FUNCTORIAL AND ALGEBRAIC PROPERTIES OF BROWN'S P FUNCTOR
LUIS-JAVIER HERNANDEZ-PARICIO
Transmitted by Ronald Brown
ABSTRACT. In 1975 E. M. Brown constructed a functorP which carries the tower
of fundamental groups of the end of a (nice) space to the Brown-Grossman fundamental group. In this work, we study this functor and its extensions and analogues dened for pro-sets, pro-pointed sets, pro-groups and pro-abelian groups. The new versions of theP functor are provided with more algebraic structure. Examples given in the paper
prove that in general the P functors are not faithful, however, one of our main results
establishes that the restrictions of the correspondingP functors to the full subcategories
of towers are faithful. We also prove that the restrictions of the P functors to the
corresponding full subcategories of nitely generated towers are also full. Consequently, in these cases, the towers of objects in the categories of sets, pointed sets, groups and abelian groups, can be replaced by adequate algebraic models (M-sets,M-pointed sets,
near-modules and modules.) The article also contains the construction of left adjoints for the P functors.
1. Introduction
This article contains a detailed study of the properties of the P functor. An interesting
consequence is that in many cases we can replace an inverse system of sets or groups by special algebraic models that contain all the information of the corresponding inverse systems. Firstly we recall the context in which L. R. Taylor and E.M. Brown dened the -homotopy groups and the P functor.
In 1971, L.R. Taylor Tay] dened the fundamental -group of a space by taking a set of base points such that any innite path-component of the complement of a compact subspace contains base points. He used the fundamental groups of these path-components based at these base points to dene the fundamental -group of a space.
In 1975, E.M. Brown Br] dened the proper fundamental group of a space with a base ray by using the string of 1{spheres, B
S
1, which is dened by attaching one 1{
sphere S
1 at each positive integer of the half line 0 +
1). Given a space X with a
base ray, he considered the proper fundamental group B 1 (
X) = B
S 1
X]
1 as the set
of germs at innity of proper maps from B S
1 to
X, modulo germs at innity of proper
homotopies. Let X be a well rayed space (the inclusion of the base ray is a cobration)
and suppose that = K 0
K
1
K
2
is a conal sequence of compact subsets.
Denote by X i =
cl(X ;K i)
=ray and by"X =fX i
g the associated end tower of X. The
Received by the editors 9 March 1995. Published on 14 April 1995
AMS subject classication (1990): 18B15,18E20, 18A40, 16Y30, 55N05, 55N07, 55Q52 .
Key words and phrases: Category of fractions, Pro-category, Monoid, M-set, Brown's P functor,
Tower, Pro-object, Near-ring, Near-module, Generator, Pro-set, Pro-group, Pro-abelian group . c Luis-Javier Hernandez-Paricio 1995. Permission to copy for private use granted.
fundamental pro-group 1
"X of X is isomorphic to the tower f 1
X i
g: Brown dened a
functor P:tow Gps ;! Gps satisfying P 1
"X =
B 1 (
X). If one takes a one ended space
having a countable base of neighbourhoods at innity, a base ray gives a set of base points and it can be checked that the fundamental -group of this space agrees with the global fundamental Brown group, which is dened by considering global proper maps and global proper homotopies instead of proper germs, see H-P].
Later Grossman Gr.1, Gr.2, Gr.3] developed a homotopy theory for towers of sim-plicial sets and dened the analogues of Brown's groups for towers of simsim-plicial sets. Using a well known exact sequence, he proved that his notion of fundamental group,
G 1(
Y), was isomorphic to tow Gps(cZ f 1
Y i
g), where cZ is dened by (cZ)
i = ji
Z j,
where Z j
= Z and denotes the coproduct in the category of groups. Applying the
Edwards-Hastings embedding theorem E-H] we have that B 1 (
X)
=
G 1(
"X). Therefore
the Edwards-Hastings embedding relates the Brown denition of the P functor and the
functor tow Gps(cZ ;). It is not hard to see that P and tow Gps(cZ ;) are isomorphic
for any tower of groups and then theP functor can be seen as a representable hom-group
functor on the category tow Gps.
We have considered this last formulation of theP functor in order to dene our more
algebraically structured version of the P functor. Next we give some notation and our
denition of the P functor and afterwards we establish some of the main results of this
work.
Given an objectH of a category C, we can consider the class of objects generated from
it by taking arbitrary sums of copies of H and eective epimorphisms. When this class
contains all the objects of the category it is said that H is a generator for the categoryC.
Notice that the one-point set is a generator for the category Set of sets, the two-point
setS
0 is a generator for the category Set
of pointed sets, the innite cyclic group Z is a
generator for the category Gr pof groups and the innite cyclic abelian group, denoted in
this paper byZ
a, is a generator for the category
Abof abelian groups. We will denote by C one of the categories: Set Set
Gr p Ab. The generator ofC will be denoted by G.
Associated with the category C, one has the category, tow C, of towers inC and the
category, pr oC, of pro-objects in C. The object G induces a pro-object cG:N ;! C
dened by
(cG) i =
ji
G i2N:
Given a pro-object X inpr 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
Gr p Ab, we note that pr oC(cG X) is a set, a
pointed set, a group and an abelian group, respectively. On the other hand, PcG = pr oC(cG cG) has respectively the structure of a monoid, a 0-monoid (see section 3),
1) If C =Set and G=, C
PcG will be the category of
PcG-sets. A PcG-set consists
of a set X together with an action of the monoid PcG=pr oC(cG cG).
2) If C = Set and
G = S 0,
C
PcG will be the category of
PcG-pointed sets. In this
case, the structure is given by an action of a 0-monoid on a pointed set. 3) IfC =Gr pandG=Z,C
PcG will be the category of
PcG-groups. Now the structure
is given by an action of a near-ring on a group, see sections 3 and 4. 4) If C =Aband G=Z
a, C
PcG will be the category of
PcG-abelian groups (modules
over the unitary ring PcG).
Using this notation, given a pro-object X inpr oC, PX =pr oC(cG X) provided with
the action of pr oC(cG cG) determines an object of the category C
PcG. This denes the
functor P:pr oC ;!C PcG.
Next we introduce some of the main results of the paper:
Theorem 4.4 P:tow C ;!C
PcG is a faithful functor.
This establishes that the restriction of the P functor to the full subcategory of towers
inCis a faithful functor. It is interesting to note that the extendedP functor, for instance
from pro-abelian groups to modules is not faithful, see Corollary 7.16.
Another important result of the paper states that the restriction of the P functor to
nitely generated towers is also full.
Theorem 4.11 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=fg ;!C
PcG is a full
embedding, where tow C=fg denotes the full subcategory of tow C determined by nitely
generated towers.
For nitely generated towers, we are able to replace a tower of objects by a single object with some additional algebraic structure. In this paper we have only considered this kind of construction for towers of sets and towers of groups, however, many of the poofs are established by using very general functorial methods. Therefore part of the constructions and results can be extended to towers and pro-objects in more general categories.
For the case C =Gr p, the main results of Chipman Ch.1, Ch.2] stated for towers of
nitely generated groups, are obtained from Theorem 4.11 as corollaries. We point out that the class of nitely generated towers of groups is larger than the class of towers of nitely generated groups.
Fortunately, some very important examples of towers of sets are nitely generated, for example, the tower of
0's of the end of a locally compact,
-compact Hausdor space or
the tower of
0's obtained by the Cech nerve for a compact metric space. In these cases,
it is easy to prove that the fundamental pro-groups are nitely generated and therefore the P functor will work nicely on this kind of pro-group. In the abelian case the towers
of singular homology groups coming from proper homotopy and shape theory are nitely generated. However, we do not know if for towers of higher homotopy groups, the abelian version of the P functor is going to be a full embedding.
of Pc-sets. We also obtain similar embeddings for the corresponding categories of
topo-logical groups and topotopo-logical abelian groups.
In this work, we also give the construction of left adjoints for the P functors. Theorem 6.3 Thefunctor P:pr oC ;!C
PcG
has a left adjoint L:C PcG
;!pr oC :
This left adjoint, for instance for the abelian case, constructs the pro-abelian group associated with a module over the ring PcZ
a. It is interesting to note that PcZ
a is
isomorphic to the ring of locally nite matrices modulo nite matrices which was used by Farrell-Wagoner F-W.1, F-W.2] to dene the Whitehead torsion of an innite complex.
Next we include some additional remarks about other results and constructions devel-oped in this paper.
In section 2, we analyse some nice properties of categories of the form pr oC.
Giv-en a strongly conite set I, we prove in Theorem 2.4 that the full subcategory pr o I
C,
determined by the objects of pr oC indexed byI, is equivalent to a category of right
frac-tions C
I;1 in the sense of Gabriel and Zisman G{Z]. When the directed set of natural
numbers I = N is considered, the category pr o N
C is usually denoted by tow C. From
Theorem 2.4 we have that tow C can be obtained as a category of right fractions of C N,
however, for this case we also prove in Theorem 2.9 that the category tow C is equivalent
to a category of left fractions ;;1 C
N. This fact has the following nice consequence: The
hom-settow C(cG X) can be 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 -homotopy group \at innity" given by Taylor.
When we are working with categories of sets, pointed sets, groups and abelian group-s we ugroup-sually congroup-sider free, forgetful and abelianization functorgroup-s. For the corregroup-sponding \towcategories" and \procategories", we also have analogous induced functors. The dier-ent versions ofP are functors into categories ofPc-sets,PcS
0- pointed sets,
PcZ-groups
and PcZ
a-abelian groups. For these categories, we analyse, in section 5, the denition
and properties of the analogues of this kind of functor. For example, the left adjoint of the natural inclusion of the category ofPcZ
a-abelian groups into the category of
PcZ-groups
is a kind of \distributivization" functor. Given a PcZ-group X, a quotientdX is dened
by considering the relations which are necessary to obtain a right distributive action. An important result of section 5 is Theorem 5.8. In terms of proper homotopy, The-orem 5.8 proves that the P functor sends the abelianization of the tower of fundamental
groups to the \distributivization" of the fundamental Brown-Grossman group. It is not hard to nd topological examples where the abelianization of the fundamental Brown-Grossman group produces a type of rst homology group, which is not naturally isomor-phic to the \distributivization" of the Brown-Grossman group.
(tow Ab Ab) nor tow Abare equivalent to a category of modules.
2. Procategories and categories of fractions
The category pr oC, where C is a given category, was introduced by A. Grothendieck
Gro]. A study of some properties of this category can be seen in the appendix of A{M], the monograph of E{H] or in the books of M{S] and C-P].
The objects of pr oC are functors X:I !C, whereI is a small left ltering category,
and the set of morphisms fromX:I !C to Y:J !C is dened by the formula pr oC(X Y) = lim
j
colim
i
C(X i
Y j)
:
A morphism from X to Y can be represented by (' ff j
g) where ':J ! I is a map
and f j:
X 'j
!Y
j is a morphism of
C such that if j ! j
0 is a morphism of
J, there are i 2I and morphisms i!'j, i !'j
0 such that the composite X
i ! X
'j !Y
j ! Y
j 0 is
equal to the composite X i
!X 'j
0 !Y
j 0.
Notice that if I is a strongly directed set and J is just a set, thenMaps(J I) is also
a directed set. It is easy to see that if I is a strongly directed set and J is a strongly
conite directed set, then the natural inclusionI J
!Maps(J I) is a conal functor, see
M{S, page 9], where I
J denotes the strongly directed set of functors from
J to I. As a
consequence of this fact, ifI J are strongly conite directed sets, any morphism of pr oC
from X to Y can be represented by (' f) where ' 2 I J and
f j:(
X') j
! Y
j is a level
morphism. Let C
scd denote the category whose objects are functors
X:I ! C, where I 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, where X' is the
composition of the functors ' and X. Given a strongly conite directed set I, we also
consider the subcategoryC I of
C
scd given by objects indexed by
I and morphisms of the
form (' f) where'=id I.
If J, I are strongly conite directed sets, then I
J the set of functors from
J to I is a
strongly directed set (' 2 I J
' if '(j) (j) j 2 J) and can be considered as
a category. The evaluation functor e:C I
I J
!C
J is dened by
e(x ') =X'=X
'.
A xed'induces a functor ;
':C I
!C
J, which sends
f:X !Y to f
':X
';!Y
',
and a xed X induces a functor X
;:I J
!C
J sending
' to X
' :
X
';!X
.
Letpr o scd
C denote the full subcategory ofpr oC dened by the objects ofpr oC indexed
by strongly conite directed sets. If X:I ! C and Y:J !C are objects in pr o scd
C, we
can take into account that I
J is conal in
Maps(J I) to see that: pr o
scd
C(X Y) =pr oC(X Y)
= colim
'2I J
C J(
X
' Y)
That is, pr oC(X Y) is the colimit of the functor
(I J)
op X
; ;;!(C
J)op C
Edwards and Hastings E{H] gave the construction (the Mardesic trick, see also M{S], page 15]) of a functorM:pr oC ;!pr o
scd
C that together with the inclusionpr o scd
C ;! ;! pr oC give an equivalence of categories. We note that the category pr o
scd
C is a
quotient of the category C
scd that is, pr o
scd
C(X Y) is a quotient of C scd(
X Y).
We include the following result about conal subsets of I I.
2.1. Lemma. Let I be a strongly conite directed set and consider I
I id=
f'2I I
j 'idg, then 1) the inclusion I
I id
;!I I
isconal,
2) for the case I = N the directed set of non-negative integers, if In(N N) =
f' 2 N
N
j ' is injective g and In(N N id) =
N N id
\ In(N N)
, then In(N N)
;! N N
and In(N
N id)
;!N N
are conal.
Next, for a given strongly conite directed set I, we analyse the relationship between C
I and pr o
I
C, the full subcategory of pr o scd
C dened by the objects indexed by a xed I. We are going to see thatpr o
I
C is a category of right fractions ofC
I. For this purpose,
rst we recall, see G{Z], under which conditions a class of morphisms of a category C
admits a calculus of left (or right) fractions.
A class of morphisms ofCadmits a calculus of left fractions if satises the following
properties:
a) The identities of C are in .
b) If u:X !Y and v:Y !Z are in , their compositionvu:X !Z is in .
c) For each diagram X 0
s ;X
u
;!Y where s is in , there is a commutative square X
u
;;;;! Y s
? ? ? ? y
? ? ? ? y t X
0
;;;;! u
0 Y
0
where t is in .
d) If f g:X ! Y are morphisms of C and if s:X 0
! X is a morphisms of such that fs =gs, there exists a morphismt:Y !Y
0 of such that
tf =tg.
If we replace the conditions c) and d) by the conditions c0) and d0) below, the class
is said to admit a calculus of right fractions. c0) For each diagram
X 0
u 0 ;!Y
0 t
;Y wheret is in , there is a diagram X 0
s ;X
u ;!Y
such that u 0
s=tuand s is in .
d0) If
f g:X ! Y are morphisms of C and if t:Y ! Y
0 is a morphism of such that tf =tg, there exists a morphism s:X
0
!X of such that fs =gs.
Now for a given strongly conite directed set I, consider the class of C
I dened by
the morphisms of the formX
' id:
X
2.2. Proposition. admits a calculus of right fractions.
Proof. It is clear that X
and b). By considering the evaluation functor C I
the commutative square
X
' are morphisms such that (Y
Therefore it follows that f(X
Given a strongly conite directed set I, we denote by C scd
I and pr o
I
C the full
sub-categories of C
scd and
pr oC, respectively, determined by objects indexed by I. Now to
compareC
I;1 and pr o
I
C, we consider the diagram C
In order to have an induced functor, it su ces to see that a morphism of is sent to an isomorphism of pr o
I
also consider the morphism id
this notation we have:
2.3. Lemma. The morphisms X
X) induce an isomorphism in pr o I
C. Proof. Consider the composites
We have that
Because the following diagram is commutative
(X
it follows that (id I
On the other hand, we have (id
Since the diagram
X
is commutative, we have that (' X
2.4. Theorem. The induced functorC I ;1
;!pr o I
C is an equivalence of categories. Proof. Since I
I
id is conal in I
I by Lemma 2.1, a morphism
X ! Y in pr o I
C can be
represented in C scd
;1. Therefore C
I;1(
X Y)!pr o I
C(X Y) is surjective.
C such that (' f) = ( g) in proIC, there is a 2I
id such that the diagram
X' .
x ? ? ? ?
X
'
& f
X;;;; X
Y
-? ? ? ? y
X
% g
X
is commutative in CI. Therefore in CI;1 we have that
f(X ' id)
;1 =f(X
')(X
id)
;1 =g(X
)(X
id)
;1 =g(X
id)
;1:
HenceCI;1(X Y ) = pro
IC(X Y ).
Next, we study a particular but important case. We consider the strongly conite directed set N of natural numbers. In this case, the category pro
NC is usually denoted
bytowC. We are going to prove that if C has a nal object, the categorytowC can also
be obtained from CN as a category of left fractions. As before N
N denotes the strongly
directed set of functors fromNto N. We will use the following notation N + =
f;1gN,
(N +)+ =
N ++ =
f;2gN
+ and in:
N ! N
+ denotes the inclusion. We also have the
strongly directed sets:
Cof(N
N) =
f'2N N
j' is conalg
Cof((N
+)N) =
f2(N +)N
j is conalg:
Notice that we have the relations: In(N
N id)
In(N N)
Cof(N N)
N
N:
Given a '2 Cof(N
N), dene !':
N! N
+ as follows: If n < '(0), then !'(n) = ;1.
Otherwise, there is an i 2 N such that '(i) n < '(i + 1), and then dene !'(n) = i.
Suppose that nm, we have that if n < '(0), then !'(n) '(m), otherwise '(i)! n <
'(i + 1), n m, and we again have that !'(n) '(m). Similarly, it is easy to check!
that idN=in and that if '
, then !' !. Therefore we have dened a contravariant
functor
;:Cof( N
N)
If C is a category with a nal object , we also consider the functor
which sends an object X inC N to
Now we dene the functor e
as the composite: C
' we also use the shorter notation e (
have the contravariant functor X
;:Cof(N N)
;!C N.
2.5. Proposition. Let C be a category with a nal object.
1) If ' 2 In(N
formation C N(
id), the following diagram is commutative X
+. Therefore for a given
objectY in C
transformation is, in this case, the identity. To dene the inverse transformation b:
By considering the formulas
it follows that the transformations above dene a natural isomorphism. 2) The commutativity of the diagram is proved by the following equalities
f
3) It follows from the following relations (f(X
'. Therefore we have:
1) (Y
N that admits a calculus of left fractions.
The category ;;1 C
N will be equivalent to
tow C =pr o N
C.
Consider the class ; dened by the morphisms of the form Y '
2.7. Proposition. ; admits a calculus of left fractions.
c) For each diagram X
where Y
'
id is in ;.
d) Consider a diagram X X
Applying Lemma 2.6 we have (Y
Therefore there exists Y
'
id in ; satisfying the desired relation.
If we consider the diagram
C
we can see that a morphismX
2.8. Lemma. The morphisms X
X) give an isomorphism in the category pr o
N
C =tow C.
Proof. We have that
Since the following diagram is commutative
On the other hand we have (X
2.9. Theorem. For a category C with nal object, the induced functor ; ;1
C is an equivalence of categories.
Proof. It su ces to dualize the proof of Theorem 2.4.
2.10. Remark. 1) We can also prove this theorem taking into account the denition of
the hom{set, see G{Z], and Proposition 2.5 ;;1
2) The equivalence of categoriesC N;1
Consider the class C of morphisms of C
a nite category and C
D denotes the category of functors of the form
D ! C we can
consider the class C
D of morphisms of (C
category of fractions is denoted by tow(C
D). Notice that we also have the equivalence
of categories (C D)
N ! (C
N) D
and the functor C N
! tow C induces the natural functor
(C N)
D
!(tow C) D.
2.11. Proposition. If D is a nite category, then there is a diagram
which is commutative up to isomorphism and is such that the induced functortow(C D)
;!
(tow C)
D is a full embedding.
Proof. LetX denote an object of (C D)
N
and the corresponding object in (C N)
X is thought of as an object
in (C
d. By considering a map
'2In(N
we can represent f
d for all
is commutative in CN. Since the set of morphisms of D is nite, we nally obtain
representatives maps Xd
; X '
0
... r
f
d ;!Y
d such that X
; X '
0
r f
;;;;!Y is a diagram in (C D)
N
.
This diagram represents a morphisms from X to Y in tow(CD) that is sent to
f:X !Y by the functor tow(C D)
;!(tow C) D.
Now iff g:X !Y are maps in (C D)
N
such thatf = g in (tow C)D, we have for each
d 2Ob D a map ' d
2 In(N N
id) such that f d(Xd
' d id) =g
d(Xd '
d
id). If '
'
d d
2 Ob D,
then fd(Xd ' id) = g
d(Xd '
id). Therefore f(X
'
id) = g(X
'
id). This implies that f = g in
tow(CD).
2.12. Remark. Meyer Mey] has proved that if the category C has nite limits, then the functor pro(CD)
;!(proC)
D is an equivalence of categories.
3. Preliminaries on monoids, near-rings and rings
In this section, we establish the notation and properties of monoids, near-rings and rings that will be used in next sections. We usually consider the categories of sets, pointed sets, groups and abelian groups which are denoted by Set, Set, Grp and Ab, respectively.
A monoid consists of a set M and an associative multiplication : M M ;! M
with unit element 1 (1m = m = m1, for every m 2M). If M has also a zero element 0
(m0 = 0 = 0m, for every m2M) it will be called a 0-monoid.
A setR with two binary operations `+' and `', is a unitary (left) near{ring if (R +) is
a group (the additive notation does not imply commutativity), (R ) is a semigroup and
the operations satisfy the left distributive law:
x(y + z) = xy + xz x y z 2R:
A near{ring R satises that x0 = 0 andx(;y) =;(xy), but in general, it is not true
that 0x = 0 for all x 2 R. If the near-ring also satises the last condition it is called
a zero-symmetric near-ring. In this paper we will only work with zero-symmetric unitary near-rings. In this case, (R ) is a 0-monoid.
If a zero-symmetric unitary near-ring also satises the right distributive law: (x + y)z = xz + yz x y z2R
then (R +) is abelian and R becomes a unitary ring.
3.1. Example. If C is a category and X is an object of C, then C(X X) is a monoid
with the composition of morphisms: (g f) !g f. In next sections C will be one of the
categories proC or towC:
3.2. Example. If C is a category with a zero object, the monoid C(X X) has a zero
element 0:X ! X and C(X X) is a 0{monoid. If C has a zero object, the categories
3.3. Example. Let F be a free group generated by a set X, then the set of
endomor-phisms ofF, End(F), becomes a zero-symmetric unitary left near-ring if the operation + is dened by
(f + g)x = fx + gx f g 2End(F) x2X:
3.4. Example. IfA is an abelian group or an object in an abelian category, then End(F)
is an unitary ring.
Let M be a monoid and C a category. A leftM{object X in C consists of an object
X of C and a monoid homomorphism M ;! C(X X) : m ! m : X~ ;! X. If M is a
0{monoid andC has a zero object, we will also assume that an M{object X inC satises
the additional condition ~0 = 0. We denote by M
C the category whose objects are the
(left)M{objects inC. By considering monoid \antimorphisms" M ;!C(X X) we have
the notion of right M{object in C and the category C M.
For the caseC =Set (C =Set
) we have the notion ofM-set (M-pointed set) and the
categories MSet, SetM (MSet, SetM). If X is a group, then Set(X X) has a natural
structure of zero-symmetric unitary right near-ring. If R is a zero-symmetric unitary right near-ring, a structure of left R-group on X (left module) is given by a near-ring homomorphism R ;! Set
(X X). If R is a zero-symmetric unitary left near-ring
and R ;! Set
(X X) is a group homomorphism and a monoid antimorphism, then X
is said to have a structure of right R-group. We denote by RGrp the category of left
R-groups and by GrpR the category of rightR-groups.
If X is an abelian group, then Ab(X X) has a natural structure of unitary ring. If R is a ring, a structure of leftR-abelian group (R-module) is given by a ring homomorphism R ;! Ab(X X). If R ;! Ab(X X) is a group homomorphism and a monoid
antimor-phism, then X is said to be a right R-abelian group (R-module). We denote byRAb the
category of leftR-abelian groups and by AbR the category of rightR-abelian groups.
3.5. Example. Let C be a category. For each object X of C, we have the monoid
End(X) = C(X X) and C(X ;) : C ;! Set
End(X) is a functor which associates to
an object Y the right End(X){object dened by End(X) ;! Set(C(X Y ) C(X Y )) :
' ;! ' ~'(f) = f' f~ 2 C(X Y ). If C has a zero object, we also have the functor: C(X ;):C ;!Set
End(X).
3.6. Example. LetF be a free group generated by the set X. We have noted in Example
3) above thatEnd(F) is a left near-ring. Is is easy to see that for any group Y , Grp(F Y ) has a natural structure of right End(F)-group. Therefore there is an induced functor Grp(F ;) :Grp;!Grp
End(F).
3.7. Example. IfX is an object in an abelian categoryA, thenA(X ;) denes a functor
fromA to the category of End(X)-abelian groups ( End(X)-modules).
Recall that in this paper we are using the following unied notation. We denote by C one of the categories: Set, Set,Grp, Ab. The small projective generator of C is denoted
by G. We also denote by , S 0,
Z, Z
Because the categories of the examples above have some common properties, we will use the following notation:
1) If C=Setand R is a monoid,C
R denotes the category of right
R-sets.
2) If C =Set and
R is a 0-moniod,C
R denotes the category of right
R-pointed sets.
3) If C = Gr p and R is a zero-symmetric unitary left near-ring, C
R denotes the
category of right R-groups (R-near-modules). 4) If C = Ab and R is an unitary ring, C
R denotes the category of right
R-abelian
groups (R-modules).
It is interesting to note that C R and
C are algebraic categories and there is a natural
forgetful functor U:C R
;! C which has a left adjoint functor
F = ; R:C ;! C R. If
C = Set and X is a set, then X R = X R and if C =Set
, then
XR =X R=(( R)(X 0)). 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:B!A, then it is well known
that F preserves colimits and that U preserves limits. A functor U:B!A re$ects nite
limits if it veries the following property: IfX is the \cone" over a nite diagramD inB
and UX is the limit of UD, then X is the limit of D.
We summarise some properties of the functors above in the following: 3.8. Proposition. The forgetful functor U:C
R
;! C has a left adjoint functor F = ; R:C ;! C
R. Moreover, the functor
U preserves and re ects nite limits,
in particular if Uf is an isomorphism, then f is also isomorphism.
Proof. It su ces to check that U re$ects nite products and dierence kernels. IfY 1
Y 2
are objects inC
R, then UY
1 UY
2 admits an action of
Rdened by (y 1
y 2)
r= (y 1
r y 2)
r,
if r 2 R. Now it is easy to check that for the dierent cases, C = Set Set
Gr p Ab,
this action satises the necessary properties to dene an object Y 1
Y 2 in
C
R such that U(Y
1 Y
2) = UY
1 UY
2. Similarly if
f g:Y ! Y
0 are morphisms in C
R, then the
dierence kernel K(Uf Ug) is dened by K(Uf Ug) = fx 2 UY j Ufx = Ugxg. In
this case the action of R on Y induces an action on K(Uf Ug) that satises the necessary
properties, and therefore denes an objectK(f g) such that UK(f g) =
K(Uf Ug).
Given a morphism R 0
! R
1, there is an induced functor V:C
R1
;! C
R0 which has
a left adjoint functor ; R
0 R
1 : C
R 0
;! C R
1. It is not hard to give a more explicit
denition of the functor ; R
0 R
1 for the cases
C =Set Set
Gr p Ab.
In next sections, we will consider the properties of the following construction to study the inverse limit functor.
Let s be an element of R (ifC =Gps and R a left near-module, we also assume that sis a right distributive element: (x+y)s=xs+ys), and let X be an object inC
R, dene F
s
X =fx2X jxs=xg:
This gives a functor F s:
C R
;! C which has a left adjoint ; s
R:C ;! C dened as
follows: Let X be an object of C, the functor ;R:C ;! C
R carries
X to X R.
structure and generated by the relations xr xsr. Denote the quotient object by
X
s
R, and the equivalence class of xr byx s
r. We summarise this construction in
the following
3.9. Proposition. The functor ; s
R:C ;!C R
is left adjoint to F s:
C R
;!C. 4. Brown's construction.
In this section we dene theP functor for the categories of sets, pointed sets,
pro-groups and pro-abelian pro-groups. As in the section above, C denotes one of the following
categories: Set Set
Gr p Ab.
Because the category C has products and sums, then we can dene the functors c:C ;!C
N and p:C
N
;!C by the formulas
(cX) i =
ji X
j X
j =
X j i pY =
+1
%
i=0 Y
i
It is easy to check that C N(
cX Y)
=C(X pY), therefore we have:
4.1. Proposition. The functor c:C;!C N
isleft adjoint to p:C N
;!C.
Associated with the generator G of C, we have the pro-object cG and the
endomor-phism set PcG =pr oC(cG cG) which has the following properties:
1) If C=Set, the morphism composition gives to Pca monoid structure.
2) If C=Set ,
PcS
0 is a 0-monoid (see section 3 .)
3) If C=Gr p, PcZis a zero-symmetric unitary left near-ring.
4) If C=Ab,PcZ
a is a ring.
For any object X of pr oC, we consider the natural action
pr oC(cG X) pr oC(cG cG) ;!pr oC(cG X)
which applies (f ') to f', if f 2pr oC(cG X) and '2pr oC(cG cG).
The morphism set pr oC(cG X) has the following properties:
1) If C = Set, pr oC(cG X) admits a natural structure of PcG-set. Thus the action
satises
(f) =f() f1 =f f 2pr oC(cG X) 12pr oC(cG cG).
2) If C=Set ,
pr oC(cG X) and pr oC(cG cG) have zero morphisms that satisfy f0 = 0 f 2pr oC(cG X)
that is,PcG is a 0-monoid (see section 3) and pr oC(cG X) is a PcG-pointed set.
3) If C = Gr p, we also have that pr oC(cG X) has a group structure and the action
satises the left distributive law:
f(+) =f+f f 2pr oC(cG X) 2PcG
Notice that the sum + need not be commutative. In this case, PcG becomes a
zero-symmetric unitary left near-ring and pr oC(cG X) is a right PcG-group (near-module),
see Mel] and Pilz].
4) If C=Ab, we also have a right distributive law:
(f +g)=f+g f g 2pr oC(cG X) 2PcG:
Now PcG becomes a unitary ring and pr oC(cG X) is a right PcG-abelian group (PcG
-module).
In order to have a unied notation, C
PcG denotes one of the following categories:
1) If C=Set,C
PcG is the category of
PcG-sets.
2) If C=Set ,
C
PcG is the category of
PcG-pointed sets.
3) If C=Gr p, C
PcG is the category of
PcG-groups (near-modules).
4) If C=Ab,C
PcG is the category of
PcG-abelian groups (modules).
Using the notation above we can dene a functor P:pr oC ;! C
PcG as the
repre-sentable functor
PX =pr oC(cG X)
together with the natural action of PcG, whereG is the small projective generator of C.
For the full subcategory tow C we will also consider the restriction functor P:tow C ;! C
PcG.
Because C has sums, products and a nal object , for any object Y of C, we can
consider the direct system %
i0
Y !( % i1
Y) !( % i2
Y) !
where the bonding maps are induced by the identityid:Y !Y and the zero mapY !.
The reduced product IY ofY is dened to be the colimit of the direct system above. We
also recall the forgetful functorU:C PcG
;!C considered in section 3 which will be used
in the following
4.2. Proposition. The functor UP:pr oC ;!C has the following properties: 1) If X =fX
j
g is an object of pr oC, then UPX
= limj IX
j : 2) If X = fX
j
j j 2 Jg is an object of pr oC, where J is a strongly conite directed set, then
UPX
= colim '2N
J C
J( cG
3) If X is an object in tow C, then UPX
= colim
'2In(N N id
) p(X
'):
Proof. For 1), it su ces to consider the denition of the hom{set inpr oC:
UPX = lim
j
colim
i
C((cG) i
X j)
= lim
j
colim
i %
k i X
j
= limj IX
j
2) follows sinceN, J are strongly conite directed sets.
3) By Remark 1) after Theorem 2.9 and Proposition 4.1, we get
UPX = tow C(cG X)
= colim
'2In(N N id
) C
N( cG X
')
= colim
'2In(N N id
)
C(G p(X
'))
= colim
'2In(N N id
) p(X
')
4.3. Proposition. The functor P:pr oC ;!C
PcG preserves nite limits.
Proof. We have that UP preserves nite limits since UP = pr oC(cG ;) is a
repre-sentable functor, see Pa Th 1, Sect 9, Ch.2]. By Proposition 3.8, we have that U re$ects
nite limits. Therefore we get that P preserves nite limits.
Now we recall that Grossman in Gr.3] proved that the functor UP:tow C ;! C
re$ects isomorphisms. Since P preserves nite limits we have:
4.4. Theorem. P:tow C ;!C
PcG is a faithful functor.
Proof. Letf g:X !Y be two morphisms intow C. If we consider the dierence kernel i:K(f g) ;! X, the Proposition above implies that PK(f g)
= K(Pf Pg). Suppose
that Pf = Pg, then K(Pf Pg)
= PX and Pi:PK(f g) ;! PX is an isomorphism.
Applying the forgetful functorU:C PcG
;!C, we have thatUPiis an isomorphism. Now
Grossman's result implies thati is also an isomorphism. Therefore f =g.
4.5. Remark. 1) Since P:tow C ;! C
PcG is a representable faithful functor, we have
that P preserves monomorphisms and re$ects monomorphisms and epimorphisms.
4.6. Proposition. P:tow C ;!C
PcG and
UP:tow C ;!C preserve epimorphisms. Proof. Let q
0: X
0
;! Y
0 be an epimorphism in
tow C, by the Remarks at the end of
ChII,x2.3 of M{S] it follows that q
0 is isomorphic in
Maps(tow C) to q:X ;!Y where q is a level map fq
i: X
i
;! Y i
g and each q i:
X i
;!Y
i is a surjective map. Now we have
that UPq= colim '
p(q
'), and since;
' p(;) colim
' preserve epimorphisms, we get
that UPq and Pq are epimorphisms.
4.7. Definition. Let S be a full 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
PcG(
PX PY):f !Pf
is bijective. If S =pr oC, X is said to be admissible.
4.8. Proposition. The object cG is admissible. Proof. We have the following natural isomorphisms
pr oC(cG X)
= UPX
= C(G UPX)
= C
PcG(
GPcG PX)
= C
PcG(
PcG PX)
which send a mapf:cG !X to Pf:PcG !PX.
4.9. Proposition. Let p:X ! Y be an epimorphism in tow C. If X is admissible in tow C, then Y is also admissible in tow C.
Proof. We can suppose thatpis a level mapfp i:
X i
;!Y i
gsuch that eachp i:
X i
;!Y i
is a surjective map. LetX i
Y i
X
i denote the equivalence relation associated with p
i that
is, X i
Yi X
i = f(x x
0) 2X
i X
i j p
i x=p
i x
0
g. Then the diagram X
Y X
pr 1 ;;;;! ;;;;!
pr 2
X p ;;;;! Y
is a dierence cokernel intow C, where X Y
X =fX i
Y i
X i
g:
Let Z be an object in tow C. Given a morphism:PY ;! PZ in C
PcG, since X is
admissible intow C, there is a morphismf:X !Z intow C such that Pp=Pf. P(f pr
1) =
Pf Ppr 1 =
PpPpr 1 =
P(p pr 1) = P(ppr
2) =
PpPpr 2 =
Pf Ppr 2 =
P(f pr 2)
:
BecauseP is faithful, it follows that f pr 1 =
f pr
2. Now we can use that
pis a dierence
cokernel to obtain a morphism g:Y ! Z such that gp = f. We have that Pg Pp = Pf = Pp. By Proposition 4.6,Pp is an epimorphism, then we have thatPg =. This
4.10. Definition. An object X of tow C is said to be nitely generated if there is an
(eective) epimorphism of the form
finite
cG;!X.
4.11. 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=fg ;!C
PcG is a full
embedding, where tow C=fg denotes the full subcategory of tow C determined by nitely
generated towers.
Proof. It is easy to check that finite
cG is isomorphic to cG. By Proposition 4.8, it
follows that
finite
cG is admissible. Because X is nitely generated, there is an eective
epimorphism
finite
cG ;! X. Now taking into account Proposition 4.9, we obtain that X is also admissible.
4.12. Proposition. Let Y = f ! Y 2
;! Y 1
;! Y 0
g be an object in tow C, where
the bonding morphisms are denoted by Y l k:
Y l
;! Y k
l k. If for each i 0, there is
a nite set A i
Y
i such that for each
n 0, jn
Y j n
A
j generates Y
n, then
Y is nitely
generated.
Proof. Dene X n =
jn
A j
G and consider the diagram
;!X n+1
;! X n
;! pn+1
? ? ? ? y
? ? ? ? y
pn ;!Y
n+1
;! Y n
;!
where the restriction of p
n to A
j
G is induced by the map Y j n:
A j
! Y
n. It is clear that X
=cG and p:X !Y is an epimorphism. Therefore Y is nitely generated.
4.13. Corollary. 1) A tower of nitely generated objects of C is a nitely generated
tower.
2) A tower of nite objects of C is nitely generated.
3) The restricted functors
P:tow(C=fg);!C PcG P:tow(C=f) ;!C
PcG
are a full embeddings, where C=fg and C=f denote the full subcategories determined by
4.14. Remark. A particular case of Corollary 4.13 are Theorem 3.4 and Corollary 3.5
of Ch.2].
Next we study some relations between the P functor and the lim functor.
Recall that for i 0, (cG) i =
k i G
k, where G
k is a copy of the generator
G. The
identity of Ginduces a mapG k
!G k+1,
ki. We denote bysh:cG! cGthe level map fsh
i: k i
G k
;! k i
G k
g induced by the maps G k
!G k+1
:
Given an object Y inC
PcG, we denote by F
sh
Y the object of C dened by F
sh
Y =fy2Y jysh =yg
Notice that F
sh denes a functor from C
PcG to C.
4.15. Theorem. The following diagram
tow C
lim
;;;;! C
P & %F
sh C
PcG
is commutative up to natural isomorphism. That is, limX
=fx2PX jxsh=xg. Proof. For each '2 In(N
N
id), there is a map
SH:P(X
');! P(X
') which applies x= (x
'(0) x
'(1) x
'(2)
) to the element xSH = ((xSH)
'(0) ( xSH)
'(1) ),
where for i0 (xSH)
'(i) = X
'(i+1) '(i)
x
'(i+1). Notice that if
xSH = x, then X
'(i+1) '(i)
x
'(i+1) = x
'(i). Therefore
x2limX
'.
Associated with the map X ;!X
', we have the commutative diagram
limX ;;;;! P(X)
SH ;;;;! ;;;;!
id
P(X) ?
? ? ? y
? ? ? ? y
? ? ? ? y
lim(X
') ;;;;! P(X
')
SH ;;;;! ;;;;!
id
P(X
')
where limX is the dierence kernel of SH and id, and similarly for lim(X
'). Because X ;! X
' is an isomorphism in tow C it follows that limX ;! lim(X
') is an
that
lim X ;!PX sh ;;;;! ;;;;!
id
PX
is a dierence kernel. Therefore limX
=fx2PX jxsh=xg=F sh
PX :
Now we can use that ; sh
PcG:C ;! C
PcG is left adjoint to the functor F
sh: C
PcG
;!C to obtain the following result:
4.16. Corollary. The functor lim:tow C ;!C can also be represented as follows:
limX
=C PcG(
G
sh
PcG PX)
limX =
C PcG(
P(conG) PX)
Moreover, there is a natural map G sh
PcG;!P(conG), where conG denotes the
level-wise constant tower f !G id ;!Gg.
Proof. This follows, because con:C ;! tow C is left adjoint to lim:tow C ;! C and
;
sh
PcG:;! C
PcG is left adjoint to F
sh: C
PcG
;! C, see Proposition 3.9. It is also
necessary to take into account the fact that conG is admissible in tow C. This follows
because conG is a tower of nitely generated objects, see Corollary 4.13 and Theorem
4.11.
4.17. Remark. 1) Theorem 4.15 gives a relation between the P functor and the lim
functor for the case of towers. If X = fX i
g is a tower, then PX
= lim IX
i, where IX
i
is the reduced countable power. For a more general pro-object X :J !C, Porter Por.1]
uses more general reduced powers to \compute" the lim and limq functors.
2) For a tower of groups X, an action of PX on PX can be dened by x
y=x+y;xsh x y2PX :
It is easy to check that the space of orbits of this action is isomorphic to the pointed set lim1
X. The dierence of two elements of the same orbit is of the formx+y;xsh;y.
Notice that the quotient group obtained by dividing by the normal subgroup generated by the relations x+y;xsh;y, satises that the action of sh is trivial and it is an
abelian group.
3) For a tower X of abelian groups, we get isomorphisms
lim1 X
= Ext1( Z
a
sh PcZ
a PX)
lim1 X
= Ext1(
P(conZ a)
PX)
In this case, we also have that lim1
X is obtained from PX by dividing by the subgroup
generated by the relations x;xsh for allx2PX.
4) A global version of Brown'sP functor can be dened for global category (pr oC C)
(for the denition of (pr oC C) see E-H]). If X is an object in (pr oC C), then P g
dened to be the hom-set P
gX = (proC C)(cG X), where cG is considered as an object
in (proC C), provided with the structure given by the action of P
gcG. We note that for
the global version of the P functor, ifX is an object in (towC C), then
UP gX
= colim
'2In 0
(N N id
)
P(X')
where In0( N
N id) =
f'2In(N N id)
j'(0) = 0g.
5. Applications and properties of the P functor
In this section, rstly we obtain some consequences of the main Theorems of section 4. We also analyse the structure of the endomorphism set PcG for the dierent cases
C = Set Set Grp Ab. Finally, we study some additional properties of the
P functor for
the cases C = Grp Ab.
IfC is one of the categories: Set Set Grp Ab, we will denote by TC the
correspond-ing topological category. That is,TC will respectively be one the categories: topological spaces, topological pointed spaces, topological groups or topological abelian groups. We denote by zcmTC the full subcategory of TC determined by zero-dimensional compact metrisable topologies.
Let X be an object in C. Consider the set of quotient objects of the form p:X ;!F
p where Fp is a nite discrete object in C. Given two quotients of this form
p:X ;!F
p and p 0:X
;!F p
0, we say that p p
0 if there is a commutative diagram
X
p. &p 0
Fp
;! F
p 0
It is easy to check that & =fp:X ;!F p
j F
p is a nite discrete quotient object
gwith ,
is a directed set. Therefore we can dene the functor TC ;!pro(C=f):X ;! fF p
g p2 ,
where C=f denotes the full subcategory of C determined by nite objects. If X has a zero{dimensional compact metrisable topology, then there is a sequencepi:X
;!F i such
that pi+1 p
i and for any p of &, there is i
0 such thatp i
p: Hencefp i
g
i0 is conal
in &, and the tower fF i
g
i0 is isomorphic to fF
p g
p2 :
Consequently it is clear that:
5.2. Remark. T. Porter has pointed out to me that Theorem 5.1 is closely connected
with a famous theorem of M.H. Stone Sto] which gives category equivalences between the category of Boolean spaces, the category of Boolean algebras and the category of Boolean rings.
5.3. Theorem. There is a full faithful functor zcmTC ;!C PcG
.
5.4. Remark. 1) There is a full embedding " from the proper homotopy category of
-compact locally compact Hausdor spaces into the homotopy category of prospaces considered by Edwards-Hastings E-H]. If X is a -compact locally compact simplicial complex, then there is a conal sequencefK
i
gof compact subsets ofX such that for every
i 0, 0(cl(X
;K
i)) is a nite set. Therefore 0"X = f
0(cl(X ;K
i))
g is admissible
in towSet. If : 0 1) ! X is a proper ray, then determines a path-component of
0(cl(X ;K
i)) and0"X can be considered as an object in towSet. If X is a simplicial
complex as above, we will suppose that is a simplicial injective map. In this case, the fundamental pro-group can be dened by 1"(X ) =
f 1(cl(X
;K i)
Im (0))g. If
X has one Freudenthal end, it is easy to check that 1"(X ) is admissible in towGrp.
Finally, we also note that for q 0, the tower H
q"X = fH
q(cl(X ;K
i))
g is admissible
in towAb, where Hq denotes the singular homology.
2) Let X be a compact metrisable pointed space. Denote by CX the pro-pointed simplicial set of the Cech nerves associated with the directed set of open coverings of X. In this case, it is easy to check that 0CX is isomorphic to an admissible object in
towSet,1CX is isomorphic to an admissible object in towGrp, and HqCX is isomorphic
to an admissible object intowAb.
3) As a consequence of Theorems 5.1 and 5.3, for the category of connected locally nite countable simplicial complexes, the following categories are adequate for modelling the proper 0{type. The category of zero-dimensional compact metrisable spaces and the Freudenthal end functore, the category tow(finite sets) and the 0" functor and SetPcS
0
and the Brown{Grossman 0{homotopy group BG
0 . The relations between these functors
are given by e = lim 0 ", 0 BG =
P
0 ". Similarly, for the shape 0-type of compact
metrisable spaces, we have the functors lim0C, 0C and P
0C.
Next we study the dierent structures of the endomorphism set PcG = proC(cG cG)
for the dierent casesC = Set Set Grp Ab.
1) C = Set
The monoidPccan be represented as follows: Consider the setR
of matrices of the
form A = (Aij), with i j
2 f0 1 2 g, where either A
ij = 0 or Aij = 1, satisfying the
following properties:
a) For each j 0, the cardinality of fijA ij = 1
g is 1.
b) For each i 0, there exists j i such that for 0 k < i and j l a
k l = 0.
Write 'A(i) = min
fj jj i and if 0k < i and j l then a k l = 0
g.
Dene the equivalence relation by declaring A A
0 if there exists j
0 such that
for j l the l{column of A agrees with the l{column of A
0 that is, A and A0 dier only
Matrix multiplication induces over R
a monoid structure that is compatible with
the relation . Therefore the quotient R 1
inherits a monoid structure from R
. If we
(also) denote by N the set of natural numbers provided with the discrete topology, we
can consider the monoid P(N N) of proper maps N ! N and the monoid of germs of
proper maps P 1(
N N). Given a matrix A of R
, we can dene a proper map that will
again be denoted by A:N ;! N as follows: if j 0 the j{columm of A has only one
elementA
ij = 1, dene
N N). The isomorphism R
The monoid PcS
0 can also be represented as a matrix monoid R
1 S
0 as follows: We
consider matricesA= (A
ij) as above satisfying properties a
0) and b), where a0) is obtained
by modifyinga).
a0) For each
j 0, the cardinality of fijA ij = 1
g is at most one.
Denote by N =
Nfg the Alexandro compactication of N by a point and
consider the monoid Top ((
germs at of continuous maps (N
);!(N
). Given a matrixA, we can dene the
continuous map A:(N
) ;! (N
) such that if j 0 and the j{column of A has a
unique elementA
ij = 1, then
)). The isomorphism R
Notice that R 1
LetF be the free group over the countable set of lettersfx 0
x 1
x 2 ...
g. The
multipli-cation ofF will be denoted by +, then a typical word of F is of the form 2x
we note that an additive notation does not imply commutativity. LetR
Z denote the set
whose elements are of the form (w 0
2 F, satisfying the
following property:
For each i0, there existsj isuch thatx 0
x
i;1are not letters of the reduction
of w
i;1 are not letters of the reduction of w
l for
The sum is dened by components
r) a word whose reduction has the letters x n
1
x n
r. The
product is dened by substitution as follows (w0 w1 w2 ...)(w
It is easy to check that + andgive the structure of a zero{symmetricnear{ring toR Z.
The zero element is (0 0 0 ...) and the unit is represented by (x0 x1 x2 ...). Another
distinguished element is the shift operator (x1 x2 x3 ...) that plays an important role in
connection with the inverse limit functor. Let IZ be the subset of
R
Z dened by the elements w = (w0 w1 w2 ...) such that
there exists m 0 such that w
l = 0 for l > m. Then it is easy to check that IZ is a
normal subgroup of R Z (
Z and we can consider the quotient near{ring R
Z we have the near{ring isomorphisms R
a denote the ring of integer matrices A = (a
ij) where i and j are non negative
integers and each row and each column have nitely many non zero elements. If A is a matrix of R
Za be the subset of R
Za dened by the nite matrices. Then it is easy to
check that IZ
a is an ideal of R
Z
a and we can consider the quotient ring R
We also have the canonical ring isomorphisms:
R
dened by
A ;! (0-column of A, 1-column of A, 2-column of A ...)],
where (0-column ofA, 1-column of A, 2-column of A ...)2p(cZ a
'
A) and'A is the map
Let F be the free abelian group generated by the countable set fx
Set ;! Ab denotes the free abelian
func-tor. Consider the following sequence of subgroups : F a
g, etc. This family of subgroups denes on F
a the structure of a
topological abelian group. Denote by TAbthe category of topological abelian groups.
LetEnd TAb(
F a
F
a) denote the ring of continuous endomorphisms of F
a is a continuous homomorphism, because x
This implies that we have a canonical isomorphism R Z
Given two continuous homomorphisms f g:F a
;! F
a we say that
f and g have
the same germ if there exists n
0 such that for every
n n
a) denote the ring of selfgerms of F
a, it is also clear that R
1
Za is isomorphic
to End
Next we compare the dierent P functors for the cases C =Set
Gr p Ab.
5.5. Proposition. Consider the diagram
tow Gps P tow Set
where the functor f:tow Set
! tow Gps is induced by the free functor Set
PcZ is the free functor associated with the algebraic \forgetful" functor Gps
PcZ
! Set PcS
0 (see Pa th 1 of 3.4]). Then the unit X ;! ufX (of the pair of
adjoint functors: f:tow Set
;! tow Gps, u:tow Gps ;! tow Set
) induces a natural
and epimorphic transformation X:
fPX ;!PfX.
Proof. The unit transformation Y ;! ufY induces the transformation PY ;! PufY = uPfY. By adjointness we obtain the desired transformation fPY
;;;;!PfY.
An element of PfY can be represented as a sequence of words: a= (("
where " k
It is clear that (b)w = a, then (bw) = a. Therefore X:
fPY ;! PfY is a surjective
map.
5.6. Corollary. Consider the functor f:tow Set
;! tow Gps. If X is admissible in tow Set
, then
fX is admissible in tow Gr p.
Proof. We use the facts that P is a faithful functor and fPX ;!PfX is an
epimor-phism to obtain:
tow Gps(fX Y)Gps PcZ(
PfX PY)Gps PcZ(
fPX PY) =
=Set PcS
0(
PX uPY) =Set PcS
0(
PX PuY) =
=tow Set (
X uY) =tow Gps(fX Y):
where \" denotes an injective map and \=" denotes an isomorphism. Because the
composite is the identity, we havetow Gps(fX Y)
=Gps PcZ(
PfX PY).
We include here some additional properties of theP functor for the category of towers
of abelian groups.
5.7. Proposition. The functor P:tow Ab;!Ab PcZ
a preserves nite colimits.
Proof. In an abelian category the product and coproduct of X and Y are both given
by an object Z and morphisms i:X !Z, j:Y ! Z, p:Z ! X and q:Z ! Y such that pi = id, qj = id, qi = 0, pj = 0 and ip+jq = id. Since tow Ab is an abelian category,
see A{M], and P is an additive functor, it follows that P preserves nite coproducts.
Given a morphism f:X ! Y in tow Ab, f factorizes as X g ;!X
0 k
;!Y, where g is an
epimorphism and k is a monomorphism. It is easy to check that coker f
= coker k.
By Remark 1) after Theorem 4.4 and by Proposition 4.6, we have that P preserves
monomorphisms and epimorphisms, so we also obtain that cokerPf
= cokerPk. Because UP
=tow Ab(cZ a
;), we have the exact sequence: UPX
0
;!UPY ;!UPcoker k ;!Ext tow Ab(
cZ a
X 0)
SincecZ
ais a projective object, see He.1], we have that Exttow Ab( cZ
a X
0)
= 0. Therefore 0 ;! UPX
0
;! UPY ;! UPcoker k ;! 0 is a short exact sequence. Since U
re$ects monomorphisms, epimorphisms and kernels, we also have that 0 ;! PX 0
;! PY ;! Pcoker k ;! 0 is a short exact sequence. Then Pcoker f
= Pcoker k =
coker Pk
= cokerPf.
We next consider the inclusion functor i:Ab ;! Gps and the abelianization functor a:Gps ;! Ab which is the left adjoint of i that is, Ab(aX Y)
= Gps(X iY). We shall
also consider the unitary near{ring epimorphismPcZ;!PcZ
athat induces an inclusion
functori:Ab PcZ
a
;!Gps
PcZ which has a left adjoint
d:Gps PcZ
;!Ab PcZ
check that the diagram
tow Ab
P
;;;;! Ab PcZa i
? ? ? ? y
i ? ? ? ? y tow Gps ;;;;!
P
Gps PcZ
is commutative up to natural isomorphism. The following result proves thatPa=dP.
5.8. Theorem. Consider the diagram
tow Ab
P
;;;;! Ab PcZa
U
;;;;! Ab a
x ? ? ? ?
x ? ? ? ? d
x ? ? ? ? a tow Gps ;;;;!
P
Gps PcZ
;;;;! U
Gps
where a and d are left adjoint to the corresponding inclusion functors. Then
1) There is a natural equivalence dPX ;! PaX induced by the unit transformation X ;!iaX (dPX ;!dPiaX
=
diPaX =
PaX).
2) The natural transformation aUY ;!UdY is epimorphic.
Proof. Given an object X in tow Gps, consider the following diagram, where several
notational abuses are made in order to have a shorter notation:
P X X] ;;;;! PX ;;;;! PaX
x ? ? ? ?
id
x ? ? ? ? D PX ;;;;! PX ;;;;! dPX
x ? ? ? ?
id
x ? ? ? ? PX PX] ;;;;! PX ;;;;! aPX
where ifX =fX i
g X X] = fX i
X i]
g and , ] denotes the normal subgroup generated
(4.4), the rst row of the diagram above is exact. In the second row D PX is the sub
where " i
An elementa of P X X] can be represented by a sequence of words a = (("
where" k
2f;1 0 1g y 0
y r
0 are basic commutators of X
1 are basic
commutators of X +
'(1), etc.
If you take, one by one, the basic commutators of these words, you obtain an element of PX PX]
6. The left adjoint for the P functor.
In this section, we construct a left adjoint functor for the P functor.
First we introduce some notation and a technical result (Proposition 6.1) that gives the construction of the left adjoint. Applying this proposition to the P functor, we have
the desired result.
Assume that A B are categories with innite sums,P:A;!B is a given functor and H is an object of A such that for any X of A
P:A(H X);!B(PH PX)
is a bijection.
Let S be the category whose objects are objects of B with a given decomposition
of the form
2A PH
, where
A is an index set, denotes the sum or coproduct in B
and H
denote the canonical \inclusion" into the coproduct, where
it is assumed that H =
H for any 2 B. Applying the functor P and the universal
property of the sum we have the morphisms:
Next we are going to construct a functor l:S ;!A. Given an object 2A
PH of
S,
dene
l( 2A
PH
) = 2A
H
where H =
H for any 2A.
If u: 2A
PH
;!
2B PH
is a morphism of
S, then u =
2A u
, where u
= u in
PH and
in PH
: PH
;! 2A
PH
are the canonical \inclusions".
For each2 A, consider the composition PH
u ;;;;!
2B PH
2B Pin
H ;;;;;;!P(
2B H
) :
Since P:A(H
2B H
)
;! B(PH
P( 2B
H
)) is a bijection, there is a unique l u
: H
;! 2B
H
such that Pl u
= ( 2B
Pin H
) u
. Then dene l u=
2A l u
Next, we check that l:S ;! A is a functor. We start by showing that l preserves
identities.
The canonical \inclusions" in H
: H
;! 2A
H
are such that the diagram
P( 2A
H ) Pin
H %
x ? ? ?
?
2A Pin
H PH
;;;;! in
PH
2A PH
is commutative, therefore
l(id) =l( 2A
in
PH) = 2A
l in
PH =
2A in
H = id:
Given two morphisms
2A PH
u ;!
2B PH
v ;!
2C PH
we have the commutative diagram
P( 2B
H )
P(lv
)
;;;;! P( 2C
H ) Plu
% Pin
H
x ? ? ? ?
Plv
%
x ? ? ? ?
Pin H
PH
;;;;! u
2B PH
;;;;! v =v
2C PH
Therefore l(vu
) = ( 2B
l v )
l u
. Then we have: l(vu) =l(
2A
(vu) ) =
l( 2A
vu
) = 2A
l(vu
) = 2A
(
2B l v
) l u
=
=
2A l v l u
=
l v ( 2A
l u ) = (
l v)(l u):
This implies that l:S;!A is a functor.
The following properties of l will also be used
a) The transformation
A(l( 2A
PH )
Y);!B( 2A
PH
PY)
given by f = 2A
f
;! 2A
Pf
is a bijection.
b) Given morphismsu: 2A
PH
;! 2B
PH and
g:Y ;!Y
0, the following diagram
is commutative
A(l( 2A
PH )
Y 0)
;;;;! B( 2A
PH
PY 0) x
? ? ? ?
x ? ? ? ? A(l(
2B PH
)
Y) ;;;;! B( 2B
PH
PY)
that is, for a given f:l( 2B
PH )
;!Y, we have
2A
P(gfu) =
Pg( 2B
Pf )
u
6.1. Proposition. Suppose that A B are two categories with innite sums and
dier-ence cokernels 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:A(H X);!B(PH PX):f ;!Pf
is a bijection.
b) Two functorsF 1
F 0:
B ;!S, whereS is the category dened above, and two natural
transformations u v:F 1
;!F
0 such that the functor difcoker(
u v):B;!B dened by
difcoker(u v)B = difcoker(F 1
B u
B ;;;;! ;;;;!
v B
F 0
B)
is equivalent to the identity functor of B.
Then the functor P:A;!B has a left adjoint L:B;!A.
Proof. By considering the functor l:S ;!B dened above, we deneL:B;!A by LB= difcoker(l F
1 B
lu B ;;;;! ;;;;!
lv B
l F 0
B):
Now we have
A(LB A) = A(difcoker(l F 1
B lu
B ;;;;! ;;;;!
lv B
l F 0
B) A) = difker(
A(l F 0
B A) (lu
B )
;;;;! ;;;;!
(lv B
)
A(l F 1
B A)) = difker(
B(F 0
B PA) u
B ;;;;! ;;;;!
v B
B(F 1
B PA)) =
B(difcoker(F 1
B u
B ;;;;! ;;;;!
v B
F 0
B) PA) =
B(B PA):
To apply Proposition 6.1, 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.
6.2. Lemma. If C has innite coproducts then pr oC is also provided with innite
Proof. Suppose we have, for each i 2 I, a pro-object X i:
J i
;! C. Consider the
left ltering small category %
i2I
: Associated with the projections p i: %
the mapsin i(
i) that dene the \inclusions" in
i veries the universal property of the coproduct in pr oC :
Recall that C denotes one of the categories: Set Set
Gr p Ab, and C
PcG
respec-tively denotes one of the categories:Set Pc
To shorten notation, the composition of forgetful functors C PcG
;! Set and the composition of free functors Set
The main result of this section is the following: 6.3. Theorem. The functor P:pr oC ;!C
PcG
has a left adjoint L:C PcG
;!pr oC : Proof. We are going to check that the conditions of Proposition 6.1 are satised. Take
A=pr oC and B =C PcG
:
a) By Proposition 4.8, the object H =cG is admissible. Then for anyX in pr oC, pr oC(cG X)
= C
PcG(
PcG PX):
In the cases we are considering for any object B of C
PcG, the natural transformation p
B:
gvB ;!B is a (surjective) epimorphism. By Lemma 3 and Corollary 4 of section 3.4
of Pa], we have that if we consider the bre product
(gvB
is a dierence cokernel. Since gv(gvB B
gvB);!gvB B
gvB is an epimorphism, it also
follows that
gv(gvB
Then we can dene the functors F 0
F 1:
C PcG
;!S by F
0
B =gvB F
1
B =gv(gvB B
gvB)
and the natural transformations by u=pr 1
p and v=pr 2
p.
Now we are under the conditions of Proposition 6.1, to obtain that P:pr oC ;!C PcG
has a left adjoint L:C PcG
;!pr oC.
6.4. Corollary. The functors P:pr oSet ;! Set Pc
P:pr oSet
;! Set PcS
0 P:pr oGps ;!Gps
PcZ and
P:pr oAb;!Ab
PcZa have left adjoints.
6.5. Remark. M.I.C. Beattie Be] has constructed an equivalence between the cate-gory of nitely presented towers of abelian groups and nitely presented PcZ
a{abelian
groups orPcZ
a{modules. This equivalence is also given by the restrictions of the functor P:pr oAb ;!Ab
PcZ
a and its left adjoint L:Ab
PcZ a
;! pr oAb to the corresponding full
subcategories.
7. Global towers and topological abelian groups.
In this section we analyse some relations between towers of abelian groups and topological abelian groups. As a consequence of these relations, we prove that the categories of towers and global towers of abelian groups do not have countable sums. This implies that neither category is equivalent to a category of modules.
7.1. Definition. Let N be a neighbourhood of the zero element 0 of a topological group B. We will say that N is structured if N is also a subgroup of B. A topological abelian
group is said to be locally structured if it has a neighbourhood base at 0 of structured neighbourhoods.
Let TAbdenote the category of topological abelian groups and STAbthe full
subcat-egory determined by locally structured topological abelian groups.
Next we dene two functors L:(pr oAb Ab) ;! STAb and N:STAb ;! (pr oAb Ab)
such that Lis left adjoint to N.
Recall that an object X of (pr oAb Ab) is a morphism X = ( 1
X ;!X
0) where 1
X
is an object of pr oAb and X
0 is an object of the category
Abwhich can be considered as
a full subcategory of pr oAb. A morphismf:X ;!Y in (pr oAb Ab) consists of a pair of
morphismsf = ( 1
f f
0) such that the following diagram 1
X 1
f ;;;;!
1 Y ?
? ? ? y
? ? ? ? y X
0
;;;;! f
0
