on a p ointed set

If C Gr p and G Z C will b e the category of P cGgroups Now the structure

P cG

is given by an action of a nearring on a group see sections and

If C Ab and G Z C will b e the category of P cGab elian groups mo dules

a P cG

over the unitary ring P cG

Using this notation given a proob ject X in pr oC P X pr oC cG X provided with

the action of pr oC cG cG determines an ob ject of the category C This denes the

P cG

functor P pr oC C

P cG

Next weintro duce some of the main results of the pap er

Theorem P tow C C is a faithful functor

P cG

This establishes that the restriction of the P functor to the full sub category of towers

in C is a faithful functor It is interesting to note that the extended P functor for instance

from proab elian groups to mo dules is not faithful see Corollary

Another imp ortant result of the pap er states that the restriction of the P functor to

nitely generated towers is also full

Theorem Let X be an object in tow C If X is nitely generated then X is

admissible in tow C Consequently the restriction functor P tow C f g C is a ful l

P cG

embedding where tow C f g denotes the ful l subcategory of tow C determined by nitely

generated towers

For nitely generated towers we are able to replace a tower of ob jects by a single ob ject

with some additional algebraic structure In this pap er wehave only considered this kind

of construction for towers of sets and towers of groups however many of the p o ofs are

established byusingvery general functorial metho ds Therefore part of the constructions

and results can b e extended to towers and proob jects in more general categories

For the case C Gr p the main results of Chipman Ch Ch stated for towers of

nitely generated groups are obtained from Theorem as corollaries We p ointout

that the class of nitely generated towers of groups is larger than the class of towers of

nitely generated groups

Fortunatelysomevery imp ortant examples of towers of sets are nitely generated for

example the tower of s of the end of a lo cally compact compact Hausdor space or

the tower of s obtained bythe Cech nerve for a compact metric space In these cases

it is easy to prove that the fundamental progroups are nitely generated and therefore

the P functor will work nicely on this kind of progroup In the ab elian case the towers

of singular homology groups coming from prop er homotopy and shap e theory are nitely

generated However we do not knowiffortowers of higher homotopy groups the ab elian

version of the P functor is going to b e a full embedding

As a consequence of Theorem wegetafullemb edding of the category of zero

dimensional compact metrisable spaces and continuous maps into the algebraic category

Theory and Applications of Categories Vol No

of P csets We also obtain similar emb eddings for the corresp onding categories of top o

logical groups and top ological ab elian groups

In this work we also give the construction of left adjoints for the P functors

Theorem The functor P pr oC C has a left adjoint L C pr oC

P cG P cG

This left adjoint for instance for the ab elian case constructs the proab elian group

asso ciated with a mo dule over the ring P cZ It is interesting to note that P cZ is

a a

isomorphic to the ring of lo cally nite matrices mo dulo nite matrices whichwas used by

FarrellWagoner FW FW to dene the Whitehead torsion of an innite complex

Next we include some additional remarks ab out other results and constructions devel

op ed in this pap er

In section we analyse some nice prop erties of categories of the form pr oC Giv

en a strongly conite set I weprove in Theorem that the full sub category pr o C

I

determined by the ob jects of pr oC indexed by I isequivalent to a category of right frac

I

tions C in the sense of Gabriel and Zisman GZ When the directed set of natural

numb ers I N is considered the category pr o C is usually denoted by tow C From

N

N

Theorem wehave that tow C can b e obtained as a category of right fractions of C

however for this case we also prove in Theorem that the category tow C is equivalent

N

to a category of left fractions C This fact has the following nice consequence The

homset tow C cG X can b e expressed as a colimit and this gives the denition of the P

functor given by Brown or if we use the standard denition given as a limit of colimits

we nearly obtain the denition of at innity given byTaylor

When we are working with categories of sets p ointed sets groups and ab elian group

swe usually consider free forgetful and ab elianization functors For the corresp onding

towcategories and pro categories wealsohave analogous induced functors The dier

entversions of P are functors into categories of P csets P cS pointed sets P cZgroups

and P cZ ab elian groups For these categories we analyse in section the denition

a

and prop erties of the analogues of this kind of functor For example the left adjoint of the

natural inclusion of the category of P cZ ab elian groups into the category of P cZgroups

a

is a kind of distributivization functor Given a P cZgroup X a quotient dX is dened

by considering the relations which are necessary to obtain a right distributive action

An imp ortant result of section is Theorem In terms of prop er homotopyThe

orem proves that the P functor sends the ab elianization of the tower of fundamental

groups to the distributivization of the fundamental BrownGrossman group It is not

hard to nd top ological examples where the ab elianization of the fundamental Brown

Grossman group pro duces a typ e of rst homology group which is not naturally isomor

phic to the distributivization of the BrownGrossman group

Finallywehave develop ed section to solve some of the theoretical questions that

have arisen from writing the pap er We see that the full sub category of lo cally structured

top ological ab elian groups admits a full emb edding into the category of global proob jects

of ab elian groups As a consequence of the relations b etween these categories we obtain

Corollaries and which are the main results of the section In these corollaries it

is proved that neither tow Ab Ab nor tow Ab havecountable sums and therefore neither

Theory and Applications of Categories Vol No

tow Ab Ab nor tow Ab are equivalent to a category of mo dules

Pro categories and categories of fractions

The category pr oC where C is a given categorywas intro duced by A Grothendieck

Gro A study of some prop erties of this category can b e seen in the app endix of AM

the monograph of EH or in the b o oks of MS and CP

The ob jects of pr oC are functors X I C where I is a small left ltering category

and the set of morphisms from X I C to Y J C is dened by the formula

pr oC X Y limcolim C X Y

i j

j i

A morphism from X to Y can b e represented by ff g where J I is a map

j

and f X Y is a morphism of C such that if j j is a morphism of J there are

j j j

i I and morphisms i j i j such that the comp osite X X Y Y is

i j j j

equal to the comp osite X X Y

i j j

Notice that if I is a strongly directed set and J is just a set then MapsJI is also

a directed set It is easy to see that if I is a strongly directed set and J is a strongly

J

conite directed set then the natural inclusion I MapsJI is a conal functor see

J

MS page where I denotes the strongly directed set of functors from J to I Asa

consequence of this fact if I J are strongly conite directed sets any morphism of pr oC

J

from X to Y can b e represented by f where I and f X Y is a level

j j j

morphism

scd

Let C denote the category whose ob jects are functors X I C whereI is a

strongly conite directed set and a morphism from X I C to Y J C is given

by a functor J I and by a natural transformation f X Y whereX is the

comp osition of the functors and X Given a strongly conite directed set I we also

I scd

consider the sub category C of C given by ob jects indexed by I and morphisms of the

form f where id

I

J

If J I are strongly conite directed sets then I the set of functors from J to I is a

J

strongly directed set I if j j j J and can b e considered as

I J J

a category The evaluation functor e C I C is dened by ex X X

I J

A xed induces a functor C C which sends f X Y to f X Y

J J

and a xed X induces a functor X I C sending to X X X

Let pr o C denote the full sub category of pr oC dened by the ob jects of pr oC indexed

scd

by strongly conite directed sets If X I C and Y J C are ob jects in pr o C we

scd

J

can takeinto account that I is conal in M apsJI to see that

J

colim pr o C X Y pr oC X Y C X Y

scd

J

I

That is pr oC X Y is the colimit of the functor

J

C Y

X

op

J op J

C Set I

Theory and Applications of Categories Vol No

Edwards and Hastings EH gave the construction the Mardesic trick see also MS

page of a functor M pr oC pr o C that together with the inclusion pr o C

scd scd

pr oC give an equivalence of categories We note that the category pr o C is a

scd

scd scd

quotient of the category C thatispr o C X Y is a quotientof C X Y

scd

I

We include the following result ab out conal subsets of I

Lemma Let I be a strongly conite directed set and consider

I I

I f I j idg then

id

I I

the inclusion I I is conal

id

N

for the case I N the directed set of nonnegative integers if InN f

N N N N N N

N j is injective g and InN N InN then InN N and

id id

N N

InN N areconal

id

Next for a given strongly conite directed set I we analyse the relationship b etween

I

C and pr o C the full sub category of pr o C dened by the ob jects indexed bya xed

I scd

I

I We are going to see that pr o C is a category of right fractions of C For this purp ose

I

rst we recall see GZ under which conditions a class of morphisms of a category C

admits a calculus of left or right fractions

A class of morphisms of C admits a calculus of left fractions if satises the following

prop erties

a The identities of C arein

b If u X Y and v Y Z are in their comp osition vu X Z is in

s u

c For each diagram X X Y where s is in there is a commutative square

u

X Y

s t

y y

X Y

u

where t is in

d If f g X Y are morphisms of C and if s X X is a morphisms of such that

fs gs there exists a morphism t Y Y of such that tf tg

If we replace the conditions c and d by the conditions c and d below the class

is said to admit a calculus of right fractions

u t s u

c For each diagram X Y Y where t is in there is a diagram X X Y

such that u s tu and s is in

d If f g X Y are morphisms of C and if t Y Y is a morphism of such that

tf tg there exists a morphism s X X of such that fs gs

I

Now for a given strongly conite directed set I consider the class of C dened by

I

the morphisms of the form X X X where I

id id

Theory and Applications of Categories Vol No

Proposition admits a calculus of right fractions

id

Proof It is clear that X id and that X X X These imply a

X

id

id id id

I I I

and b By considering the evaluation functor C I C X X we obtain

id

the commutative square

f

X Y

X Y

id id

y y

X Y

f

Hence wehavethatc is satised

Finallyiff g X Y are morphisms such that Y f Y g wehave that

id id

g g Y f Y f Y Y

id id id id

Y f f X

id id

Y g g X

id id

Therefore it follows that f X g X and so d is also satised

id id

scd

I

Given a strongly conite directed set I we denote by C and pr o C the full sub

I

scd

categories of C and pr oC resp ectively determined by ob jects indexed by I Nowto

I

compare C and pr o C we consider the diagram

I

I scd

I

C C

y y

I

C pr o C

I

In order to have an induced functor it suces to see that a morphism of is sentto

I

an isomorphism of pr o C A morphism of is of the form X X X I We

I

id id

scd id

I

X id X X in the category C Using also consider the morphism

X

this notation wehave

id

Lemma The morphisms X X induce an isomorphism in pr o C

I

id

Proof Consider the comp osites

id X

id

I

X

id

X X X

id X

id

I

X

id

X X X

Theory and Applications of Categories Vol No

Wehavethat

id id X X

X I

id id

Because the following diagram is commutative

X

X X

X X

id id

y y

X X

id

X

inpr o C it follows that id id X

I I X

id

On the other hand wehave

id X id X

I X

id id

Since the diagram

X

X X

X X

id id

y y

X X

id

X

is commutative wehavethat X id id inpr o C

I X I

id

I

Theorem The induced functor C pr o C is an equivalenceofcategories

I

I I

Proof Since I is conal in I by Lemma a morphism X Y in pr o C can b e

I

id

scd

I

by represented in C

id

X

f

X Y X

I id

where f is in C By Lemma in the category pr o C wehavethat f X

I

I

Therefore C X Y pr o C X Y is surjective f X

I

id

X

X

id

g id

f

Given two morphisms in

X X Y

X X Y

Theory and Applications of Categories Vol No

I I

C suchthat f ginpr o C there is a I such that the diagram

I

id

X

x

f

X

X X Y

X

g

y

X

I I

is commutativein C Therefore in C wehavethat

f X f X X g X X g X

id id id id

I

pr o C X Y Hence C X Y

I

Next we study a particular but imp ortant case We consider the strongly conite

directed set N of natural numb ers In this case the category pr o C is usually denoted

N

by tow C We are going to prove that if C has a nal ob ject the category tow C can also

N N

b e obtained from C as a category of left fractions As b efore N denotes the strongly

directed set of functors from N to NWe will use the following notation N fgN

N N fg N and in N N denotes the inclusion We also have the

strongly directed sets

N

N

Cof N f N j is conal g

N

N

Cof N f N j is conalg

Notice that wehave the relations

N N N N

InN InN Cof N N

id

N

Given a Cof N dene N N as follows If n then n

Otherwise there is an i N suchthat i ni and then dene ni

Supp ose that n mwehavethatifn then n m otherwise i n

i n m and we again havethatn m Similarlyitiseasytocheck

id in and that if then Therefore wehave dened a contravariant that

N

functor

N N

Cof N Cof N

Theory and Applications of Categories Vol No

N N

If C is a category with a nal ob ject we also consider the functor C C

N

X for n and X which sends an ob ject X in C to X dened by X

n

n

Nowwe dene the functor e as the comp osite

e

N

N N N N

C Cof N C Cof N C

we also use the shorter notation e X so e X eX X

N N

X For a xed wehave a functor C C and for a xed X we

N N

have the contravariant functor X Cof N C

Proposition Let C bea category with a nal object

N

If InN then is left adjoint to that is there is a natural trans

id

N b N

C X Y that wil l be denotedbyf f g g formation C X Y

N

If InN the fol lowing diagram is commutative

id

b

f

Y X

x x

Y X

id id

X Y

f

X

f

N

If InN for the diagram X X Y we have that

id

b

b

f X Y f

N

Proof We note that if InN in N N Therefore for a given

id

N

ob ject Y in C wehave

Y Y Y Y in Y

N N

Dene C X Y C X Y by g g It is clear that the counit

transformation is in this case the identity

b N N

To dene the inverse transformation C X Y C X Y wehave that

in since is injective and idWealsohavethat

X X in

X X X X

b in

that is it is the comp osite Now given f X Y dene f f X

in

X

f



X X Y

Theory and Applications of Categories Vol No

By considering the formulas

in

in b in

f X f f X f f X

in

b

b

in in

g Y g g g X

in in

in

g g g Y Y g Y

in

it follows that the transformations ab ove dene a natural isomorphism

The commutativity of the diagram is proved by the following equalities

in

in in b

X X in f X X f f X

in

id id

in

in in in

f X f X f X Y f

f Y

id

It follows from the following relations

b

in in

X X f X f X f X

in in

f X X f X

in in

f X X X f X

in b

Y f X f Y

N

then Lemma Let C beacategory with nal object and InN

id

Y Y

id

Therefore wehave Proof It is easy to check that in

Y Y Y Y

in Y Y

Y Y

in

in

Y Y Y Y

id id

in in

Y Y Y Y

in

N

Nowwe dene a class of morphisms in C that admits a calculus of left fractions

N

The category C will b e equivalentto tow C pr o C

N

Consider the class dened by the morphisms of the form Y Y Y where

id

N N

Y is an ob ject in C and InN C has a nal ob ject

id

Proposition admits a calculus of left fractions

id

id Proof a It is clear that Y

Y

id

b By Lemma wehave

Y Y Y Y Y

id id id id

Theory and Applications of Categories Vol No

X

f

id

X X Y

c For each diagram wehave the

f

X Y

X Y

id id

y y

X Y

f

where Y is in

id

f

X

id

X Y X

such that f X d Consider a diagram g X

id id

g

Applying Lemma wehave

f X f X Y f f X f X

id id id id

g Y g X g X g X

id id id

in satisfying the desired relation Therefore there exists Y

id

If we consider the diagram

scd

N

N

C C

y y

N

pr o C C

N

X X of has an inverse X inpr o C Dene we can see that a morphism X

N

id id

scd

N

X id inC

X

id

Lemma The morphisms X and X give an isomorphism in the category

id id

pr o C tow C

N

Proof Wehave that

id X id X

X

N

id id

Theory and Applications of Categories Vol No

Since the following diagram is commutative

X

X X

X X

id id

y y

id

X X

id id wehavethat X

X

N

id

On the other hand wehave

in in in

X X X X X

in id

id id

id id X X X

X

N

id id id

Wehave already seen in the pro of of Lemma that X id id

X

N

id

N

Theorem For a category C with nal object the inducedfunctor C

pr o C is an equivalenceofcategories

N

Proof It suces to dualize the pro of of Theorem

Remark We can also prove this theorem taking into account the denition of

the homset see GZ and Prop osition

N N N

C X Y colim colim pr o C X Y C X Y C X Y

N

N N

InN InN

id id

N N

The equivalence of categories C C is given by

b

X Y

f f

id id

X X Y X Y Y

N

Consider the class of morphisms of C of the from X X X If D is

C

id

D

a nite category and C denotes the category of functors of the form D C wecan

N

D

consider the class D of morphisms of C of the form A A The corresp onding

C

D

category of fractions is denoted by tow C Notice that wealsohave the equivalence

D

N

D N N

of categories C tow C induces the natural functor C and the functor C

D

N D

tow C C

With this notation wehave the following result

Theory and Applications of Categories Vol No

Proposition If D is a nite category then there is a diagram

D

N

D N

C C

y y

D

D

tow C tow C

D

which is commutative up to isomorphism and is such that the induced functor tow C

D

tow C is a ful l embedding

D

N

D N

Proof Let X denote an ob ject of C and the corresp onding ob ject in C IfX

N

D D

is an ob ject in C X n denotes a diagram of C and if X is thoughtofasanobject

D

N N

in C then X is an ob ject in C We note that X n X n

d d d

D

N

D N

Now supp ose that X Y are ob jects in C or in C andf X Y is a

D

morphism in tow C Then for each d an ob ject of D we can represent f for all d by

d

f

d

N d d

Y By considering a map InN suchthat d Ob D X X

d d d

id

f

d

we can represent f for all d by X X Y Thenwehavethat X X is

d d d d

N



D

d is a morphism in D then a morphism in C However if d

f

d



X Y

d d

 

X Y





y y

Y X

d d

 

f

d



N

such that is only commutativein tow C Nevertheless wecancho ose InN

id 

f

d



Y X

d d

  

y y

X Y

d d

  

f

d 

Theory and Applications of Categories Vol No

N

is commutativein C Since the set of morphisms of D is nite we nally obtain

f

d

Y suchthatX X representatives maps X X

d d

r

 

f

N

D

Y is a diagram in C

r

D

This diagram represents a morphisms from X to Y in tow C thatissentto

D D

f X Y by the functor tow C tow C

N

D D

Nowiff g X Y are maps in C such that f g in tow C wehave for each

d d

d d N

If d Ob D g X d Ob D a map InN such that f X

d d d d

id id id

then f X g X Therefore f X g X This implies that f g in

d d d d

id id id id

D

tow C

Remark Meyer Mey has proved that if the category C has nite limits then

D D

the functor pr oC pr oC is an equivalence of categories

Preliminaries on monoids nearrings and rings

In this section we establish the notation and prop erties of monoids nearrings and rings

that will b e used in next sections We usually consider the categories of sets p ointed sets

groups and ab elian groups which are denoted by Set Set Gr p and Ab resp ectively

A monoid consists of a set M and an asso ciativemultiplication M M M

with unit element m m m for every m M If M has also a zero element

m m for every m M it will b e called a monoid

A set R with two binary op erations and is a unitary left nearring if R is

a group the additive notation do es not imply commutativity R is a semigroup and

the op erations satisfy the left distributivelaw

x y z x y x z x y z R

A nearring R satises that x andx y x y but in general it is not true

that x for all x R If the nearring also satises the last condition it is called

a zerosymmetric nearring In this pap er we will only work with zerosymmetric unitary

nearrings In this case R is a monoid

If a zerosymmetric unitary nearring also satises the right distributivelaw

x y z x z y z x y z R

then R is ab elian and R b ecomes a unitary ring

Example If C is a category and X is an ob ject of C thenC X X is a monoid

with the comp osition of morphisms g f g f In next sections C will b e one of the

categories pr oC or towC

Example If C is a category with a zero ob ject the monoid C X X has a zero

elementX X and C X X is a monoid If C has a zero ob ject the categories

pr oC and tow C also have zero ob jects

Theory and Applications of Categories Vol No

Example Let F b e a free group generated by a set X then the set of endomor

phisms of F EndF b ecomes a zerosymmetric unitary left nearring if the op eration

is dened by

f g x fx gx f g EndF x X

Example If A is an ab elian group or an ob ject in an ab elian categorythenEndF

is an unitary ring

Let M b e a monoid and C a category A left M ob ject X in C consists of an ob ject

X of C and a monoid M CX X m m X X IfM is a

monoid and C has a zero ob ject we will also assume that an M ob ject X in C satises

the additional condition We denote by C the category whose ob jects are the

M

left M ob jects in C By considering monoid antimorphisms M CX X wehave

the notion of right M ob ject in C and the category C

M

For the case C Set C Set wehave the notion of M set M p ointed set and the

categories Set Set Set Set If X is a group then Set X X has a natural

M M M M

structure of zerosymmetric unitary right nearring If R is a zerosymmetric unitary

right nearring a structure of left Rgroup on X left nearmo dule is given by a near

ring homomorphism R Set X X If R is a zerosymmetric unitary left nearring

and R Set X X is a and a monoid antimorphism then X

is said to have a structure of right Rgroup We denote by Gr p the category of left

R

Rgroups and by Gr p the category of right Rgroups

R

If X is an ab elian group then AbX X has a natural structure of unitary ring If R

is a ring a structure of left Rab elian group Rmo dule is given by a ring homomorphism

R AbX X If R AbX X is a group homomorphism and a monoid antimor

phism then X is said to b e a right Rab elian group Rmo dule We denote by Ab the

R

category of left Rab elian groups and by Ab the category of right Rab elian groups

R

Example Let C b e a category For each ob ject X of C wehave the monoid

EndX C X X andC X C Set is a functor which asso ciates to

EndX

an ob ject Y the right EndX ob ject dened by EndX SetC X Y C X Y

f f f CX Y If C has a zero ob ject wealsohave the functor

C X C Set

EndX

Example Let F b e a free group generated by the set X Wehave noted in Example

ab ovethat EndF is a left nearring Is is easy to see that for any group Y Gr pF Y

has a natural structure of right EndF group Therefore there is an induced functor

Gr pF Gr p Gr p

EndF

Example If X is an ob ject in an ab elian category AthenAX denes a functor

from A to the category of EndX ab elian groups EndX mo dules

Recall that in this pap er we are using the following unied notation We denote by C

one of the categories Set Set Gr p Ab The small pro jective generator of C is denoted

by GWe also denote by S Z Z the corresp onding generators of these categories a

Theory and Applications of Categories Vol No

Because the categories of the examples ab ovehave some common prop erties we will use

the following notation

If C Set and R is a monoid C denotes the category of right Rsets

R

If C Set and R is a monio d C denotes the category of right Rp ointed sets

R

If C Gr p and R is a zerosymmetric unitary left nearring C denotes the

R

category of right Rgroups Rnearmo dules

If C Ab and R is an unitary ring C denotes the category of right Rab elian

R

groups Rmo dules

It is interesting to note that C and C are algebraic categories and there is a natural

R

U C C which has a left adjoint functor

R

F R C C If C Set and X is a set then X R X R and if

R

C Set then X R X RR X It is also easy to dene R for

the cases C Gr p and C Ab

If the functor F A B is left adjoint to the functor U BAthenitiswell known

that F preserves colimits and that U preserves limits A functor U BAreects nite

limits if it veries the following prop erty If X is the cone over a nite diagram D in B

and UX is the limit of UDthenX is the limit of D

We summarise some prop erties of the functors ab ove in the following

Proposition The forgetful functor U C C has a left adjoint functor

R

F R C C Moreover the functor U preserves and reects nite limits

R

in particular if Uf is an isomorphism then f is also isomorphism

Proof It suces to checkthat U reects nite pro ducts and dierence kernels If Y Y

are ob jects in C then UY UY admits an action of R dened byy y r y ry r

R

if r R Nowitiseasytocheck that for the dierent cases C Set Set GrpAb

this action satises the necessary prop erties to dene an ob ject Y Y in C such that

R

U Y Y UY UY Similarly if f g Y Y are morphisms in C then the

R

dierence kernel K Uf Ugisdenedby K Uf Ugfx UY j Ufx UgxgIn

this case the action of R on Y induces an action on K Uf Ug that satises the necessary

prop erties and therefore denes an ob ject K f gsuch that UKf g K Uf Ug

whichhas C Given a morphism R R there is an induced functor V C

R R

 

a left adjoint functor R C C It is not hard to give a more explicit

R R R

  

denition of the functor R for the cases C Set Set Gr p Ab

R



In next sections we will consider the prop erties of the following construction to study

the inverse limit functor

Let s b e an elementof R if C Gps and R a left nearmo dule we also assume that

s is a right distributiveelement x y s xs ys and let X b e an ob ject in C dene

R

F X fx X j x s xg

s

s

This gives a functor F C C which has a left adjoint R C C dened as

s R

follows Let X b e an ob ject of C the functor R C C carries X to X R

R

Consider on X R the equivalence relation compatible with the corresp onding algebraic

Theory and Applications of Categories Vol No

structure and generated by the relations x r x sr Denote the quotient ob ject by

s s

X R and the equivalence class of x r by x r We summarise this construction in

the following

s

Proposition The functor R C C is left adjoint to F C C

R s R

Browns construction

In this section we dene the P functor for the categories of prosets prop ointed sets pro

groups and proab elian groups As in the section ab ove C denotes one of the following

categories Set Set Gr p Ab

Because the category C has pro ducts and sums then we can dene the functors

N N

c C C and p C C by the formulas

cX X X X j i

i j j

j i

pY Y

i

i

N

C X pY therefore wehave It is easy to check that C cX Y

N N

Proposition The functor c C C is left adjoint to p C C

Asso ciated with the generator G of C wehave the proob ject cG and the endomor

phism set P cG pr oC cG cGwhich has the following prop erties

If C Set the morphism comp osition gives to P c a monoid structure

If C Set P cS is a monoid see section

If C Gr p P cZ is a zerosymmetric unitary left nearring

If C Ab P cZ is a ring

a

For anyobjectX of pr oC we consider the natural action

pr oC cG X pr oC cG cG pr oC cG X

which applies f to fiff pr oC cG X and pr oC cG cG

The morphism set pr oC cG X has the following prop erties

If C Set pr oC cG X admits a natural structure of P cGset Thus the action

satises

f f

f f

f pr oC cG X pr oC cG cG

If C Set pr oC cG X andpr oC cG cGhave zero morphisms that satisfy

f f pr oC cG X

PcG

Theory and Applications of Categories Vol No

that is P cG is a monoid see section and pr oC cG X isaP cGp ointed set

If C Gr pwealsohave that pr oC cG X has a group structure and the action

satises the left distributivelaw

f f f f pr oC cG X PcG

Notice that the sum need not b e commutative In this case P cG b ecomes a zero

symmetric unitary left nearring and pr oC cG X isaright P cGgroup nearmo dule

see Mel and Pilz

If C Abwe also have a right distributivelaw

f g f g f g pr oC cG X PcG

Now P cG b ecomes a unitary ring and pr oC cG X is a right P cGab elian group P cG

mo dule

In order to have a unied notation C denotes one of the following categories

P cG

If C Set C is the category of P cGsets

P cG

If C Set C is the category of P cGp ointed sets

P cG

If C Gr p C is the category of P cGgroups nearmo dules

P cG

If C Ab C is the category of P cGab elian groups mo dules

P cG

Using the notation ab ovewe can dene a functor P pr oC C as the repre

P cG

sentable functor

P X pr oC cG X

together with the natural action of P cG where G is the small pro jective generator of C

For the full sub category tow C we will also consider the restriction functor P tow C

C

P cG

Because C has sums pro ducts and a nal ob ject forany ob ject Y of C wecan

consider the direct system

Y Y Y

i i i

where the b onding maps are induced by the identity id Y Y and the zero map Y

The reduced pro duct IY of Y is dened to b e the colimit of the direct system ab ove We

also recall the forgetful functor U C C considered in section whichwillbeused

P cG

in the following

Proposition The functor U P pr oC C has the fol lowing properties

If X fX g is an object of pr oC then

j

U P X lim IX

j j

If X fX j j J g is an object of pr oC where J is a strongly conite directed

j

set then

J

colim U P X C cG X

J

N

Theory and Applications of Categories Vol No

If X is an object in tow C then

colim pX U P X

N

InN

id

Proof For it suces to consider the denition of the homset in pr oC

U P X lim colim C cG X

i j

j i

lim colim X

j

j i

k i

lim IX

j

j

follows since N J are strongly conite directed sets

By Remark after Theorem and Prop osition weget

U P X tow C cG X

N

colim C cG X

N

InN

id

colim C G pX

N

InN

id

colim pX

N

InN

id

Proposition The functor P pr oC C preserves nite limits

P cG

Proof Wehavethat U P preserves nite limits since U P pr oC cG is a repre

sentable functor see Pa Th Sect Ch By Prop osition wehavethat U reects

nite limits Therefore we get that P preserves nite limits

Nowwe recall that Grossman in Gr proved that the functor U P tow C C

reects isomorphisms Since P preserves nite limits wehave

Theorem P tow C C is a faithful functor

P cG

Proof Let f g X Y be two morphisms in tow C Ifwe consider the dierence kernel

i K f g X the Prop osition ab ove implies that P K f g K P f P g Supp ose

that P f P g then K P f P g P X and P i P K f g PX is an isomorphism

Applying the forgetful functor U C C wehave that U P i is an isomorphism Now

P cG

Grossmans result implies that i is also an isomorphism Therefore f g

Remark Since P tow C C is a representable faithful functor wehave

P cG

that P preserves monomorphisms and reects monomorphisms and epimorphisms

Notice that the pro of given do es not work for the larger category pr oC

Theory and Applications of Categories Vol No

Proposition P tow C C and U P tow C C preserve epimorphisms

P cG

Proof Let q X Y b e an epimorphism in tow C by the Remarks at the end of

ChI I x of MS it follows that q is isomorphic in M apstow C to q X Y where

q is a level map fq X Y g and each q X Y is a surjectivemapNowwehave

i i i i i i

that U P q colim pq and since p colim preserve epimorphisms weget

that U P q and P q are epimorphisms

Definition Let S be a ful l subcategory of pr oC An object X in S is said to be

admissible in S if for every Y of S the transformation

P pr oC X Y C P X P Y f Pf

P cG

is bijective If S pr oC X is said to be admissible

Proposition The object cG is admissible

Proof Wehave the following natural isomorphisms

U P X pr oC cG X

C G U P X

C G PcG P X

P cG

C P cG P X

P cG

which send a map f cG X to P f P cG PX

Proposition Let p X Y be an epimorphism in tow C IfX is admissible in

tow C then Y is also admissible in tow C

Proof We can supp ose that p is a level map fp X Y g such that each p X Y

i i i i i i

is a surjective map Let X X denote the equivalence relation asso ciated with p that

i i i

Y

i

is X X fx x X X j p x p x g Then the diagram

i i i i i i

Y

i

pr



p

X X X Y

Y

pr



X g is a dierence cokernel in tow C where X X fX

i i

Y Y

i

Let Z b e an ob ject in tow C Given a morphism P Y PZ in C since X is

P cG

admissible in tow C there is a morphism f X Z in tow C suchthat P p P f

P fprP f P pr P p P pr P ppr

P ppr P p P pr P f P pr P fpr

Because P is faithful it follows that fpr fprNowwe can use that p is a dierence

cokernel to obtain a morphism g Y Z such that gp f Wehave that P g P p

P f P p By Prop osition P p is an epimorphism then wehavethat P g This

implies that Y is also admissible in tow C

Theory and Applications of Categories Vol No

Definition Anobject X of tow C is said to be nitely generated if thereisan

eective epimorphism of the form cG X

f inite

Theorem Let X beanobject in tow C If X is nitely generated then X is

admissible in tow C Consequently the restriction functor P tow C f g C is a ful l

P cG

embedding where tow C f g denotes the ful l subcategory of tow C determined by nitely

generated towers

cG is isomorphic to cG By Prop osition it Proof It is easy to check that

f inite

follows that cG is admissible Because X is nitely generated there is an eective

f inite

cG X Now taking into account Prop osition we obtain that epimorphism

finite

X is also admissible

Proposition Let Y f Y Y Y g be an object in tow C where

l

the bonding morphisms are denotedbyY Y Y l k If for each i thereis

l k

k

j

a nite set A Y such that for each n Y A generates Y then Y is nitely

i i j n

n

j n

generated

G and consider the diagram Proof Dene X

n

j n A

j

X X

n n

p p

n

n

y y

Y Y

n n

j

where the restriction of p to G is induced by the map Y A Y It is clear that

n j n

n

A

j

cG and p X Y is an epimorphism Therefore Y is nitely generated X

Corollary A tower of nitely generated objects of C is a nitely generated

tower

A tower of nite objects of C is nitely generated

The restricted functors

P tow Cf g C

P cG

P tow Cf C

P cG

are a ful l embeddings where Cf g and Cf denote the ful l subcategories determinedby

nitely generated objects and nite objects respectively

Theory and Applications of Categories Vol No

Remark A particular case of Corollary are Theorem and Corollary

of Ch

Next we study some relations b etween the P functor and the lim functor

Recall that for i cG G where G is a copy of the generator G The

i k k

k i

identityof G induces a map G G k iWe denote by sh cG cG the level map

k k

G g induced by the maps G G G fsh

k k k k i

k i k i

Given an ob ject Y in C we denote by F Y the ob ject of C dened by

P cG sh

F Y fy Y j y sh y g

sh

Notice that F denes a functor from C to C

sh P cG

Theorem The fol lowing diagram

lim

tow C C

P F

sh

C

P cG

is commutative up to natural isomorphism That is limX fx PX j x sh xg

N

Proof For each InN there is a map SH P X P X whichapplies

id

x x x x to the element

xS H xS H xS H

i

X Notice that if xS H x then where for i xS H x

i i

i

i

X x x Therefore x limX

i i

i

Asso ciated with the map X X wehave the commutative diagram

SH

lim X P X P X

id

y y y

SH

limX P X P X

id

Because where limX is the dierence kernel of SH and id and similarly for limX

X X is an isomorphism in tow C it follows that limX limX isaniso

morphism Nowtakinginto account that colim preserves dierence kernels we obtain

Theory and Applications of Categories Vol No

that

sh

lim X PX P X

id

is a dierence kernel Therefore limX fx PX j x sh xg F P X

sh

sh

Nowwe can use that P cG C C is left adjoint to the functor

P cG

F C C to obtain the following result

sh P cG

Corollary The functor lim tow C C can also berepresented as fol lows

sh

limX C G P cG P X

P cG

limX C P conG P X

P cG

sh

Moreover there is a natural map G P cG PconG where conG denotes the level

id

wise constant tower f G Gg

Proof This follows b ecause con C tow C is left adjointtolim tow C C and

sh

P cG C is left adjointto F C C see Prop osition It is also

P cG sh P cG

necessary to takeinto account the fact that conG is admissible in tow C This follows

b ecause conG is a tower of nitely generated ob jects see Corollary and Theorem

Remark Theorem gives a relation b etween the P functor and the lim

functor for the case of towers If X fX g is a tower then P X lim IX whereIX

i i i

is the reduced countable p ower For a more general proob ject X J C Porter Por

q

uses more general reduced p owers to compute the lim and lim functors

For a tower of groups X an action of P X on P X can b e dened by

x

y x y x sh x y PX

It is easy to check that the space of orbits of this action is isomorphic to the p ointed

X The dierence of twoelements of the same orbit is of the form x y x sh y set lim

Notice that the quotient group obtained by dividing by the normal subgroup generated

by the relations x y x sh y satises that the action of sh is trivial and it is an

ab elian group

For a tower X of ab elian groups we get isomorphisms

sh

lim X Ext Z P cZ P X

a a

Ext P conZ P X lim X

a

In this case wealsohavethatlim X is obtained from P X by dividing by the subgroup

generated by the relations x xsh for all x PX

A global version of Browns P functor can b e dened for global category pr oC C

for the denition of pr oC C see EH If X is an ob ject in pr oC C then P X is g

Theory and Applications of Categories Vol No

dened to b e the homset P X pr oC C cG X where cG is considered as an ob ject

g

in pr oC C provided with the structure given by the action of P cGWe note that for

g

the global version of the P functor if X is an ob ject in tow C C then

U P X colim P X

g

N

In N



id

N N

where In N f InN j g

id id

Applications and prop erties of the P functor

In this section rstly we obtain some consequences of the main Theorems of section

We also analyse the structure of the endomorphism set P cG for the dierent cases

C Set Set GrpAb Finallywe study some additional prop erties of the P functor for

the cases C Gr p Ab

If C is one of the categories S et S et GrpAbwe will denote by TC the corresp ond

ing top ological category That is TC will resp ectively b e one the categories top ological

spaces top ological p ointed spaces top ological groups or top ological ab elian groups We

denote by z cmT C the full sub category of TC determined by zerodimensional compact

metrisable top ologies

Let X b e an ob ject in C Consider the set of quotient ob jects of the form

p X F where F is a nite discrete ob ject in C Given two quotients of this form

p p

p X F and p X F wesaythat p p if there is a commutative diagram

p p

X

p p

F F

p p

It is easy to check that fp X F j F is a nite discrete quotient ob ject g with

p p

is a directed set Therefore we can dene the functor TC pr oCf X fF g

p p

where Cf denotes the full sub category of C determined by nite ob jects If X has a

zerodimensional compact metrisable top ology then there is a sequence p X F such

i i

that p p and for any p of there is i suchthat p p Hence fp g is conal

i i i i i

in and the tower fF g is isomorphic to fF g

i i p p

Consequently it is clear that

Theorem The category tow Cf of towers of nite objects in C is equivalent to

the category z cmT C of objects in TC which have a zerodimensional compact metrisable

Theory and Applications of Categories Vol No

Remark T Porter has p ointed out to me that Theorem is closely connected

with a famous theorem of MH Stone Sto whichgives category equivalences b etween the

category of Bo olean spaces the category of Bo olean algebras and the category of Bo olean

rings

Theorem There is a ful l faithful functor zcmT C C

P cG

Remark There is a full emb edding from the prop er homotopy category of

compact lo cally compact Hausdor spaces into the homotopy category of prospaces

considered by EdwardsHastings EH If X is a compact lo cally compact simplicial

complex then there is a conal sequence fK g of compact subsets of X such that for every

i

i cl X K is a nite set Therefore X f cl X K g is admissible

i i

in tow S et If X is a prop er raythen determines a pathcomp onentof

cl X K and X can b e considered as an ob ject in tow S et IfX is a simplicial

i

complex as ab ove we will supp ose that is a simplicial injective map In this case the

fundamental progroup can b e dened by X f cl X K Im gIf

i

X has one Freudenthal end it is easy to check that X is admissible in towGrp

Finallywe also note that for q the tower H X fH cl X K g is admissible

q q i

in tow Abwhere H denotes the singular homology

q

Let X b e a compact metrisable p ointed space Denote by CX the prop ointed

simplicial set of the Cech nerves asso ciated with the directed set of op en coverings of

X In this case it is easy to checkthat CX is isomorphic to an admissible ob ject in

CX is isomorphic CX is isomorphic to an admissible ob ject in tow Gr pandH tow S et

q

to an admissible ob ject in tow Ab

As a consequence of Theorems and for the category of connected lo cally

nite countable simplicial complexes the following categories are adequate for mo delling

the prop er typ e The category of zerodimensional compact metrisable spaces and the

 Freudenthal end functor e the category tow f inite sets and the functor and Set

P cS

BG

and the BrownGrossman homotopy group The relations b etween these functors

BG

are given by e lim P Similarly for the shap e typ e of compact

metrisable spaces wehave the functors lim C C and P C

Next we study the dierent structures of the endomorphism set P cG pr oC cG cG

for the dierent cases C S et S et GrpAb

C Set

The monoid P c can b e represented as follows Consider the set R of matrices of the

form A A with i j f g where either A or A satisfying the

ij ij ij

following prop erties

a For each j the cardinalityof fi j A g is

ij

b For each i there exists j i suchthatfor kiand j l a

kl

Write i minfj j j i and if kiand j l then a g

A kl

Dene the equivalence relation by declaring A A if there exists j such that

for j l the l column of A agrees with the l column of A thatisA and A dier only

on a nite numb er of columns

Theory and Applications of Categories Vol No

Matrix multiplication induces over R a monoid structure that is compatible with

the relation Therefore the quotient R inherits a monoid structure from R Ifwe

also denote by N the set of natural numb ers provided with the discrete top ologywe

can consider the monoid P N N of prop er maps N N and the monoid of germs of

prop er maps P N N Given a matrix A of R we can dene a prop er map that will

again b e denoted by A N N as follows if j the j columm of A has only one

element A dene Aj i This gives monoid isomorphisms R P N Nand

ij

R P N N The isomorphism

P c colim pc R

N

InN

id

is given by A AAA pc

A

Given an ob ject X of tow S et the action P X PcPX can b e dened as follows

take x pX and A R then xA y where y pX is dened by

A

A j A j

y X x X x A

l lj

j A j

j j

A

A A

l

C Set

The monoid P cS can also b e represented as a matrix monoid R as follows We



S

consider matrices A A asabove satisfying prop erties a and b where a is obtained

ij

by mo difying a

a For each j the cardinalityof fi j A g is at most one

ij

Denote by N N fg the Alexandro compactication of N bya point and

consider the monoid Top N N and the monoid Top N N of

germs at of continuous maps N N Given a matrix Awe can dene the

continuous map AN N such that if j and the j column of A has a

unique element A then Aj i otherwise Aj This denes isomorphisms

ij



Top N N The isomorphism R Top N N and R



S

S

colim pcS is given by A AAA pcS R



A

S

N

InN

id

Notice that R is a submonoid of R



S

C Gr p

Let F b e the free group over the countable set of letters fx x x g The multipli

cation of F will b e denoted bythenatypical word of F is of the form x x x

we note that an additive notation do es not imply commutativity Let R denote the set

Z

whose elements are of the form w w w where for i w F satisfying the

i

following prop erty

For each i there exists j i suchthatx x are not letters of the reduction

i

of w for l j For a given element w w w w of R write

l

Z

i minfj j j i and x x are not letters of the reduction of w for l j g

w i l

Theory and Applications of Categories Vol No

The sum is dened by comp onents

w w w w w w w w w w w w

The a word whose reduction has the letters x x Denote by w x x

n n n n

r r

 

pro duct is dened by substitution as follows

     

w x x w x x w w w w x x

n n n

n n n

r r r

  

  

     

w w w w w w w w w

n n n

n n n

r r r

  

  

It is easy to checkthat and give the structure of a zerosymmetric nearring to R

Z

The zero elementis and the unit is represented byx x x Another

distinguished element is the shift op erator x x x that plays an imp ortant role in

connection with the inverse limit functor

Let I b e the subset of R dened by the elements w w w w such that

Z Z

there exists m suchthat w forlm Then it is easy to check that I is a

l

Z

normal subgroup of R R I I and r is rs I for all i I rs R

Z Z Z Z Z Z Z

Then I is a ideal of R and we can consider the quotient nearring R R I

Z Z Z Z

Z

For R and R wehave the nearring isomorphisms

Z

Z

R colim pcZ

Z

N

In N



id

R colim pcZ

Z

N

InN

id

dened by w w w w w w w where w w w pcZ

w

and is the map dened ab ove

w

C Ab

Let R denote the ring of integer matrices A a where i and j are non negative

ij

Z

a

integers and eachrow and each column have nitely many non zero elements

If A is a matrix of R for each i there exists j i such that a for l j

kl

Z

a

and kiFor a given matrix Awrite i minfj j j i and if kiand l j then

A

a g Let I b e the subset of R dened by the nite matrices Then it is easy to

kl

Z Z

a a

checkthat I is an ideal of R and we can consider the quotientringR R I

Z Z Z Z

a a a a

Z

a

We also have the canonical ring isomorphisms

R colim pcZ

a

Z

a

N

In N



id

R pcZ colim

a

Z

a

N

InN

id

dened by

A column of A column of A column of A

where column of A column of A column of A pcZ and is the map

a A A

dened ab ove

Theory and Applications of Categories Vol No

a

Let F b e the free ab elian group generated by the countable set fx x x g

a

That is F f fx x x g where f Set Ab denotes the free ab elian func

a a

a a a

tor Consider the following sequence of subgroups F F F f fx x g

a

a a

F f fx x getc This family of subgroups denes on F the structure of a

a

top ological ab elian group Denote by TAb the category of top ological ab elian groups

a a a a

Let End F F denote the ring of continuous endomorphisms of F Iff F

TAb

a

F is a continuous homomorphism b ecause x wehavethatw fx

i i i

a a

This implies that wehave a canonical isomorphism R End F F

TAb

Z

a

a a

Given two continuous f g F F wesay that f and g have

the same germ if there exists n such that for every n n f x g x Let

n n

a a a

End F F denote the ring of selfgerms of F it is also clear that R is isomorphic

TAb

Z

a

a a

to End F F

TAb

Next we compare the dierent P functors for the cases C Set GrpAb

Proposition Consider the diagram

P

tow Gps Gps

P cZ

x x

f f



tow S et Set

P cS

P

where the functor f tow S et tow Gps is induced by the free functor Set Gps and



Gps is the free functor associated with the algebraic forgetful functor f Set

P cZ P cS



Gps Set see Pa th of Then the unit X uf X of the pair of

P cS

P cZ

adjoint functors f tow S et tow Gps u tow Gps tow S et induces a natural

and epimorphic transformation f P X PfX

X

Proof The unit transformation Y uf Y induces the transformation

P Y P uf Y uP fY By adjointness we obtain the desired transformation

f P Y P fY

An elementof P fY can b e represented as a sequence of words

a y y y y

r r r r r r

     

where f g y y Y y y Y etc

k r r r

  

If you take one by one the letters of these words you obtain an elementof f P Y

b y y y y y

r r r r

   

If you replace the y s of a by xs you will haveanelementof P cZ

w x x x x

r r r r r r

     

Theory and Applications of Categories Vol No

It is clear that bw a then bw a Therefore f P Y PfY is a surjective

X

map

Corollary Consider the functor f tow S et tow GpsIfX is admissible in

tow S et then fX is admissible in towGrp

Proof We use the facts that P is a faithful functor and f P X PfX is an epimor

phism to obtain

tow GpsfXY Gps P fX P Y Gps f P X P Y

P cZ P cZ

 

Set P X uP Y Set P X P uY

P cS P cS

tow S et X uY tow GpsfXY

where denotes an injective map and denotes an isomorphism Because the

comp osite is the identitywehave tow GpsfXY Gps P fX P Y

P cZ

We include here some additional prop erties of the P functor for the category of towers

of ab elian groups

Proposition The functor P tow Ab Ab preserves nite colimits

P cZ

a

Proof In an ab elian category the pro duct and copro duct of X and Y arebothgiven

byanobjectZ and morphisms i X Z j Y Z p Z X and q Z Y suchthat

pi id qj id qi pj and ip jq idSincetow Ab is an ab elian category

see AM and P is an additive functor it follows that P preserves nite copro ducts

g

k

Given a morphism f X Y in tow Ab f factorizes as X X Y where g is an

epimorphism and k is a monomorphism It is easy to checkthatcoker f coker k

By Remark after Theorem and by Prop osition wehavethat P preserves

coker P k Because monomorphisms and epimorphisms so we also obtain that coker P f

tow AbcZ wehave the exact sequence U P

a

U P X U P Y U P coker k Ext cZ X

tow Ab a

Since cZ is a pro jective ob ject see He wehavethatExt cZ X Therefore

a tow Ab a

U P X U P Y U P coker k is a short exact sequence Since U

reects monomorphisms epimorphisms and kernels wealsohavethat PX

P Y Pcoker k is a short exact sequence Then P coker f P coker k

coker P k coker P f

We next consider the inclusion functor i Ab Gps and the ab elianization functor

a Gps Ab which is the left adjointofithatisAbaX Y GpsX iY Weshall

also consider the unitary nearring epimorphism P cZ PcZ that induces an inclusion

a

functor i Ab Gps which has a left adjoint d Gps Ab Itiseasyto

P cZ P cZ P cZ P cZ

a a

Theory and Applications of Categories Vol No

check that the diagram

P

tow Ab Ab

P cZ

a

i i

y y

tow Gps Gps

P cZ

P

is commutative up to natural isomorphism The following result proves that P a dP

Theorem Consider the diagram

P U

tow Ab Ab Ab

P cZ

a

x x x

a d a

tow Gps Gps Gps

P cZ

P U

where a and d are left adjoint to the corresponding inclusion functors Then

There is a natural equivalence dP X PaX induced by the unit transformation

P aX diP aX X iaX dP X dP iaX

The natural transformation aU Y UdY is epimorphic

Proof Given an ob ject X in tow Gps consider the following diagram where several

notational abuses are made in order to have a shorter notation

P X X P X P aX

x x

id

D P X P X dP X

x x

id

P X P X P X aP X

where if X fX g X X fX X g and denotes the normal subgroup generated

i i i

by the commutators x y x y x y By Remark after Theorem and

Theory and Applications of Categories Vol No

the rst row of the diagram ab ove is exact In the second row D P X is the sub

P cZgroup generated by xw yw x y w where x y are elements of P X and w PcZ

Notice that if w wehave x y x y Therefore D P X contains the commutator

subgroup P X P Y Recall that if H isagroupand y H H theny y y

r r

where f g and y a b with a b H

i i i i i i

An element a of P X X can b e represented bya sequenceofwords

a y y y y

r r r r r r

     

are basic y y are basic commutators of X where f g y y

r r k r

  

commutators of X etc

If you take one by one the basic commutators of these words you obtain an element

of P X P X

y y y b y y

r r r r

   

and by replacing the y s of a by xs we get an elementof P cZ

x w x x x

r r r r r r

    



satisfying bw a Since D P X is a subP cZgroup a D P X Therefore D P X

P X X and this implies that P aX dP X

The left adjoint for the P functor

In this section we construct a left adjoint functor for the P functor

First weintro duce some notation and a technical result Prop osition that gives

the construction of the left adjoint Applying this prop osition to the P functor wehave

the desired result

Assume that A B are categories with innite sums P AB is a given functor and

H is an ob ject of A suchthatforany X of A

P AH X BP H P X

is a bijection

Let S b e the category whose ob jects are ob jects of B with a given decomp osition

P H where A is an index set denotes the sum or copro duct in B of the form

A

and H H for all A The morphismset from P H to P H is given by

A B

B P H P H

A B

H denote the canonical inclusion into the copro duct where Let in H

H

B

it is assumed that H H for any B Applying the functor P and the universal

prop erty of the sum wehave the morphisms

H P in P H P

H

B

H P H P P in

H

B B B

Theory and Applications of Categories Vol No

Next we are going to construct a functor l SAGiven an ob ject P H of S

A

dene

l P H H

A A

where H H for any A

u where P H is a morphism of S then u P H If u

A B A

P H and in u uin P H are the canonical inclusions

P H P H

A

For each A consider the comp osition

P in

H

B

u

P H P H P H

B B

H is a bijection there is a unique H B P H P Since P AH

B B

lu H H suchthat P lu P in u Then dene

H

B B

lu lu

A

Next wecheck that l SAis a functor We start by showing that l preserves

identities

H are such that the diagram The canonical inclusions in H

H

A

P H

A

x

P in P in

H H

A

P H P H

in

A

P H

is commutative therefore

in lin in l idl id

P H P H H

A A A

Given two morphisms

v u

P H P H P H

C B A

Theory and Applications of Categories Vol No

wehavethecommutative diagram

P lv

P H P H

C

B

x x

P in P in

H H

P lu P lv

P H P H P H

u

B v v C

lv lu Then wehave Therefore l vu

B

lv lu l vu vu vu l l vul

B A A A A

lu lv lu lv lu lv

A A

This implies that l SA is a functor

The following prop erties of l will also b e used

a The transformation

Al P H Y B P H P Y

A A

P f is a bijection f given by f

A A

P H and g Y Y the following diagram P H b Given morphisms u

B A

is commutative

Al P H Y B P H P Y

A A

x x

Al P H Y B P H P Y

B B

P H Y wehave that is for a given f l

B

P gf u P g P f u

A B

Using this notation and the prop erties of l SAwecanprove

Theory and Applications of Categories Vol No

Proposition Suppose that A B are two categories with innite sums and dier

encecokernels and P A B a functor Assume that we have

a An object H of A such that for any X of A the map

P AH X BP H P X f Pf

is a bijection

b Two functors F F BS where S is the category denedabove and two natural

transformations u v F F such that the functor difcokeru v BB denedby

u

B

difcokeru v B difcokerF B F B

v

B

is equivalent to the identity functor of B

Then the functor P AB has a left adjoint L BA

Proof By considering the functor l SB dened ab ove wedeneL BA by

lu

B

LB difcokerlF B lF B

lv

B

Nowwehave

lu

B

lF B A ALB A AdifcokerlF B

lv

B

lu

B

AlF B A difkerAlF B A

lv

B

u

B

B F B P A difkerB F B P A

v

B

u

B

B difcokerF B F B P A

v

B

B B P A

To apply Prop osition we need to have a category with innite sums and dierence

cokernels The category pr oC has dierence cokernels and the following Lemma shows

that it also has innite sums

Lemma If C has innite then pr oC is also provided with innite co

products

Theory and Applications of Categories Vol No

Proof Supp ose wehave for each i I a proob ject X J C Consider the

i i

left ltering small category J and dene X J C by X j

i i i i i

iI

iI iI iI iI

X j j J Asso ciated with the pro jections p J J wehave

i i i iI i i i i

iI iI iI

the maps in p j X j X j that dene the inclusions in X X

i i i iI i i i i i i

iI iI

It is easy to check that X veries the universal prop erty of the copro duct in pr oC

i

iI

Recall that C denotes one of the categories Set Set GrpAband C resp ec

P cG

tively denotes one of the categoriesSet Set  Gps Ab

P c P cZ

P cZ

P cS

a

U u

C Set will To shorten notation the comp osition of forgetful functors C

P cG

be denoted by v uU C Set and the comp osition of free functors

P cG

f

F

SetC C by g Ff Set C

P cG P cG

The main result of this section is the following

Theorem The functor P pr oC C has a left adjoint L C pr oC

P cG P cG

Proof We are going to check that the conditions of Prop osition are satised Take

A pr oC and B C

P cG

a By Prop osition the ob ject H cG is admissible Then for any X in pr oC

C P cG P X pr oC cG X

P cG

In the cases we are considering for anyobjectB of C the natural transformation

P cG

p gvB B is a surjective epimorphism By Lemma and Corollary of section

B

of Pa wehave that if we consider the bre pro duct

pr



gvB gvB gvB

B

p pr



B

y y

B gvB

p

B

then

pr



gvB B gvB gvB

p

B

B

pr



is a dierence cokernel Since gvgvB gvB gvB gvB is an epimorphism it also

B B

follows that

pr p



B gvB gvgvB gvB

p

B

B

pr p



is also a dierence cokernel

Theory and Applications of Categories Vol No

Then we can dene the functors F F C S by

P cG

F B gvB

F B gvgvB gvB

B

and the natural transformations by u pr p and v pr p

Nowwe are under the conditions of Prop osition to obtain that P pr oC C

P cG

has a left adjoint L C pr oC

P cG



Corollary The functors P pr oS et Set P pr oS et Set

P c

P cS

P pr oGps Gps and P pr oAb Ab have left adjoints

P cZ P cZa

Remark MIC Beattie Be has constructed an equivalence b etween the cate

gory of nitely presented towers of ab elian groups and nitely presented P cZ ab elian

a

groups or P cZ mo dules This equivalence is also given by the restrictions of the functor

a

P pr oAb Ab and its left adjoint L Ab pr oAb to the corresp onding full

P cZ P cZ

a a

sub categories

Global towers and top ological ab elian groups

In this section we analyse some relations b etween towers of ab elian groups and top ological

ab elian groups As a consequence of these relations we prove that the categories of towers

and global towers of ab elian groups do not havecountable sums This implies that neither

category is equivalent to a category of mo dules

Definition Let N be a neighbourhood of the zero element of a topological group

B We wil l say that N is structuredif N is also a subgroup of B Atopological abelian

group is said to belocal ly structured if it has a neighbourhoodbase at of structured

neighbourhoods

Let TAb denote the category of top ological ab elian groups and ST Ab the full sub cat

egory determined by lo cally structured top ological ab elian groups

Next we dene two functors Lpr oAb Ab ST Ab and N ST Ab pr oAb Ab

such that L is left adjointtoN

Recall that an ob ject X of pr oAb Ab is a morphism X X X where X

is an ob ject of pr oAb and X is an ob ject of the category Ab which can b e considered as

a full sub category of pr oAb A morphism f X Y in pr oAb Ab consists of a pair of

morphisms f f f such that the following diagram

f

X Y

y y

Y X

f 

Theory and Applications of Categories Vol No

is commutativein pr oAb

Any ob ject of pr oAb Ab can b e represented up to isomorphism by a functor

X Ab where is a directed set with a nal element If

let X X X denote the corresp onding b onding morphism Asso ciated with X we

have X X Ab which is an ob ject of pr oAb X whichisanobjectofAb or a

constant proab elian group and the natural morphism X X given by the identity

X X

Next we use this notation to dene a functor Lpr oAb Ab ST Ab Given an

ob ject X of pr oAb Ab LX is dened to b e the ab elian group X together with the lo cally

structured top ology dened by the subgroups ImX where X X X are b onding

maps of X Notice that given there exists suchthatImX ImX ImX

o

This implies that the neighbourhood local base fImX g denes a top ology on X

If f X Y is a morphism in pr oAb Ab then the functor L is dened by

Lf f X Y Wemust checkthat f is continuous Assume that f X Y

is given by a map and homomorphisms f X Y If

Y X

ImY is a neighbourhood at Y there is a such that the following diagram is

X

commutative

f

Y X

X Y



y

X Y

f



ImY Therefore Lf LX LY is a continuous homo This implies that f ImX

morphism

To dene a functor N ST Ab pr oAb Ab for a given ob ject B of ST Ab consider

the directed set fS j S is a subgroup of B and S is a nbh at g whichhasanal

element S B Now dene NB Ab by NB S S Notice that NB B

S

Proposition Consider the functors L and N denedabove then

Lpr oAb Ab ST Ab is left adjoint to N ST Ab pr oAb Ab

The unit B LN B induced by the pair of adjoint functors is a natural equiv

alence Then ST Ab can beconsidered as a ful l subcategory of pr oAb Ab

Proof Let X b e an ob ject of pr oAb Aband Y an ob ject of ST AbIff LX Y

is a continuous homomorphism for each structured neighbourhood S of Y there exists

S S

a structured neighbourhood ImX at such that f ImX S forS Y wetake

b b b b

S Dene f X NY by f f wheref X S is the comp osition

S

S S

S S

b

X f f jImX

S

For a given g X NY dene g LX Y by g g Nowitiseasytocheck

b

b

that f f and g g

Theory and Applications of Categories Vol No

If S T Abf c denotes the full sub category of ST Ab determined by rst countable top o

logical ab elian groups we also have

Proposition The restriction functors

Ltow Ab Ab S T Abf c and

N S T Abf c tow Ab Ab satisfy

L is left adjoint to N

The unit B LN B is a natural equivalenceandS T Abf c can beconsideredas

a ful l subcategory of tow Ab Ab

We also consider the following functors g ST Ab Ab that forgets the top ology and

the functor t Ab ST Ab dened as follows If A is an ab elian group tA is the ab elian

group A together with the trivial top ology Notice that t is also a functor of the form

t Ab S T Abf cWehave the following prop erties

Proposition The functors above satisfy

g ST Ab Ab is left adjoint to t Ab ST Ab

g S T Abf c Ab is left adjoint to t Ab S T Abf c

Next weprove that the category S T Abf c do es not havecountable sums To do this

we take X LcZ that is given by the free ab elian group

a

X X f fx x x g

a

and the lo cal neighb ourho o d base at given by

X nf fx x x g

a n n n

In the category ST Abwe can consider the countable sum S X together the

i

i

top ology given by the following lo cal base For each sequence n n n n

N N N we consider

S nX n X n X n

It is not dicult to check that S is the countable sum in the category ST Ab

Lemma S X is a non rst countable topological abelian group

i

i

Proof Assume that wehavea countable neighb ourho o d base at This implies the

existence of a sequence m m m in N N N where

m m m m

m m m m

m m m m

such that S m S m S m is a countable neighb ourho o d base at where

S m X m X m X m

Theory and Applications of Categories Vol No

S m X m X m X m

S m X m X m X m

Now consider m m m m such that

m m

m m maxfm m g

m m maxfm m m g

Then wehave that

S mX m X m X m

i

is a neighb ourho o d at that do es not contain S m for i This contradiction comes

from the assumption that S was rst countable Therefore S is non rst countable

Corollary The ful l subcategory S T Abf c of ST Ab is not closedundercountable

sums

X where X Lemma The category S T Abf c does not have the countable sum

i

LcZ

a

X where Proof Supp ose that wehave a rst countable top ological ab elian group

i

i

X b e the canonical inclusions Nowsince X X for i Let in X

i i i i

i

g S T Abf c Ab is a left adjointby Prop osition it follows that g preserves sums

X gX such that for each i Therefore there exists an isomorphism g

i i

i

i

the following diagram is commutative

g X gX

i i

i i

gin in

i gX

i

gX

i

where gX X for i The isomorphism induces a top ology on X such

i f i

i

that in X X is continuous for each i Then X together with the

X i i i f

i

i i

inclusions is the sum in the category S T Abf c

Let X b e the sum in the category ST Ab Since in X X is continuous

i n i i i

i i

for the top ology itfollows that id X X iscontinuous Therefore

f i n i f

i i

is ner than

n f

Theory and Applications of Categories Vol No

For each n n n n consider the ab elian group X X n provided with

i i

i

the discrete top ology diswhich is rst countable As in X X X n iscontin

i i i i

i

uous for i the natural pro jection p X X X n disiscontinuous

i f i i

i i

Then p X n is an op en neighb ourho o d of for This implies that is

i f f

i

ner than As wehave is a rst countable top ology This fact contradicts

n f n n

Lemma

Corollary S T Abf c does not have countable sums

Corollary The category tow Ab Ab does not have the countable sum cZ

i

Corollary The category tow Ab Ab does not have countable sums

Corollary The category tow Ab Ab is not equivalent to a category of modules

Proof Assume that the ob ject cZ exists Since Ltow Ab Ab S T Abf c is a

i

LcZ This left adjoint functor it follows that L preserves sums Then L cZ

i i

contradicts Lemma

Corollary The category tow Ab does not have the countable sum cZ

i

Corollary The category tow Ab does not have countable sums

Corollary The category tow Ab is not equivalent to a category of modules

Proof If there is a countable sum cZintow Ab cZ cZ would b e

i i i

isomorphic to a countable sum cZ in tow Ab Ab This is not p ossible by Corollary

i

Next we use top ological ab elian groups to prove that the extended functor

is not faithful This implies that cZ is not a generator of all P pr oAb Ab

a

P cZ

a

the category pr oAb

Let S b e an innite set and let f S b e the free ab elian group generated by S Consider

a

the top ology dened on f S by the family of subgroups of the form f T where T S and

a a

S T is a countable set We are going to see that if a sequence y in f T converges to zero

k a

then there exists k such that y forevery k Assume that there is a subsequence

k

with x for every i Since y it follows that x Each x can b e x y

i k i i i k

i

written as a linear combination of nitely manyelements of S Therefore the sequence

x determines a countable set S of generators such that x f S S forevery i

i i a

However this contradicts the fact that x

i

Using the functor N ST Ab pr oAb Abwehave the global pro ob ject Nf S that

a

also denes a pro ob ject denoted in the same wayinpr oAb

Proposition pr oAbcZ Nf S

a a

Theory and Applications of Categories Vol No

Proof The homset pr oAbcZ Nf S is a quotient of pr oAb AbcZ Nf S

a a a a

ST AbLcZ f S Notice that By Prop osition pr oAb AbcZ Nf S

a a a a

ST AbLcZ f S is the set of sequences in f S converging to zero Twoconverging

a a a

sequences dene the same morphism in pr oAbcZ Nf S if and only if they have the

a a

same germ as the zero sequence Therefore it follows that pr oAbcZ Nf S

a a

Corollary The functor P pr oAb Ab is not faithful

P cZ

a

Proof Since the b onding morphisms of Nf S are non trivial wehavethat Nf S is not

a a

isomorphic to the zero ob ject This implies that the identityidof Nf S is not equal to

a

the zero map Nf S Nf S By Prop osition wehave that P idP

a a

Remark Grossmans result that P reects isomorphisms do es not work for the

extended functor P pr oAb Ab

P cZ

a

Corollary The object cZ does not generate the whole of the category pr oAb

Proof It is easy to prove that if cZ were a generator for pr oAbthenP pr oAb

Ab would b e a faithful functor

P cZ

a

Remark At present the author He is writing a pap er that contains some

top ological applications of the emb eddings given in this pap er It considers an extension

of the P functor to categories whose ob jects are towers of simplicial sets or towers of

simplicial groups One of the main results of the new pap er is the construction of a

simplicial set hoP X asso ciated with a tower of simplicial sets X This space is constructed

by considering a rightderived functor hoP of the version of Browns P functor dened for

the category of towers of simplicial sets Recall that the homotopy limit functor holim

can b e dened as the rightderived functor of the lim functor The simplicial set holimX

is a simplicial subset of hoP X Itiswell known that the Hurewicz homotopy groups of

holimX are the strong or Steenro d homotopy groups of the tower X we obtain that the

Hurewicz homotopy groups of the larger space hoP X are the BrownGrossman homotopy

groups of the tower X

Acknowledgements There are many mathematicians that help ed me to develop

this pap er I am very grateful to T Porter for explaining to me the relations b etween

prop er homotopy theory and pro categories The Chipmans preprints contain the idea

of considering the BrownGrossman groups with more structure I thank MIC Beattie

for his questions and suggestions ab out towers of ab elian groups H Baues and J Zob el

help ed me with the construction of innite sums and pro ducts to solve the problem

of construction of the left adjointoftheP functor Ihavealsohadvery interesting

discussions with Cab eza Elvira Extremiana Navarro Rivas y Quintero I also thank

Jose Antonio for helping me to typ e the pap er

The author acknowledges the nancial help given by DGICYT PBC

and Programa Acciones Integradas HispanoGermanas A

Theory and Applications of Categories Vol No

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