Theory and Applications of Categories, Vol. 1, No. 2, 1995, pp. 10–53.

FUNCTORIAL AND ALGEBRAIC PROPERTIES OF BROWN’S P FUNCTOR

LUIS-JAVIER HERNÁNDEZ-PARICIO

Transmitted by Ronald Brown

ABSTRACT. In 1975 E. M. Brown constructed a functor P 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 defined for pro-sets, pro-pointed sets, pro-groups and pro-abelian groups. The new versions of the P 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 corresponding P 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 finitely 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 defined the Δ-homotopy groups and the P functor.

In 1971, L.R. Taylor [Tay] defined the fundamental Δ-group of a space by taking a set of base points such that any infinite 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 define the fundamental Δ-group of a space.

In 1975, E.M. Brown [Br] defined the proper fundamental group of a space with a base ray by using the string of 1–spheres, ^{B}S^{1}, which is defined by attaching one 1– sphere S^{1} at each positive integer of the half line [0, +∞). Given a space X with a base ray, he considered the proper fundamental group π_{1}^{B}(X) = [^{B}S^{1}, X]_{∞} as the set of germs at infinity of proper maps from ^{B}S^{1} to X, modulo germs at infinity of proper homotopies. Let X be a well rayed space (the inclusion of the base ray is a cofibration) and suppose that ∅ = K_{0} ⊂ K_{1} ⊂ K_{2} ⊂ ⋯, is a cofinal sequence of compact subsets. Denote by X_{i} = cl(X − K_{i})/ray and by εX = {X_{i}} the associated end tower of X. The

Received by the editors 9 March 1995. Published on 14 April 1995 AMS subject classification (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 . \begin{matrix}c \\ ◯ \end{matrix} Luis-Javier Hernández-Paricio 1995. Permission to copy for private use granted.

fundamental pro-group π_{1}εX of X is isomorphic to the tower {π_{1}X_{i}}. Brown defined a functor P: towGps ⟶ Gps satisfying Pπ_{1}εX ≅ π_{1}^{B}(X). If one takes a one ended space having a countable base of neighbourhoods at infinity, a base ray gives a set of base points and it can be checked that the fundamental Δ-group of this space agrees with the global fundamental Brown group, which is defined 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 simplicial sets and defined the analogues of Brown’s groups for towers of simplicial sets. Using a well known exact sequence, he proved that his notion of fundamental group, π_{1}^{G}(Y), was isomorphic to towGps(cZ, {π_{1}Y_{i}}), where cZ is defined by (cZ)_{i} = Σ_{j≥i} Z_{j},

where Z_{j} ≅ Z and Σ denotes the coproduct in the category of groups. Applying the Edwards-Hastings embedding theorem [E-H] we have that π_{1}^{B}(X) ≅ π_{1}^{G}(εX). Therefore the Edwards-Hastings embedding relates the Brown definition of the P functor and the functor towGps(cZ, −). It is not hard to see that P and towGps(cZ, −) are isomorphic for any tower of groups and then the P functor can be seen as a representable hom-group functor on the category towGps.

We have considered this last formulation of the P functor in order to define our more algebraically structured version of the P functor. Next we give some notation and our definition of the P functor and afterwards we establish some of the main results of this work.

Given an object H of a category C, we can consider the class of objects generated from it by taking arbitrary sums of copies of H and effective epimorphisms. When this class contains all the objects of the category it is said that H is a generator for the category C. Notice that the one-point set ∗ is a generator for the category Set of sets, the two-point set S^{0} is a generator for the category Set_{∗} of pointed sets, the infinite cyclic group Z is a generator for the category Grp of groups and the infinite cyclic abelian group, denoted in this paper by Z_{a}, is a generator for the category Ab of abelian groups. We will denote by C one of the categories: Set, Set_{∗}, Grp, Ab. The generator of C will be denoted by G.

Associated with the category C, one has the category, towC, of towers in C and the category, proC, of pro-objects in C. The object G induces a pro-object cG: N ⟶ C defined by

(cG)_{i} = Σ_{j≥i} G, i ∈ N.

Given a pro-object X in proC, one has the canonical action

proC(cG, X) × proC(cG, cG) ⟶ proC(cG, X) : (f, φ) → fφ.

For the different cases C = Set, Set_{∗}, Grp, Ab, we note that proC(cG, X) is a set, a pointed set, a group and an abelian group, respectively. On the other hand, PcG = proC(cG, cG) has respectively the structure of a monoid, a 0-monoid (see section 3), a near-ring and a ring. As a consequence of this fact, we will use different algebraic categories, but because they have many common functorial properties we will use the following unified notation:

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 = proC(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) If C = Grp and G = 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 = Ab and 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 in proC, PX = proC(cG, X) provided with the action of proC(cG, cG) determines an object of the category C_{PcG}. This defines the functor P: proC ⟶ C_{PcG}.

Next we introduce some of the main results of the paper:

Theorem 4.4 P: towC ⟶ C_{PcG} is a faithful functor.

This establishes that the restriction of the P functor to the full subcategory of towers in C is a faithful functor. It is interesting to note that the extended P 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 finitely generated towers is also full.

Theorem 4.11 Let X be an object in towC. If X is finitely generated, then X is admissible in towC. Consequently, the restriction functor P: towC/fg ⟶ C_{PcG} is a full embedding, where towC/fg denotes the full subcategory of towC determined by finitely generated towers.

For finitely 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 = Grp, the main results of Chipman [Ch.1, Ch.2] stated for towers of finitely generated groups, are obtained from Theorem 4.11 as corollaries. We point out that the class of finitely generated towers of groups is larger than the class of towers of finitely generated groups.

Fortunately, some very important examples of towers of sets are finitely generated, for example, the tower of π_{0}’s of the end of a locally compact, σ-compact Hausdorff space or the tower of π_{0}’s obtained by the Čech nerve for a compact metric space. In these cases, it is easy to prove that the fundamental pro-groups are finitely 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 finitely 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.

As a consequence of Theorem 4.11, we get a full embedding of the category of zerodimensional compact metrisable spaces and continuous maps into the algebraic category of Pc∗-sets. We also obtain similar embeddings for the corresponding categories of topological groups and topological abelian groups.

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

Theorem 6.3 The functor P: proC ⟶ C_{PcG} has a left adjoint L: C_{PcG} ⟶ proC.

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 finite matrices modulo finite matrices which was used by Farrell-Wagoner [F-W.1, F-W.2] to define the Whitehead torsion of an infinite complex.

Next we include some additional remarks about other results and constructions developed in this paper.

In section 2, we analyse some nice properties of categories of the form proC. Given a strongly cofinite set I, we prove in Theorem 2.4 that the full subcategory pro_{I}C, determined by the objects of proC indexed by I, is equivalent to a category of right fractions C^{I}Σ^{−1} in the sense of Gabriel and Zisman [G–Z]. When the directed set of natural numbers I = N is considered, the category pro_{N}C is usually denoted by towC. From Theorem 2.4 we have that towC 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 towC is equivalent to a category of left fractions Γ^{−1}C^{N}. This fact has the following nice consequence: The hom-set towC(cG, X) can be expressed as a colimit and this gives the definition of the P functor given by Brown or if we use the standard definition given as a limit of colimits, we nearly obtain the definition of Δ-homotopy group “at infinity” given by Taylor.

When we are working with categories of sets, pointed sets, groups and abelian groups we usually consider free, forgetful and abelianization functors. For the corresponding “towcategories” and “procategories”, we also have analogous induced functors. The different versions of P are functors into categories of Pc∗-sets, PcS^{0}- pointed sets, PcZ-groups and PcZ_{a}-abelian groups. For these categories, we analyse, in section 5, the definition and properties of the analogues of this kind of functor. For example, the left adjoint of the natural inclusion of the category of PcZ_{a}-abelian groups into the category of PcZ-groups is a kind of “distributivization” functor. Given a PcZ-group X, a quotient dX is defined 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, Theorem 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 find topological examples where the abelianization of the fundamental Brown-Grossman group produces a type of first homology group, which is not naturally isomorphic to the “distributivization” of the Brown-Grossman group.

Finally, we have developed section 7 to solve some of the theoretical questions that have arisen from writing the paper. We see that the full subcategory of locally structured topological abelian groups admits a full embedding into the category of global pro-objects of abelian groups. As a consequence of the relations between these categories we obtain Corollaries 7.10 and 7.13, which are the main results of the section. In these corollaries it is proved that neither (towAb, Ab) nor towAb have countable sums, and therefore, neither (towAb, Ab) nor towAb are equivalent to a category of modules.

2. Procategories and categories of fractions

The category proC, 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 proC are functors X: I → C, where I is a small left filtering category, and the set of morphisms from X: I → C to Y: J → C is defined by the formula

proC(X, Y) = lim_{j} colim_{i} C(X_{i}, Y_{j}).

A morphism from X to Y can be represented by (φ, {f_{j}}) where φ: J → I is a map and f_{j}: X_{φj} → Y_{j} is a morphism of C such that if j → j′ is a morphism of J, there are i ∈ I and morphisms i → φj, i → φj′ such that the composite X_{i} → X_{φj} → Y_{j} → Y_{j′} is equal to the composite X_{i} → X_{φj′} → Y_{j′}.

Notice that if I is a strongly directed set and J is just a set, then Maps(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 cofinite directed set, then the natural inclusion I^{J} → Maps(J, I) is a cofinal 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, if I, J are strongly cofinite directed sets, any morphism of proC from X to Y can be represented by (φ, f) where φ ∈ I^{J} and f_{j}: (Xφ)_{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 cofinite directed set, and a morphism from X: I → C to Y: J → C is given by a functor φ: J → I and by a natural transformation f: Xφ → Y, where Xφ is the composition of the functors φ and X. Given a strongly cofinite directed set I, we also consider the subcategory C^{I} of C^{scd} given by objects indexed by I and morphisms of the form (φ, f) where φ = id_{I}.

If J, I are strongly cofinite directed sets, then I^{J} the set of functors from J to I is a strongly directed set (φ, ψ ∈ I^{J}, φ ≥ ψ if φ(j) ≥ ψ(j), j ∈ J) and can be considered as a category. The evaluation functor e: C^{I} × I^{J} → C^{J} is defined by e(x, φ) = Xφ = X_{∗}φ. A fixed φ induces a functor −_{∗}φ: C^{I} → C^{J}, which sends f: X → Y to f_{∗}φ: X_{∗}φ ⟶ Y_{∗}φ, and a fixed X induces a functor X_{∗}−: I^{J} → C^{J} sending φ ≥ ψ to X_{∗}_{ψ}^{φ}: X_{∗}φ ⟶ X_{∗}ψ.

Let pro_{scd}C denote the full subcategory of proC defined by the objects of proC indexed by strongly cofinite directed sets. If X: I → C and Y: J → C are objects in pro_{scd}C, we can take into account that I^{J} is cofinal in Maps(J, I) to see that:

pro_{scd}C(X, Y) = proC(X, Y) ≅ colim_{φ∈I^{J}} C^{J}(X_{∗}φ, Y)

That is, proC(X, Y) is the colimit of the functor

(I^{J})^{op} ⟶^{X_{∗}−} (C^{J})^{op} ⟶^{C^{J} (−,Y)} Set

Edwards and Hastings [E–H] gave the construction (the Mardešić trick, see also [M–S], page 15]) of a functor M: proC ⟶ pro_{scd}C that together with the inclusion pro_{scd}C ⟶ ⟶ proC give an equivalence of categories. We note that the category pro_{scd}C is a quotient of the category C^{scd}; that is, pro_{scd}C(X, Y) is a quotient of C^{scd}(X, Y).

We include the following result about cofinal subsets of I^{I}.

2.1. Lemma. Let I be a strongly cofinite directed set and consider I_{id}^{I} = {φ ∈ I^{I} | φ ≥ id}, then

1) the inclusion I_{id}^{I} ⟶ I^{I} is cofinal,

2) for the case I = N the directed set of non-negative integers, if In(N^{N}) = {φ ∈ N^{N} | φ is injective } and In(NN_{id}) = NN_{id} ∩ In(N^{N}), then In(N^{N}) ⟶ N^{N} and In(NN_{id}) ⟶ N^{N} are cofinal.

Next, for a given strongly cofinite directed set I, we analyse the relationship between C^{I} and pro_{I}C, the full subcategory of pro_{scd}C defined by the objects indexed by a fixed I. We are going to see that pro_{I}C is a category of right fractions of C^{I}. For this purpose, first we recall, see [G–Z], under which conditions a class Σ of morphisms of a category C admits a calculus of left (or right) fractions.

A class Σ of morphisms of C admits a calculus of left fractions if Σ satisfies the following properties:

a) The identities of C are in Σ.

b) If u: X → Y and v: Y → Z are in Σ, their composition vu: X → Z is in Σ.

c) For each diagram X′ ⟵^{s} X ⟶^{u} Y where s is in Σ, there is a commutative square

X ⟶^{u} Y
s ↓  ↓ t
X′ ⟶_{u′} Y′

where t is in Σ.

d) If f, g: X → Y are morphisms of C and if s: X′ → X is a morphisms of Σ such that fs = gs, there exists a morphism t: Y → Y′ of Σ such that tf = tg.

If we replace the conditions c) and d) by the conditions c′) and d′) below, the class Σ is said to admit a calculus of right fractions.

c′) For each diagram X′ ⟶^{u′} Y′ ⟵^{t} Y where t is in Σ, there is a diagram X′ ⟵^{s} X ⟶^{u} Y such that u′s = tu and s is in Σ. d′) If f, g: X → Y are morphisms of C and if t: Y → Y′ is a morphism of Σ such that tf = tg, there exists a morphism s: X′ → X of Σ such that fs = gs.

Now for a given strongly cofinite directed set I, consider the class Σ of C^{I} defined by the morphisms of the form X_{∗}_{id}^{φ}: X_{∗}φ → X where φ ∈ I_{id}^{I}.

2.2. Proposition. Σ admits a calculus of right fractions.

Proof. It is clear that X_{∗}_{id}^{id} = id_{X} and that (X_{∗}_{id}^{φ})((X_{∗}φ)_{∗}_{id}^{ψ}) = X_{∗}_{id}^{φψ}. These imply a) and b). By considering the evaluation functor C^{I} × I_{id}^{I} → C^{I}, (X, φ) → X_{∗}φ, we obtain the commutative square

X_{∗}φ ⟶^{f_{∗}φ} Y_{∗}φ
X_{∗}_{id}^{φ} ↓  ↓ Y_{∗}_{id}^{φ}
X ⟶_{f} Y

Hence we have that c′) is satisfied.

Finally, if f, g: X → Y_{∗}φ are morphisms such that (Y_{∗}_{id}^{φ})f = (Y_{∗}_{id}^{φ})g, we have that ((Y_{∗}φ)_{∗}_{id}^{φ})(f_{∗}φ) = ((Y_{∗}_{id}^{φ})_{∗}φ)(f_{∗}φ) = ((Y_{∗}_{id}^{φ})_{∗}φ)(g_{∗}φ) = ((Y_{∗}φ)_{∗}_{id}^{φ})(g_{∗}φ), ((Y_{∗}φ)_{∗}_{id}^{φ})(f_{∗}φ) = f(X_{∗}_{id}^{φ}), ((Y_{∗}φ)_{∗}_{id}^{φ})(g_{∗}φ) = g(X_{∗}_{id}^{φ}).

Therefore it follows that f(X_{∗}_{id}^{φ}) = g(X_{∗}_{id}^{φ}), and so d′) is also satisfied.

Given a strongly cofinite directed set I, we denote by C^{scd_{I}} and pro_{I}C the full subcategories of C^{scd} and proC, respectively, determined by objects indexed by I. Now to compare C^{I}Σ^{−1} and pro_{I}C, we consider the diagram

C^{I} ⟶ C^{scd_{I}}
↓  ↓
C^{I}Σ^{−1}  pro_{I}C

In order to have an induced functor, it suffices to see that a morphism of Σ is sent to an isomorphism of pro_{I}C. A morphism of Σ is of the form X_{∗}_{id}^{φ}: X_{∗}φ → X, φ ∈ I_{id}^{I}. We also consider the morphism σ_{φ}^{id}(X) = (φ, id_{X_{∗}φ}): X → X_{∗}φ in the category C^{scd_{I}}. Using this notation we have:

2.3. Lemma. The morphisms X_{∗}_{id}^{φ}, σ_{φ}^{id}(X) induce an isomorphism in pro_{I}C.

Proof. Consider the composites

X_{∗}φ ⟶^{(id_{I},X_{∗}_{id}^{φ})} X ⟶^{(φ,id_{X∗φ})} X_{∗}φ

X ⟶^{(φ,id_{X∗φ})} X_{∗}φ ⟶^{(id_{I},X_{∗}_{id}^{φ})} X.

We have that

(φ, id_{X_{∗}φ})(id_{I}, X_{∗}_{id}^{φ}) = (φ, (X_{∗}_{id}^{φ})_{∗}φ).

Because the following diagram is commutative

(X_{∗}φ)_{∗}φ ⟶^{(X_{∗}φ)_{∗}_{φ}^{φ}} (X_{∗}φ)_{∗}φ
(X_{∗}φ)_{∗}_{id}^{φ} ↓  ↓ (X_{∗}_{id}^{φ})_{∗}φ
X_{∗}φ ⟶_{id_{X∗φ}} X_{∗}φ,

it follows that (id_{I}, id_{X_{∗}φ}) = (φ, (X_{∗}_{id}^{φ})_{∗}φ) in pro_{I}C.

On the other hand, we have

(id_{I}, X_{∗}_{id}^{φ})(φ, id_{X_{∗}φ}) = (φ, X_{∗}_{id}^{φ}).

Since the diagram

X_{∗}φ ⟶^{X_{∗}_{φ}^{φ}} X_{∗}φ
X_{∗}_{id}^{φ} ↓  ↓ X_{∗}_{id}^{φ}
X ⟶_{id_{X}} X

is commutative, we have that (φ, X_{∗}_{id}^{φ}) = (id_{I}, id_{X}) in pro_{I}C.

2.4. Theorem. The induced functor C^{I} Σ^{−1} ⟶ pro_{I}C is an equivalence of categories.

Proof. Since I_{id}^{I} is cofinal in I^{I}, by Lemma 2.1, a morphism X → Y in pro_{I}C can be represented in C^{scd_{I}} by

X ⟶^{σ_{φ}^{id}(X)} X_{∗}φ ⟶^{f} Y

where f is in C^{I}. By Lemma 2.3, in the category pro_{I}C we have that f σ_{φ}^{id}(X) = f(X_{∗}_{id}^{φ})^{−1}. Therefore C^{I}Σ^{−1}(X, Y) → pro_{I}C(X, Y) is surjective.

Given two morphisms X ⟵^{X_{∗}_{id}^{φ}} X_{∗}φ ⟶^{f} Y , X ⟵^{X_{∗}_{id}^{ψ}} X_{∗}ψ ⟶^{g} Y in

C^{I}Σ^{−1} such that (φ, f) = (ψ, g) in pro_{I}C, there is a θ ∈ I_{id}^{I} such that the diagram

X_{∗}φ
↙  ↑ X_{∗}_{φ}^{θ} ↘^{f}
X⟵ X_{∗}θ  Y
↖  ↓ X_{∗}_{ψ}^{θ} ↗_{g}
X_{∗}ψ

is commutative in C^{I}. Therefore in C^{I}Σ^{−1} we have that

f(X_{∗}_{id}^{φ})^{−1} = f(X_{∗}_{φ}^{θ})(X_{∗}_{id}^{θ})^{−1} = g(X_{∗}_{ψ}^{θ})(X_{∗}_{id}^{θ})^{−1} = g(X_{∗}_{id}^{ψ})^{−1}.

Hence C^{I}Σ^{−1}(X, Y) ≅ pro_{I}C(X, Y).

Next, we study a particular but important case. We consider the strongly cofinite directed set N of natural numbers. In this case, the category pro_{N}C is usually denoted by towC. We are going to prove that if C has a final object ∗, the category towC can also be obtained from C^{N} as a category of left fractions. As before N^{N} denotes the strongly directed set of functors from N to N. We will use the following notation N^{+} = {−1}∪N, (N^{+})^{+} = N^{++} = {−2} ∪ N^{+} and in: N → N^{+} denotes the inclusion. We also have the strongly directed sets:

Cof(N^{N}) = {φ ∈ N^{N} | φ is cofinal}
Cof((N^{+})^{N}) = {ψ ∈ (N^{+})^{N} | ψ is cofinal}.

Notice that we have the relations:

In(NN_{id}) ⊂ In(N^{N}) ⊂ Cof(N^{N}) ⊂ N^{N}.

Given a φ ∈ Cof(N^{N}), define φ̄: N → N^{+} as follows: If n < φ(0), then φ̄(n) = −1. Otherwise, there is an i ∈ N such that φ(i) ≤ n < φ(i + 1), and then define φ̄(n) = i. Suppose that n ≤ m, we have that if n < φ(0), then φ̄(n) ≤ φ̄(m), otherwise φ(i) ≤ n < φ(i + 1), n ≤ m, and we again have that φ̄(n) ≤ φ̄(m). Similarly, it is easy to check that \overline{id}_{N} = in and that if φ ≤ ψ, then φ̄ ≥ ψ̄. Therefore we have defined a contravariant functor

−: Cof(N^{N}) ⟶ Cof((N^{+})^{N}).

If C is a category with a final object ∗, we also consider the functor ^{+}: C^{N} ⟶ C^{N^{+}} which sends an object X in C^{N} to X^{+}, defined by X_{−1}^{+} = ∗ and X_{n}^{+} = X_{n} for n ≥ 0.

Now we define the functor e^{∗} as the composite:

C^{N} × Cof(N^{N}) ⟶^{+_{×}−} C^{N^{+}} × Cof((N^{+})^{N}) ⟶^{e} C^{N},

so e^{∗}(X, φ) = e(X^{+}, φ̄) = X^{+}_{∗}φ̄; we also use the shorter notation e^{∗}(X, φ) = = X^{∗}φ. For a fixed φ, we have a functor −^{∗}φ: C^{N} ⟶ C^{N} and for a fixed X, we have the contravariant functor X^{∗}−: Cof(N^{N}) ⟶ C^{N}.

2.5. Proposition. Let C be a category with a final object.

1) If φ ∈ In(NN_{id}), then −_{∗}φ is left adjoint to −^{∗}φ; that is, there is a natural transformation C^{N}(X_{∗}φ, Y) ≅ C^{N}(X, Y^{∗}φ) that will be denoted by f → f^{b}, g^{#} ← g.

2) If φ ∈ In(NN_{id}), the following diagram is commutative

X ⟶^{f^{b}} Y^{∗}φ
X_{∗}_{id}^{φ} ↑  ↑ Y^{∗}_{id}^{φ}
X_{∗}φ ⟶_{f} Y

3) If φ, ψ ∈ In(NN_{id}), φ ≤ ψ, for the diagram X_{∗}ψ ⟶^{X_{∗}_{φ}^{ψ}} X_{∗}φ ⟶^{f} Y we have that (f(X_{∗}_{φ}^{ψ}))^{b} = (Y^{∗}_{φ}^{ψ})f^{b}.

Proof. 1) : We note that if φ ∈ In(NN_{id}), φ̄φ = in: N → N^{+}. Therefore for a given object Y in C^{N}, we have

(Y^{∗}φ)_{∗}φ = (Y^{+}_{∗}φ̄)_{∗}φ = Y^{+}_{∗}(φ̄φ) = Y^{+}_{∗}in = Y

Define ^{#}: C^{N}(X, Y^{∗}φ) ⟶ C^{N}(X_{∗}φ, Y) by g^{#} = g_{∗}φ. It is clear that the counit transformation is, in this case, the identity.

To define the inverse transformation ^{b}: C^{N}(X_{∗}φ, Y) ⟶ C^{N}(X, Y^{∗}φ), we have that φ^{+}φ̄ ≤ in, since φ is injective and φ ≥ id. We also have that

X = X^{+}_{∗}in

(X_{∗}φ)^{∗}φ = (X_{∗}φ)^{+}_{∗}φ̄ = (X^{+}_{∗}φ^{+})_{∗}φ̄ = X^{+}_{∗}φ^{+}φ̄

Now, given f: X_{∗}φ → Y, define f^{b} = (f^{∗}φ)(X_{∗φ^{+}φ̄}^{in}); that is, it is the composite

X ⟶^{X^{+}_{∗}_{φ^{+}φ̄}^{in}} (X_{∗}φ)^{∗}φ ⟶^{f^{∗}φ} Y^{∗}φ .

Theory and Applications of Categories, Vol. 1, No. 2 20

By considering the formulas (f^{b})^{#} = [(f^{∗}φ)(X^{+}_{∗}_{φ^{+}φ̄}^{in})]^{#} = ((f^{∗}φ)_{∗}φ)((X^{+}_{∗}_{φ^{+}φ̄}^{in})_{∗}φ) = f(X^{+}_{∗}_{φ^{+}in}^{in φ}) = f, (g^{#})^{b} = (g_{∗}φ)^{b} = ((g_{∗}φ)^{∗}φ)(X^{+}_{∗}_{φ^{+}φ̄}^{in}) = ((Y^{∗}φ)^{+}_{∗}_{φ^{+}φ̄}^{in})g = = ((Y^{+}_{∗}φ̄)^{+}_{∗}_{φ^{+}φ̄}^{in})g = ((Y^{+})^{+}_{∗}_{φ̄^{+}φ^{+}φ̄}^{φ̄^{+}in})g = ((Y^{+})^{+}_{∗}_{in^{+}φ̄}^{φ̄^{+}in})g = g,

it follows that the transformations above define a natural isomorphism.

2) The commutativity of the diagram is proved by the following equalities f^{b}(X_{∗}_{id}^{φ}) = (f^{+}_{∗}φ̄)(X^{+}_{∗}_{φ^{+}φ̄}^{in})((X_{∗}_{id}^{φ})^{+}_{∗} in) = (f^{+}_{∗}φ̄)((X^{+}_{∗}_{φ^{+}φ̄}^{in}))(X^{+}_{∗}_{in}^{φ^{+}in}) =

= (f^{+}_{∗}φ̄)(X^{+}_{∗}_{φ^{+}φ̄}^{φ^{+}in}) = (f^{+}_{∗}φ̄)((X^{+}_{∗}φ^{+})_{∗}in ; φ̄) = (f^{+}_{∗}φ̄)((X_{∗}φ)^{+}_{∗}in ; φ̄) = (Y^{+}_{∗}in ; φ̄)f =

= (Y _{id}^{∗ φ})f.

3) It follows from the following relations (f(X_{∗ φ}^{ψ}))^{b} = ((f(X_{∗}_{φ}^{ψ}))^{+}_{∗}ψ̄)(X^{+}_{∗}_{ψ^{+}ψ̄}^{in}) = ((f^{+}(X^{+}_{∗}_{φ^{+}}^{ψ^{+}}))_{∗} ψ̄)(X^{+}_{∗}_{ψ^{+}ψ̄}^{in}) = (f^{+}_{∗}ψ̄)(X^{+}_{∗}_{φ^{+}ψ̄}^{ψ^{+}ψ̄})(X^{+}_{∗}_{ψ^{+}ψ̄}^{in}) = (f^{+}_{∗} ψ̄)(X^{+}_{∗}_{ψ^{+}ψ̄}^{in})

= (f^{+}_{∗} ψ̄)(X^{+}_{∗}_{φ^{+}ψ̄}^{φ^{+}φ̄})(X^{+}_{∗}_{φ^{+}φ̄}^{in}) = (f^{+}_{∗} ψ̄)((X_{∗}φ)^{+}_{∗}φ̄ ; ψ̄)(X^{+}_{∗}_{φ^{+}φ̄}^{in})

= (Y^{+}_{∗}φ̄ ; ψ̄)(f^{+}_{∗}φ̄)(X^{+}_{∗}_{φ^{+}φ̄}^{in}) = (Y^{∗}_{φ}^{ψ}) f^{b}

2.6. Lemma. Let C be a category with final object ∗, and φ, ψ ∈ In(NN_{id}), then 1) (_{−}^{∗}φ)^{∗}ψ = _{−}^{∗}(ψφ)

2) (Y^{∗}φ)^{∗}_{id}^{ψ} = Y^{∗}_{φ}^{φψ}

Proof. It is easy to check that φ̄^{+}ψ̄ = in^{+}\overline{ψφ}. Therefore we have:

1) (Y^{∗}φ)^{∗}ψ = (Y^{+}_{∗}φ̄)^{∗}ψ = (Y^{+}_{∗}φ̄)^{+}_{∗}ψ̄ = (Y^{+})^{+}_{∗} φ̄^{+}_{∗} ψ̄ = (Y^{+})^{+}_{∗} φ̄^{+}ψ̄ = (Y^{+})^{+}_{∗} in^{+}\overline{ψφ} = = Y^{+}_{∗} \overline{ψφ} = Y^{∗}(ψφ).

2) (Y^{∗}φ)^{∗}_{id}^{ψ} = (Y^{+}_{∗}φ̄)^{∗}_{id}^{ψ} = (Y^{+}_{∗}φ̄)_{∗ ψ̄}^{+ in} = (Y^{+})^{+}_{∗} φ̄^{+}_{∗}in ; φ̄

= (Y^{+})^{+}_{∗}_{φ̄^{+}ψ̄}^{φ̄^{+}in} = (Y^{+})^{+}_{∗}_{in^{+}\overline{ψφ}}^{in^{+}φ̄} = Y^{+}_{∗}φ̄ ; \overline{ψφ} = Y^{∗}_{φ}^{ψφ}.

Now we define a class Γ of morphisms in C^{N} that admits a calculus of left fractions. The category Γ^{−1}C^{N} will be equivalent to towC = pro_{N}C.

Consider the class Γ defined by the morphisms of the form Y _{id}^{∗ φ}: Y ⟶ Y^{∗}φ, where Y is an object in C^{N} and φ ∈ In(NN_{id}) (C has a final object).

2.7. Proposition. Γ admits a calculus of left fractions.

Proof. a) It is clear that Y^{∗}_{id}^{id} = id_{Y}.

b) By Lemma 2.6, we have

((Y^{∗}φ)^{∗}_{id}^{ψ})(Y^{∗}_{id}^{φ}) = (Y^{∗}_{φ}^{ψφ})(Y^{∗}_{id}^{φ}) = Y^{∗}_{id}^{ψφ}

c) For each diagram X^{∗}φ ⟵^{X^{∗}_{id}^{φ}} X ⟶^{f} Y , we have the commutative diagram

X ⟶^{f} Y
X^{∗}_{id}^{φ} ↓  ↓ Y^{∗}_{id}^{φ}
X^{∗}φ ⟶_{f^{∗}φ} Y^{∗}φ

where Y^{∗}_{id}^{φ} is in Γ.

d) Consider a diagram X ⟶^{X^{∗}_{id}^{φ}} X^{∗}φ ⟶^{f} ; ⟶ ; g Y such that f(X^{∗}_{id}^{φ}) = g(X^{∗}_{id}^{φ}).

Applying Lemma 2.6 we have (Y^{∗}_{id}^{φ})f = (f^{∗}φ)((X^{∗}φ)^{∗}_{id}^{φ}) = (f^{∗}φ)(X^{∗}_{φ}^{φφ}) = (f^{∗}φ)((X^{∗}_{id}^{φ})^{∗}φ) = (f(X^{∗}_{id}^{φ}))^{∗}φ = = (g(X^{∗}_{id}^{φ}))^{∗}φ = (g^{∗}φ)(X^{∗}_{φ}^{φφ}) = (g^{∗}φ)(X^{∗}φ)_{id}^{φ} = (Y^{∗}_{id}^{φ})g. Therefore there exists Y^{∗}_{id}^{φ} in Γ satisfying the desired relation.

If we consider the diagram

C^{scdN} ⟵ C^{N}
↓  ↓
pro_{N}C  Γ^{−1} C^{N}

we can see that a morphism X^{∗}_{id}^{φ}: X ⟶ X^{∗}φ of Γ has an inverse τ_{id}^{φ}(X) in pro_{N}C. Define τ_{id}^{φ} (X) = (φ, id_{X}) in C^{scdN}.

2.8. Lemma. The morphisms X^{∗}_{id}^{φ} and τ_{id}^{φ}(X) give an isomorphism in the category pro_{N}C = towC.

Proof. We have that

(id_{N}, X^{∗}_{id}^{φ})(φ, id_{X}) = (φ, X^{∗}_{id}^{φ})

Since the following diagram is commutative

(X^{∗}φ)_{∗}φ ⟶^{(X^{∗}φ)_{∗}_{φ}^{φ}} (X^{∗}φ)_{∗}φ
(X^{∗}φ)_{∗}_{id}^{φ} ↓  ↓ (X^{∗}φ)_{∗}_{id}^{φ}
X^{∗}φ ⟶^{id} X^{∗}φ

we have that (φ, X^{∗}_{id}^{φ}) = (id_{N}, id_{X^{∗}φ}).

On the other hand we have

(X^{∗}_{id}^{φ})_{∗}φ = (X^{+}_{∗}in ; φ̄)_{∗}φ = X^{+}_{∗}_{φ̄φ}^{inφ} = X^{+}_{∗}_{in id}^{in φ} = X^{∗}_{id}^{φ} ,

(φ, id_{X})(id_{N}, X^{∗}_{id}^{φ}) = (φ, (X^{∗}_{id}^{φ})_{∗}φ) = (φ, X^{∗}_{id}^{φ}). We have already seen in the proof of Lemma 2.3 that (φ, X_{∗}_{id}^{φ}) = (id_{N}, id_{X}).

2.9. Theorem. For a category C with final object, the induced functor Γ^{−1}C^{N} ⟶ pro_{N}C is an equivalence of categories.

Proof. It suffices to dualize the proof of Theorem 2.4.

2.10. Remark. 1) We can also prove this theorem taking into account the definition of the hom–set, see [G–Z], and Proposition 2.5

Γ^{−1}C^{N}(X, Y) = colim_{φ∈In(NN_{id})} C^{N}(X, Y^{∗}φ) ≅ colim_{φ∈In(NN_{id})} C^{N}(X_{∗}φ, Y) ≅ pro_{N}C(X, Y),

2) The equivalence of categories C^{N}Σ^{−1} ⟶ Γ^{−1}C^{N} is given by

(X ⟵^{X_{∗}{φ \atop id}} X_{∗}φ ⟶^{f} Y) ⟶ (X ⟶^{f^{b}} Y^{∗}φ ⟶^{Y^{∗}{φ \atop id}} Y) .

Consider the class Σ_{C} of morphisms of C^{N} of the from X_{∗}_{id}^{φ}: X_{∗}φ ⟶ X. If D is a finite category and C^{D} denotes the category of functors of the form D → C we can consider the class Σ_{C^{D}} of morphisms of (C^{D})^{N} of the form A_{∗}φ → A. The corresponding 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} → towC induces the natural functor (C^{N})^{D} → (towC)^{D}.

With this notation we have the following result:

2.11. Proposition. If D is a finite category, then there is a diagram

(C^{D})^{N} ⟶ (C^{N})^{D}
↓  ↓
tow(C^{D}) ⟶ (towC)^{D}

which is commutative up to isomorphism and is such that the induced functor tow(C^{D}) ⟶ (towC)^{D} is a full embedding.

Proof. Let X denote an object of (C^{D})^{N} and the corresponding object in (C^{N})^{D}. If X is an object in (C^{D})^{N}, X(n) denotes a diagram of C^{D} and if X is thought of as an object in (C^{N})^{D}, then X_{d} is an object in C^{N}. We note that X(n)_{d} = X_{d}(n).

Now suppose that X, Y are objects in (C^{D})^{N} (or in (C^{N})^{D}) and f: X → Y is a morphism in (towC)^{D}. Then for each d an object of D, we can represent f_{d} for all d by X_{d} ⟵ (X_{d})_{∗}φ^{d} ⟶^{f′_{d}} Y_{d}. By considering a map φ ∈ In(NN_{id}) such that φ ≥ φ^{d}, d ∈ Ob D, we can represent f_{d} for all d by X_{d} ⟵ (X_{d})_{∗}φ ⟶^{f′_{d}} Y_{d}. Then we have that X ⟵ X_{∗}φ is a morphism in (C^{D})^{N}. However, if d_{0} ⟶^{α_{0}} d_{1} is a morphism in D, then

X_{d_{0}}_{∗}φ ⟶^{f′_{d_{0}}} Y_{d_{0}}
(X_{α0})_{∗}φ ↓  ↓ Y α_{0}
X_{d_{1}}_{∗}φ ⟶_{f′_{d_{1}}} Y_{d_{1}}

is only commutative in towC. Nevertheless we can choose ψ_{α_{0}} ∈ In(NN_{id}) such that

X_{d_{0}}_{∗}φ_{∗}ψ_{α_{0}}  ⟶^{f′′_{d_{0}}} Y_{d_{0}}
↓  ↓
X_{d_{1}}_{∗}φ_{∗}ψ_{α_{0}}  ⟶_{f′′_{d_{1}}} Y_{d_{1}}

is commutative in C^{N}. Since the set of morphisms of D is finite, we finally obtain

representatives maps X_{d} ⟵ X_{∗}φ_{∗}ψ_{α_{0}}_{∗}…_{∗}ψ_{α_{r}} ⟶^{f̄_{d}} Y_{d} such that X ⟵ X_{∗}φ_{∗}ψ_{α_{0}}_{∗} ⋯

∗ψ_{α_{r}} ⟶^{f̄}Y is a diagram in (C^{D})^{N}.

This diagram represents a morphisms from X to Y in tow(C^{D}) that is sent to f: X → Y by the functor tow(C^{D}) ⟶ (tow C)^{D}.

Now if f, g: X → Y are maps in (C^{D})^{N} such that f = g in (tow C)^{D}, we have for each

d ∈ Ob D a map φ^{d} ∈ In(NN_{id}) such that f_{d}(X_{d∗}_{id}^{φ^{d}}) = g_{d}(X_{d∗}_{id}^{φ^{d}}). If φ ≥ φ^{d}, d ∈ Ob D,

then f_{d}(X_{d∗}_{id}^{φ}) = g_{d}(X_{d∗}_{id}^{φ}). Therefore f(X_{∗}_{id}^{φ}) = g(X_{∗}_{id}^{φ}). This implies that f = g in tow(C^{D}).

2.12. Remark. Meyer [Mey] has proved that if the category C has finite limits, then the functor pro(C^{D}) ⟶ (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 (1·m = m = m·1, for every m ∈ M). If M has also a zero element 0 (m·0 = 0 = 0·m, for every m ∈ M) it will be called a 0-monoid.

A set R 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) = x · y + x · z, x, y, z ∈ R.

A near–ring R satisfies that x · 0 = 0 and x · (−y) = −(x · y), but in general, it is not true that 0 · x = 0 for all x ∈ R. If the near-ring also satisfies 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 satisfies the right distributive law:

(x + y) · z = x · z + y · z, x, y, z ∈ R

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 proC and towC also have zero objects.

3.3. Example. Let F be a free group generated by a set X, then the set of endomorphisms of F, End(F), becomes a zero-symmetric unitary left near-ring if the operation + is defined by

(f + g)x = fx + gx, f, g ∈ End(F), x ∈ X.

3.4. Example. If A 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 left M–object X in C consists of an object X of C and a monoid homomorphism M ⟶ C(X, X) : m → m̃ : X ⟶ X. If M is a 0–monoid and C has a zero object, we will also assume that an M–object X in C satisfies the additional condition ˜0 = 0. We denote by _{M}C the category whose objects are the (left) M–objects in C. 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 case C = Set (C = Set_{∗}) we have the notion of M-set (M-pointed set) and the categories _{M}Set, Set_{M} (_{M}Set_{∗}, Set_{∗M}). 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 near-module) is given by a nearring 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 _{R}Grp the category of left R-groups and by Grp_{R} the category of right R-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 left R-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 antimorphism, then X is said to be a right R-abelian group (R-module). We denote by _{R}Ab the category of left R-abelian groups and by Ab_{R} the category of right R-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 defined by End(X) ⟶ Set(C(X, Y), C(X, Y)) : φ ⟶ φ̃, φ̃(f) = fφ, f ∈ C(X, Y). If C has a zero object, we also have the functor: C(X, −): C ⟶ Set_{∗}_{End(X)}.

3.6. Example. Let F be a free group generated by the set X. We have noted in Example 3) above that End(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. If X is an object in an abelian category A, then A(X, −) defines a functor from A to the category of End(X)-abelian groups ( End(X)-modules).

Recall that in this paper we are using the following unified 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_{a} the corresponding generators of these categories.

Because the categories of the examples above have some common properties, we will use the following notation:

1) If C = Set and 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 = Grp 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 X ⊗ R = X × R/((∗ × R) ∪ (X × 0)). It is also easy to define − ⊗ R for the cases C = Grp 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 reflects finite limits if it verifies the following property: If X is the “cone” over a finite diagram D in B 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 reflects finite limits, in particular if Uf is an isomorphism, then f is also isomorphism.

Proof. It suffices to check that U reflects finite products and difference kernels. If Y_{1}, Y_{2} are objects in C_{R}, then UY_{1} × UY_{2} admits an action of R defined by (y_{1}, y_{2})r = (y_{1}r, y_{2})r, if r ∈ R. Now it is easy to check that for the different cases, C = Set, Set_{∗}, Grp, Ab, this action satisfies the necessary properties to define 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′ are morphisms in C_{R}, then the difference kernel K(Uf, Ug) is defined by K(Uf, Ug) = {x ∈ UY | Ufx = Ugx}. In this case the action of R on Y induces an action on K(Uf, Ug) that satisfies the necessary properties, and therefore defines an object K(f, g) such that UK(f, g) ≅ K(Uf, Ug).

Given a morphism R_{0} → R_{1}, there is an induced functor V: C_{R_{1}} ⟶ C_{R_{0}} 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 definition of the functor −⊗_{R_{0}}R_{1} for the cases C = Set, Set_{∗}, Grp, 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 (if C = Gps and R a left near-module, we also assume that s is a right distributive element: (x + y)s = xs + ys), and let X be an object in C_{R}, define

F_{s}X = {x ∈ X | x · s = x}.

This gives a functor F_{s}: C_{R} ⟶ C which has a left adjoint −⊗^{s}R: C ⟶ C defined as follows: Let X be an object of C, the functor −⊗R: C ⟶ C_{R} carries X to X ⊗ R. Consider on X ⊗ R the equivalence relation compatible with the corresponding algebraic structure and generated by the relations x ⊗ r ∼ x ⊗ sr. Denote the quotient object by X ⊗^{s} R, and the equivalence class of x ⊗ r by x ⊗^{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 define the P functor for the categories of pro-sets, pro-pointed sets, progroups and pro-abelian groups. As in the section above, C denotes one of the following categories: Set, Set_{∗}, Grp, Ab.

Because the category C has products and sums, then we can define the functors c: C ⟶ C^{N} and p: C^{N} ⟶ C by the formulas

(cX)_{i} = Σ_{j≥i} X_{j}, X_{j} = X, j ≥ i,

pY = Π_{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} is left adjoint to p: C^{N} ⟶ C.

Associated with the generator G of C, we have the pro-object cG and the endomorphism set PcG = proC(cG, cG) which has the following properties:

1) If C = Set, the morphism composition gives to Pc∗ a monoid structure.

2) If C = Set_{∗}, PcS^{0} is a 0-monoid (see section 3 .)

3) If C = Grp, PcZ is a zero-symmetric unitary left near-ring.

4) If C = Ab, PcZ_{a} is a ring.

For any object X of proC, we consider the natural action

proC(cG, X) × proC(cG, cG) ⟶ proC(cG, X)

which applies (f, φ) to fφ, if f ∈ proC(cG, X) and φ ∈ proC(cG, cG).

The morphism set proC(cG, X) has the following properties:

1) If C = Set, proC(cG, X) admits a natural structure of PcG-set. Thus the action satisfies

(fα)β = f(αβ)
f1 = f

f ∈ proC(cG, X), α, β, 1 ∈ proC(cG, cG).

2) If C = Set_{∗}, proC(cG, X) and proC(cG, cG) have zero morphisms that satisfy

f0 = 0, f ∈ proC(cG, X),
0α = 0, α ∈ PcG;

that is, PcG is a 0-monoid (see section 3) and proC(cG, X) is a PcG-pointed set.

3) If C = Grp, we also have that proC(cG, X) has a group structure and the action satisfies the left distributive law:

f(α + β) = fα + fβ, f ∈ proC(cG, X), α, β ∈ PcG

Notice that the sum + need not be commutative. In this case, PcG becomes a zerosymmetric unitary left near-ring and proC(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)α = fα + gα, f, g ∈ proC(cG, X), α ∈ PcG.

Now PcG becomes a unitary ring and proC(cG, X) is a right PcG-abelian group (PcGmodule).

In order to have a unified 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 = Grp, 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 define a functor P: proC ⟶ C_{PcG} as the representable functor

PX = proC(cG, X)

together with the natural action of PcG, where G is the small projective generator of C. For the full subcategory towC we will also consider the restriction functor P: towC ⟶ C_{PcG}.

Because C has sums, products and a final object ∗, for any object Y of C, we can consider the direct system

Π_{i≥0} Y → ( Π_{i≥1} Y) × ∗ → ( Π_{i≥2} Y) × ∗ × ∗ → ⋯

where the bonding maps are induced by the identity id: Y → Y and the zero map Y → ∗. The reduced product IY of Y is defined to be the colimit of the direct system above. We also recall the forgetful functor U: C_{PcG} ⟶ C considered in section 3 which will be used in the following

4.2. Proposition. The functor UP: proC ⟶ C has the following properties: 1) If X = {X_{j}} is an object of proC, then

UPX ≅ lim_{j}IX_{j}.

2) If X = {X_{j} | j ∈ J} is an object of proC, where J is a strongly cofinite directed set, then

UPX ≅ colim_{φ∈N^{J}} C^{J}(cG_{∗}φ, X).

3) If X is an object in towC, then

UPX ≅ colim_{φ∈In(NN_{id})} p(X^{∗}φ).

Proof. For 1), it suffices to consider the definition of the hom–set in proC:

UPX = lim_{j} colim_{i} C((cG)_{i}, X_{j})
≅ lim_{j} colim_{i} Π_{k≥i} X_{j}
≅ lim_{j} IX_{j}

2) follows since N, J are strongly cofinite directed sets.

3) By Remark 1) after Theorem 2.9 and Proposition 4.1, we get

UPX = towC(cG, X)
≅ colim_{φ∈In(NN_{id})} C^{N}(cG, X^{∗}φ)

≅ colim_{φ∈In(NN_{id})} C(G, p(X^{∗}φ))

≅ colim_{φ∈In(NN_{id})} p(X^{∗}φ)

4.3. Proposition. The functor P: proC ⟶ C_{PcG} preserves finite limits.

Proof. We have that UP preserves finite limits since UP = proC(cG, −) is a representable functor, see [Pa; Th 1, Sect 9, Ch.2]. By Proposition 3.8, we have that U reflects finite limits. Therefore we get that P preserves finite limits.

Now we recall that Grossman in [Gr.3] proved that the functor UP: towC ⟶ C reflects isomorphisms. Since P preserves finite limits we have:

4.4. Theorem. P: towC ⟶ C_{PcG} is a faithful functor.

Proof. Let f, g: X → Y be two morphisms in towC. If we consider the difference 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 functor U: C_{PcG} ⟶ C, we have that UPi is an isomorphism. Now Grossman’s result implies that i is also an isomorphism. Therefore f = g.

4.5. Remark. 1) Since P: towC ⟶ C_{PcG} is a representable faithful functor, we have that P preserves monomorphisms and reflects monomorphisms and epimorphisms.

2) Notice that the proof given does not work for the larger category proC.

4.6. Proposition. P: towC ⟶ C_{PcG} and UP: towC ⟶ C preserve epimorphisms.

Proof. Let q′: X′ ⟶ Y′ be an epimorphism in towC, by the Remarks at the end of ChII, §2.3 of [M–S] it follows that q′ is isomorphic in Maps(towC) to q: X ⟶ Y where q is a level map {q_{i}: X_{i} ⟶ Y_{i}} and each q_{i}: X_{i} ⟶ Y_{i} is a surjective map. Now we have that UPq = colim_{φ} p(q_{∗}φ), and since −^{∗}φ, p(−), colim_{φ} preserve epimorphisms, we get that UPq and Pq are epimorphisms.

4.7. Definition. Let S be a full subcategory of proC. An object X in S is said to be admissible in S if for every Y of S the transformation

P: proC(X, Y) ⟶ C_{PcG}(PX, PY): f → Pf,

is bijective. If S = proC, X is said to be admissible.

4.8. Proposition. The object cG is admissible.

Proof. We have the following natural isomorphisms

proC(cG, X) ≅ UPX
≅ C(G, UPX)
≅ C_{PcG}(G ⊗ PcG, PX)
≅ C_{PcG}(PcG, PX)

which send a map f: cG → X to Pf: PcG → PX.

4.9. Proposition. Let p: X → Y be an epimorphism in towC. If X is admissible in towC, then Y is also admissible in towC.

Proof. We can suppose that p is a level map {p_{i}: X_{i} ⟶ Y_{i}} such that each p_{i}: X_{i} ⟶ Y_{i} is a surjective map. Let X_{i} ×_{Y_{i}} X_{i} denote the equivalence relation associated with p_{i}; that

is, X_{i} ×_{Y_{i}} X_{i} = {(x, x′) ∈ X_{i} × X_{i} | p_{i}x = p_{i}x′}. Then the diagram

X ×_{Y} X ⟶^{pr_{1}} ; ⟶ ; pr_{2} X ⟶^{p} Y

is a difference cokernel in towC, where X ×_{Y} X = {X_{i} ×_{Y_{i}} X_{i}}.

Let Z be an object in towC. Given a morphism β: PY ⟶ PZ in C_{PcG}, since X is admissible in towC, there is a morphism f: X → Z in towC such that β Pp = Pf.

P(f pr_{1}) = Pf Ppr_{1} = β Pp Ppr_{1} = β P(p pr_{1}) =
β P(p pr_{2}) = β Pp Ppr_{2} = Pf Ppr_{2} = P(f pr_{2}).

Because P is faithful, it follows that f pr_{1} = f pr_{2}. Now we can use that p is a difference cokernel to obtain a morphism g: Y → Z such that gp = f. We have that Pg Pp = Pf = β Pp. By Proposition 4.6, Pp is an epimorphism, then we have that Pg = β. This implies that Y is also admissible in towC.

4.10. Definition. An object X of towC is said to be finitely generated if there is an (effective) epimorphism of the form Σ_{finite} cG ⟶ X.

4.11. Theorem. Let X be an object in towC. If X is finitely generated, then X is admissible in towC. Consequently, the restriction functor P: towC/fg ⟶ C_{PcG} is a full embedding, where towC/fg denotes the full subcategory of towC determined by finitely generated towers.

Proof. It is easy to check that Σ_{finite} cG is isomorphic to cG. By Proposition 4.8, it

follows that Σ_{finite} cG is admissible. Because X is finitely generated, there is an effective

epimorphism Σ_{finite} cG ⟶ X. Now taking into account Proposition 4.9, we obtain that

X is also admissible.

4.12. Proposition. Let Y = {⋯ → Y_{2} ⟶ Y_{1} ⟶ Y_{0}} be an object in towC, where the bonding morphisms are denoted by Y_{k}^{l}: Y_{l} ⟶ Y_{k}, l ≥ k. If for each i ≥ 0, there is a finite set A_{i} ⊂ Y_{i} such that for each n ≥ 0, ∪_{j≥n} Y_{n}^{j}A_{j} generates Y_{n}, then Y is finitely

generated.

Proof. Define X_{n} = Σ_{j≥n} Σ_{A_{j}} G and consider the diagram

⋯ ⟶ X_{n+1} ⟶ X_{n} ⟶ ⋯
p_{n+1} ↓  ↓ p_{n}
⋯ ⟶ Y_{n+1} ⟶ Y_{n} ⟶ ⋯

where the restriction of p_{n} to Σ_{A_{j}} G is induced by the map Y_{n}^{j}: A_{j} → Y_{n}. It is clear that

X ≅ cG and p: X → Y is an epimorphism. Therefore Y is finitely generated.

4.13. Corollary. 1) A tower of finitely generated objects of C is a finitely generated tower.

2) A tower of finite objects of C is finitely 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 finitely generated objects and finite objects, respectively.

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} = Σ_{k≥i} G_{k}, where G_{k} is a copy of the generator G. The

identity of G induces a map G_{k} → G_{k+1}, k ≥ i. We denote by sh: cG → cG the level map

{sh_{i}: Σ_{k≥i} G_{k} ⟶ Σ_{k≥i} G_{k}} induced by the maps G_{k} → G_{k+1}.

Given an object Y in C_{PcG}, we denote by F_{sh}Y the object of C defined by

F_{sh}Y = {y ∈ Y | y · sh = y}

Notice that F_{sh} defines a functor from C_{PcG} to C.

4.15. Theorem. The following diagram

towC ⟶^{lim} C
P ↘  ↗ F_{sh}
C_{PcG}

is commutative up to natural isomorphism. That is, limX ≅ {x ∈ PX | x · sh = x}.

Proof. For each φ ∈ In(NN_{id}), there is a map SH: P (X^{∗}φ) ⟶ P (X^{∗}φ) which applies x = (x_{\overline{φ}(0)}, x_{\overline{φ}(1)}, x_{\overline{φ}(2)}, ⋯) to the element

xSH = ((xSH)_{\overline{φ}(0)}, (xSH)_{\overline{φ}(1)}, ⋯), where for i≥0, (xSH)_{\overline{φ}(i)} = X_{\overline{φ}(i)}^{\overline{φ}(i+1)}x_{\overline{φ}(i+1)}. Notice that if xSH = x, then X_{\overline{φ}(i)}^{\overline{φ}(i+1)}x_{\overline{φ}(i+1)} = x_{\overline{φ}(i)}. Therefore x ∈ limX^{∗}φ.

Associated with the map X ⟶ X^{∗}φ, we have the commutative diagram

lim X ⟶ P (X) ⟶^{SH} ; ⟶ ; id P (X)
↓  ↓  ↓
lim(X^{∗}φ) ⟶ P (X^{∗}φ) ⟶^{SH} ; ⟶ ; id P (X^{∗}φ)

where limX is the difference kernel of SH and id, and similarly for lim(X^{∗}φ). Because X ⟶ X^{∗}φ is an isomorphism in towC it follows that limX ⟶ lim(X^{∗}φ) is an isomorphism. Now taking into account that colim_{φ} preserves difference kernels, we obtain that

lim X ⟶ PX ⟶^{sh} ; ⟶ ; id PX

is a difference kernel. Therefore limX ≅ {x ∈ PX | x · sh = x} = 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: towC ⟶ 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 levelwise constant tower {⋯ → G ⟶^{id} G}.

Proof. This follows, because con: C ⟶ towC is left adjoint to lim: towC ⟶ 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 towC. This follows because conG is a tower of finitely 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 = {X_{i}} 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 lim^{q} functors.

2) For a tower of groups X, an action of PX on PX can be defined by

xy = x + y − x · sh x, y ∈ PX.

It is easy to check that the space of orbits of this action is isomorphic to the pointed set lim^{1}X. The difference of two elements of the same orbit is of the form x+y −x·sh−y. Notice that the quotient group obtained by dividing by the normal subgroup generated by the relations x + y − x · sh − y, satisfies that the action of sh is trivial and it is an abelian group.

3) For a tower X of abelian groups, we get isomorphisms

lim^{1}X ≅ Ext^{1}(Z_{a} ⊗^{sh} PcZ_{a}, PX)
lim^{1}X ≅ Ext^{1}(P(conZ_{a}), PX)

In this case, we also have that lim^{1}X is obtained from PX by dividing by the subgroup generated by the relations x − x·sh for all x ∈ PX.

4) A global version of Brown’s P functor can be defined for global category (proC, C) (for the definition of (proC, C) see [E-H]). If X is an object in (proC, C), then P_{g}X is defined to be the hom-set P_{g}X = (proC, C)(cG, X), where cG is considered as an object in (proC, C), provided with the structure given by the action of P_{g}cG. We note that for the global version of the P functor, if X is an object in (towC, C), then

UP_{g}X ≅ colim_{φ∈In_{0}(NN_{id})} P (X^{∗}φ),

where In_{0}(NN_{id}) = {φ ∈ In(NN_{id}) | φ(0) = 0}.

5. Applications and properties of the P functor

In this section, firstly we obtain some consequences of the main Theorems of section 4. We also analyse the structure of the endomorphism set PcG for the different cases C = Set, Set_{∗}, Grp, Ab. Finally, we study some additional properties of the P functor for the cases C = Grp, Ab.

If C is one of the categories: Set, Set_{∗}, Grp, Ab, we will denote by T C the corresponding topological category. That is, T C will respectively be one the categories: topological spaces, topological pointed spaces, topological groups or topological abelian groups. We denote by zcmT C the full subcategory of T C 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 F_{p} is a finite discrete object in C. Given two quotients of this form p: X ⟶ F_{p} and p′: X ⟶ F_{p′}, we say that p ≥ p′ if there is a commutative diagram

X
p ↙  ↘ p′
F_{p} ⟶ F_{p′}

It is easy to check that Λ = {p: X ⟶ F_{p} | F_{p} is a finite discrete quotient object } with ≥, is a directed set. Therefore we can define the functor T C ⟶ pro(C/f): X ⟶ {F_{p}}_{p∈Λ}, where C/f denotes the full subcategory of C determined by finite objects. If X has a zero–dimensional compact metrisable topology, then there is a sequence p_{i}: X ⟶ F_{i} such that p_{i+1} ≥ p_{i} and for any p of Λ, there is i ≥ 0 such that p_{i} ≥ p. Hence {p_{i}}_{i≥0} is cofinal in Λ, and the tower {F_{i}}_{i≥0} is isomorphic to {F_{p}}_{p∈Λ}.

Consequently it is clear that:

5.1. Theorem. The category tow(C/f) of towers of finite objects in C is equivalent to the category zcmT C of objects in T C which have a zero–dimensional compact metrisable topology.

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 zcmT C ⟶ C_{PcG}.

5.4. Remark. 1) There is a full embedding ε from the proper homotopy category of σ-compact locally compact Hausdorff spaces into the homotopy category of prospaces considered by Edwards-Hastings [E-H]. If X is a σ-compact locally compact simplicial complex, then there is a cofinal sequence {K_{i}} of compact subsets of X such that for every i ≥ 0, π_{0}(cl(X − K_{i})) is a finite set. Therefore π_{0}εX = {π_{0}(cl(X − K_{i}))} is admissible in towSet. If α: [0, ∞) → X is a proper ray, then α determines a path-component of π_{0}(cl(X − K_{i})) and π_{0}εX can be considered as an object in towSet_{∗}. If X is a simplicial complex as above, we will suppose that α is a simplicial injective map. In this case, the fundamental pro-group can be defined by π_{1}ε(X, α) = {π_{1}(cl(X − K_{i}) ∪ Imα, α(0))}. If X has one Freudenthal end, it is easy to check that π_{1}ε(X, α) is admissible in towGrp. Finally, we also note that for q ≥ 0, the tower H_{q}εX = {H_{q}(cl(X − K_{i}))} is admissible in towAb, where H_{q} denotes the singular homology.

2) Let X be a compact metrisable pointed space. Denote by ČX the pro-pointed simplicial set of the Čech nerves associated with the directed set of open coverings of X. In this case, it is easy to check that π_{0}ČX is isomorphic to an admissible object in towSet_{∗}, π_{1}ČX is isomorphic to an admissible object in towGrp, and H_{q}ČX is isomorphic to an admissible object in towAb.

3) As a consequence of Theorems 5.1 and 5.3, for the category of connected locally finite 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 functor e, the category tow(finite sets) and the π_{0}ε functor and Set_{∗}_{PcS^{0}} and the Brown–Grossman 0–homotopy group π_{0}^{BG}. The relations between these functors are given by e = lim π_{0} ε, π_{0}^{BG} = P π_{0} ε. Similarly, for the shape 0-type of compact metrisable spaces, we have the functors limπ_{0}Č, π_{0}Č and Pπ_{0}Č.

Next we study the different structures of the endomorphism set PcG = proC(cG, cG) for the different cases C = Set, Set_{∗}, Grp, Ab.

1) C = Set

The monoid Pc∗ can be represented as follows: Consider the set R_{∗} of matrices of the form A = (A_{ij}), with i, j ∈ {0, 1, 2, ⋯}, where either A_{ij} = 0 or A_{ij} = 1, satisfying the following properties:

a) For each j ≥ 0, the cardinality of {i | A_{ij} = 1} is 1.

b) For each i ≥ 0, there exists j ≥ i such that for 0 ≤ k < i and j ≤ l, a_{kl} = 0. Write φ_{A}(i) = min{j | j ≥ i and if 0 ≤ k < i and j ≤ l, then a_{kl} = 0}.

Define the equivalence relation ∼ by declaring A ∼ A′ if there exists j ≥ 0 such that for j ≤ l the l–column of A agrees with the l–column of A′; that is, A and A′ differ only on a finite number of columns.

Matrix multiplication induces over R_{∗} a monoid structure that is compatible with the relation ∼. Therefore the quotient R_{∗}^{∞} 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^{∞}(N, N). Given a matrix A of R_{∗}, we can define 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 element A_{ij} = 1, define A(j) = i. This gives monoid isomorphisms R_{∗} ⟶ P (N, N) and R_{∗}^{∞} ⟶ P^{∞}(N, N). The isomorphism

R_{∗}^{∞} ⟶ P c ∗ = colim_{φ∈In(NN_{id})} p(c ∗^{∗} φ)

is given by A ⟶ (A(0), A(1), A(2), ⋯) ∈ p(c ∗^{∗} φ_{A}).

Given an object X of towSet, the action PX ×Pc∗ ⟶ PX can be defined as follows: take x ∈ p(X^{∗}φ) and A ∈ R_{∗}, then [x][A] = [y], where y ∈ p((X^{∗}φ)^{∗}φ_{A})) is defined by

y_{φ̄φ̄_{A}(j)} = X^{+}_{φ̄φ̄_{A}(j)}^{φ̄A(j)} x_{φ̄A(j)} = X^{+}_{φ̄φ̄_{A}(j)}^{φ̄A(j)}(Σ_{l} x_{φ̄l} A_{lj}).

2) C = Set_{∗}

The monoid PcS^{0} can also be represented as a matrix monoid R_{S^{0}}^{∞} as follows: We consider matrices A = (A_{ij}) as above satisfying properties a′) and b), where a′) is obtained by modifying a).

a′) For each j ≥ 0, the cardinality of {i | A_{ij} = 1} is at most one.

Denote by N^{∗} = N ∪ {∗} the Alexandroff compactification of N by a point ∗ and consider the monoid T op_{∗}((N^{∗}, ∗), (N^{∗}, ∗)) and the monoid (T op_{∗})^{∞}((N^{∗}, ∗), (N^{∗}, ∗)) of germs at ∗ of continuous maps (N^{∗}, ∗) ⟶ (N^{∗}, ∗). Given a matrix A, we can define the continuous map A: (N^{∗}, ∗) ⟶ (N^{∗}, ∗) such that if j ≥ 0 and the j–column of A has a unique element A_{ij} = 1, then A(j) = i, otherwise A(j) = ∗. This defines isomorphisms R_{S^{0}} ⟶ T op_{∗}(N^{∗}, ∗), (N^{∗}, ∗)) and R_{S^{0}}^{∞} ⟶ (T op_{∗})^{∞} ((N^{∗}, ∗), (N^{∗}, ∗)). The isomorphism

R_{S^{0}}^{∞} ⟶ colim_{φ∈In(NN_{id})} p(cS^{0∗}φ) is given by A ⟶ (A(0), A(1), A(2), ⋯) ∈ p(cS^{0∗}φ_{A}).

Notice that R_{∗}^{∞} is a submonoid of R_{S^{0}}^{∞}.

3) C = Grp

Let F be the free group over the countable set of letters {x_{0}, x_{1}, x_{2},…}. The multiplication of F will be denoted by +, then a typical word of F is of the form 2x_{2} + 3x_{1} − x_{0}, we note that an additive notation does not imply commutativity. Let R_{Z} denote the set whose elements are of the form (w_{0}, w_{1}, w_{2}, ⋯), where for i ≥ 0 w_{i} ∈ F, satisfying the following property:

For each i ≥ 0, there exists j ≥ i such that x_{0}, ⋯, x_{i−1} are not letters of the reduction of w_{l} for l ≥ j. For a given element w = (w_{0}, w_{1}, w_{2}, ⋯) of R_{Z}, write φ_{w}(i) = min{j | j ≥ i and x_{0}, ⋯, x_{i−1} are not letters of the reduction of w_{l} for l ≥ j}.

The sum is defined by components

(w_{0}, w_{1}, w_{2}, ⋯) + (w′_{0}, w′_{1}, w′_{2},…) = (w_{0} + w′_{0} w_{1} + w′_{1} w_{2} + w′_{2}, ⋯)

Denote by w(x_{n_{1}},…, x_{n_{r}}) a word whose reduction has the letters x_{n_{1}}, ⋯, x_{n_{r}}. The product is defined by substitution as follows

(w_{0}, w_{1}, w_{2},…)(w′_{0}(x_{n_{1}^{0}},…, x_{n^{0}_{r}_{0}}), w′_{1}(x_{n_{1}^{1}},…, x_{n^{1}_{r}_{1}}), w′_{2}(x_{n_{1}^{2}},…, x_{n^{2}_{r}_{2}}),…) = = (w′_{0}(w_{n_{1}^{0}},…, w_{n^{0}_{r}_{0}}), w′_{1}(w_{n_{1}^{1}},…, w_{n^{1}_{r}_{1}}), w′_{2}(w_{n_{1}^{2}},…, w_{n^{2}_{r}_{2}}),…) It is easy to check that + and · give the structure of a zero–symmetric near–ring to R_{Z}.

The zero element is (0, 0, 0,…) and the unit is represented by (x_{0}, x_{1}, x_{2},…). Another distinguished element is the shift operator (x_{1}, x_{2}, x_{3},…) that plays an important role in connection with the inverse limit functor.

Let I_{Z} be the subset of R_{Z} defined by the elements w = (w_{0}, w_{1}, w_{2},…) such that there exists m ≥ 0 such that w_{l} = 0 for l > m. Then it is easy to check that I_{Z} is a normal subgroup of R_{Z}, (R_{Z})(I_{Z}) ⊆ I_{Z} and (r + i)s − rs ∈ I_{Z} for all i ∈ I_{Z}, r, s ∈ R_{Z}. Then I_{Z} is a ideal of R_{Z} and we can consider the quotient near–ring RZ^{∞} = R_{Z}/I_{Z}.

For R_{Z} and RZ^{∞} we have the near–ring isomorphisms

R_{Z} ⟶ colim_{φ∈In_{0}(NN_{id})} p(cZ^{∗}φ)

RZ^{∞} ⟶ colim_{φ∈In(NN_{id})} p(cZ^{∗}φ)

defined by w = (w_{0}, w_{1}, w_{2},…) ⟶ [(w_{0}, w_{1}, w_{2},…)], where (w_{0}, w_{1}, w_{2},…) ∈ p(cZ^{∗}φ_{w}) and φ_{w} is the map defined above.

4) C = Ab

Let R_{Z_{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 finitely many non zero elements.

If A is a matrix of R_{Z_{a}} for each i ≥ 0, there exists j ≥ i such that a_{kl} = 0 for l ≥ j and k < i. For a given matrix A, write φ_{A}(i) = min{j | j ≥ i and if k < i and l ≥ j, then a_{kl} = 0 }. Let I_{Z_{a}} be the subset of R_{Z_{a}} defined by the finite matrices. Then it is easy to check that I_{Z_{a}} is an ideal of R_{Z_{a}} and we can consider the quotient ring RZ_{a}^{∞} = R_{Z_{a}}/I_{Z_{a}}.

We also have the canonical ring isomorphisms:

R_{Z_{a}}⟶ colim_{φ∈In_{0}(NN_{id})} p(cZ_{a}^{∗}φ)

RZ_{a}^{∞} ⟶ colim_{φ∈In(NN_{id})} p(cZ_{a}^{∗}φ)

defined by

A ⟶ [(0-column of A, 1-column of A, 2-column of A,…)], where (0-column of A, 1-column of A, 2-column of A,…) ∈ p(cZ_{a}^{∗}φ_{A}) and φ_{A} is the map defined above.

Let F^{a} be the free abelian group generated by the countable set {x_{0}, x_{1}, x_{2},…}. That is, F^{a} = f_{a}{x_{0}, x_{1}, x_{2},…}, where f_{a}: Set ⟶ Ab denotes the free abelian functor. Consider the following sequence of subgroups : F_{0}^{a} = F^{a}, F_{1}^{a} = f_{a}{x_{1}, x_{2},…}, F_{2}^{a} = f_{a}{x_{2}, x_{3},…}, etc. This family of subgroups defines on F^{a} the structure of a topological abelian group. Denote by T Ab the category of topological abelian groups.

Let End_{T Ab}(F^{a}, F^{a}) denote the ring of continuous endomorphisms of F^{a}. If f: F^{a} ⟶ ⟶ F^{a} is a continuous homomorphism, because x_{i} ⟶ 0, we have that w_{i} = fx_{i} ⟶ 0. This implies that we have a canonical isomorphism R_{Z_{a}} ⟶ End_{T Ab}(F^{a}, F^{a}).

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_{0} f(x_{n}) = g(x_{n}). Let End_{T Ab}^{∞}(F^{a}, F^{a}) denote the ring of selfgerms of F^{a}, it is also clear that RZ_{a}^{∞} is isomorphic to End_{T Ab}^{∞}(F^{a}, F^{a}).

Next we compare the different P functors for the cases C = Set_{∗}, Grp, Ab.

5.5. Proposition. Consider the diagram

towGps ⟶^{P} Gps_{PcZ}
f ↑  ↑ f
towSet_{∗} ⟶_{P} Set_{∗PcS^{0}}

where the functor f: towSet_{∗} → towGps is induced by the free functor Set_{∗} → Gps and f: Set_{∗PcS^{0}} → Gps_{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: towSet_{∗} ⟶ towGps, u: towGps ⟶ towSet_{∗}) induces a natural and epimorphic transformation η_{X}: fPX ⟶ PfX.

Proof. The unit transformation Y ⟶ ufY induces the transformation PY ⟶ PufY = uPfY. By adjointness we obtain the desired transformation

fPY ⟶^{η}PfY.

An element of PfY can be represented as a sequence of words:

a = [((ε_{0}y_{0} + … + ε_{r_{0}}y_{r_{0}})_{φ̄(0)}, (ε_{r_{0}+1}y_{r_{0}+1} + … + ε_{r_{1}}y_{r_{1}})_{φ̄(1)},…)]

where ε_{k} ∈ {−1, 0, 1}, y_{0},…, y_{r_{0}} ∈ Y_{φ̄(0)}^{+}, y_{r_{0}+1},…, y_{r_{1}} ∈ Y_{φ̄(1)}^{+}, etc.

If you take, one by one, the “letters” of these words, you obtain an element of fPY

b = [(y_{0},…, y_{r_{0}}, y_{r_{0}+1},…, y_{r_{1}}, y_{r_{1}+1},…)].

If you replace the y’s of a by x’s, you will have an element of PcZ

w = [((ε_{0}x_{0} + … + ε_{r_{0}}x_{r_{0}})_{φ̄(0)}, (ε_{r_{0}+1}x_{r_{0}+1} + … + ε_{r_{1}}x_{r_{1}})_{φ̄(1)},…)]

It is clear that (ηb)w = a, then η(bw) = a. Therefore η_{X}: fPY ⟶ PfY is a surjective map.

5.6. Corollary. Consider the functor f: towSet_{∗} ⟶ towGps. If X is admissible in towSet_{∗}, then fX is admissible in towGrp.

Proof. We use the facts that P is a faithful functor and fPX ⟶ PfX is an epimorphism to obtain:

towGps(fX, Y) ⊂ Gps_{PcZ}(PfX, PY) ⊂ Gps_{PcZ}(fPX, PY) =

= Set_{∗PcS^{0}}(PX, uPY) = Set_{∗PcS^{0}}(PX, PuY) =

= towSet_{∗}(X, uY) = towGps(fX, Y). where “⊂” denotes an injective map and “=” denotes an isomorphism. Because the composite is the identity, we have towGps(fX, Y) ≅ Gps_{PcZ}(PfX, PY).

We include here some additional properties of the P functor for the category of towers of abelian groups.

5.7. Proposition. The functor P: towAb ⟶ Ab_{PcZ_{a}} preserves finite 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 towAb is an abelian category, see [A–M], and P is an additive functor, it follows that P preserves finite coproducts. Given a morphism f: X → Y in towAb, f factorizes as X ⟶^{g} X′ ⟶^{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 coker Pf ≅ coker Pk. Because UP ≅ towAb(cZ_{a}, −), we have the exact sequence:

UPX′ ⟶ UPY ⟶ UPcoker k ⟶ Ext_{towAb}(cZ_{a}, X′)

Since cZ_{a} is a projective object, see [He.1], we have that Ext_{towAb}(cZ_{a}, X′) ≅ 0. Therefore 0 ⟶ UPX′ ⟶ UPY ⟶ UPcoker k ⟶ 0 is a short exact sequence. Since U reflects monomorphisms, epimorphisms and kernels, we also have that 0 ⟶ PX′ ⟶ PY ⟶ Pcoker k ⟶ 0 is a short exact sequence. Then Pcoker f ≅ Pcoker k ≅ coker Pk ≅ coker Pf.

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 epimorphism PcZ ⟶ PcZ_{a} that induces an inclusion functor i: Ab_{PcZ_{a}} ⟶ Gps_{PcZ} which has a left adjoint d: Gps_{PcZ} ⟶ Ab_{PcZ_{a}}. It is easy to check that the diagram

towAb ⟶^{P} Ab_{PcZ_{a}}
i ↓  i ↓
towGps ⟶_{P} Gps_{PcZ}

is commutative up to natural isomorphism. The following result proves that Pa = dP.

5.8. Theorem. Consider the diagram

towAb ⟶^{P} Ab_{PcZ_{a}}  ⟶^{U} Ab
a ↑  ↑ d  ↑ a
towGps ⟶_{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 towGps, consider the following diagram, where several notational abuses are made in order to have a shorter notation:

P[X, X] ⟶ PX ⟶ PaX
∪  ↑ id  ↑
DPX ⟶ PX ⟶ dPX
∪  ↑ id  ↑
[PX, PX] ⟶ PX ⟶ aPX

where if X = {X_{i}}, [X, X] = {[X_{i}, X_{i}]} and [ , ] denotes the normal subgroup generated by the commutators [x, y] = x + y − x − y. By (4.1), Remark 1) after Theorem 4.4 and (4.4), the first row of the diagram above is exact. In the second row DPX is the sub PcZ–group generated by xw+yw−(x+y)w where x, y are elements of PX and w ∈ PcZ. Notice that if w = −1, we have −x − y + x + y. Therefore DPX contains the commutator subgroup [PX, PY]. Recall that if H is a group and y ∈ [H, H] then y = ε_{1} y_{1} +⋯+ε_{r} y_{r} where ε_{i} ∈ {−1, 0, 1} and y_{i} = [a_{i}, b_{i}] with a_{i}, b_{i} ∈ H.

An element a of P[X, X] can be represented by a sequence of words

a = [((ε_{0} y_{0} + ⋯ + ε_{r_{0}} y_{r_{0}})_{\overline{φ}(0)}, (ε_{r_{0}+1} y_{r_{0}+1} + ⋯ + ε_{r_{1}} y_{r_{1}})_{\overline{φ}(1)}, ⋯)] where ε_{k} ∈ \{−1, 0, 1\}, y_{0}, ⋯, y_{r_{0}} are basic commutators of X_{\overline{φ}(0)}^{+}, y_{r_{0}+1}, ⋯, y_{r_{1}} are basic commutators of X_{\overline{φ}(1)}^{+}, etc.

If you take, one by one, the basic commutators of these words, you obtain an element of [PX, PX]

b = [(y_{0}, ⋯, y_{r_{0}}, y_{r_{0}+1}, ⋯, y_{r_{1}}, y_{r_{1}+1}, ⋯)] and by replacing the y’s of a by x’s, we get an element of PcZ,

w = [((ε_{0} x_{0} + ⋯ + ε_{r_{0}} x_{r_{0}})_{\overline{φ}(0)}, (ε_{r_{0}+1} x_{r_{0}+1} + ⋯ + ε_{r_{1}} x_{r_{1}})_{\overline{φ}(1)}, ⋯)] satisfying bw = a. Since DPX is a sub–PcZ–group, a ∈ D(PX). Therefore DPX = P[X, X] and this implies that PaX = dPX.

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 infinite 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 Σ_{α∈A} PH_{α}, where A is an index set, Σ denotes the sum or coproduct in B

and H_{α} = H for all α ∈ A. The morphism–set from Σ_{α∈A} PH_{α} to Σ_{β∈B} PH_{β} is given by

B( Σ_{α∈A} PH_{α}, Σ_{β∈B} PH_{β})

Let in_{H_{β}}: H_{β} ⟶ Σ_{β∈B} H_{β} denote the canonical “inclusion” into the coproduct, where

it is assumed that H_{β} = H for any β ∈ B. Applying the functor P and the universal property of the sum we have the morphisms:

Pin_{H_{β}}: PH_{β} ⟶ P( Σ_{β∈B} H_{β})

Σ_{β∈B} Pin_{H_{β}}: Σ_{β∈B} PH_{β} ⟶ P( Σ_{β∈B} H_{β})

Next we are going to construct a functor l: S ⟶ A. Given an object Σ_{α∈A} PH_{α} of S, define

l( Σ_{α∈A} PH_{α}) = Σ_{α∈A} H_{α}

where H_{α} = H for any α ∈ A.

If u: Σ_{α∈A} PH_{α} ⟶ Σ_{β∈B} PH_{β} is a morphism of S, then u = Σ_{α∈A} u_{α}, where

u_{α} = u in_{PH_{α}} and in_{PH_{α}}: PH_{α} ⟶ Σ_{α∈A} PH_{α} are the canonical “inclusions”.

For each α ∈ A, consider the composition

PH_{α}⟶^{u_{α}} Σ_{β∈B} PH_{β}\underrightarrow{Σ_{β∈B} Pin_{H_{β}}}P( Σ_{β∈B} H_{β}).

Since P: A(H_{α}, Σ_{β∈B} H_{β}) ⟶ B(PH_{α}, P( Σ_{β∈B} H_{β})) is a bijection, there is a unique

lu_{α}: H_{α} ⟶ Σ_{β∈B} H_{β} such that Plu_{α} = ( Σ_{β∈B} Pin_{H_{β}})u_{α}. Then define

lu = Σ_{α∈A} lu_{α}

Next, we check that l: S ⟶ A is a functor. We start by showing that l preserves identities.

The canonical “inclusions” in_{H_{α}}: H_{α} ⟶ Σ_{α∈A} H_{α} are such that the diagram

P( Σ_{α∈A} H_{α})

Pin_{H_{α}} ↗  ↑ Σ_{α∈A} Pin_{H_{α}}
PH_{α}  ⟶_{in_{PHα}}  Σ_{α∈A} PH_{α}

is commutative, therefore

l(id) = l( Σ_{α∈A} in_{PH_{α}}) = Σ_{α∈A} l in_{PH_{α}} = Σ_{α∈A} in_{H_{α}} = id.

Given two morphisms

Σ_{α∈A} PH_{α} ⟶^{u} Σ_{β∈B} PH_{β} ⟶^{v} Σ_{γ∈C} PH_{γ}

we have the commutative diagram

P( Σ_{β∈B} H_{β}) ⟶^{P(Σlv_{β})} P( Σ_{γ∈C} H_{γ})
Plu_{α} ↗  ΣPin_{H_{β}} ↑  _{ΣPlv_{β}} ↗  ↑ ΣPin_{Hγ}
PH_{α}⟶_{u_{α}}  Σ_{β∈B} PH_{β}  ⟶_{v=Σv_{β}}  Σ_{γ∈C} PH_{γ}

Therefore l(vu_{α}) = ( Σ_{β∈B} lv_{β})lu_{α}. Then we have:

l(vu) = l( Σ_{α∈A}(vu)_{α}) = l( Σ_{α∈A} vu_{α}) = Σ_{α∈A} l(vu_{α}) = Σ_{α∈A}( Σ_{β∈B} lv_{β})lu_{α} =

= Σ_{α∈A} lv lu_{α} = lv ( Σ_{α∈A} lu_{α}) = (lv)(lu).

This implies that l: S ⟶ A is a functor.

The following properties of l will also be used

a) The transformation

A(l( Σ_{α∈A} PH_{α}), Y) ⟶ B( Σ_{α∈A} PH_{α}, PY)

given by f = Σ_{α∈A} f_{α} ⟶ Σ_{α∈A} Pf_{α} is a bijection.

b) Given morphisms u: Σ_{α∈A} PH_{α} ⟶ Σ_{β∈B} PH_{β} and g: Y ⟶ Y′, the following diagram is commutative

A(l( Σ_{α∈A} PH_{α}), Y′) ⟶ B( Σ_{α∈A} PH_{α}, PY′)
↑  ↑
A(l( Σ_{β∈B} PH_{β}), Y) ⟶ B( Σ_{β∈B} PH_{α}, PY)

that is, for a given f: l( Σ_{β∈B} PH_{β}) ⟶ Y, we have

Σ_{α∈A} P(gfu)_{α} = Pg( Σ_{β∈B} Pf_{β})u

Using this notation and the properties of l: S ⟶ A, we can prove:

6.1. Proposition. Suppose that A, B are two categories with infinite sums and difference 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 functors F_{1}, F_{0}: B ⟶ S, where S is the category defined above, and two natural transformations u, v: F_{1} ⟶ F_{0} such that the functor difcoker(u, v): B ⟶ B defined 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 defined above, we define L: B ⟶ A by

LB = difcoker(lF_{1}B⟶^{lu_{B}} ; ⟶ ; lv_{B}lF_{0} B).

Now we have

A(LB, A) = A(difcoker(lF_{1}B⟶^{lu_{B}} ; ⟶ ; lv_{B}lF_{0}B), A)

≅ difker(A(lF_{0}B, A)⟶^{(lu_{B})^{∗}} ; ⟶ ; (lv_{B})^{∗}A(lF_{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 infinite sums and difference cokernels. The category proC has difference cokernels and the following Lemma shows that it also has infinite sums.

6.2. Lemma. If C has infinite coproducts then proC is also provided with infinite coproducts.

Proof. Suppose we have, for each i ∈ I, a pro-object X_{i}: J_{i} ⟶ C. Consider the left filtering small category Π_{i∈I} J_{i} and define Σ_{i∈I} X_{i}: Π_{i∈I} J_{i} ⟶ C by Σ_{i∈I} X_{i} ((j_{i})_{i∈I}) =

= Σ_{i∈I} X_{i}(j_{i}), ((j_{i})_{i∈I}) ∈ Π_{i∈I} J_{i}. Associated with the projections p_{i}: Π_{i∈I} J_{i} ⟶ J_{i} we have

the maps in_{i}(p_{i}((j_{i})_{i∈I})): X_{i}(j_{i}) → Σ_{i∈I} X(j_{i}) that define the “inclusions” in_{i}: X_{i} ⟶ Σ_{i∈I} X_{i}.

It is easy to check that Σ_{i∈I} X_{i} verifies the universal property of the coproduct in proC.

Recall that C denotes one of the categories: Set, Set_{∗}, Grp, Ab, and C_{PcG} respectively denotes one of the categories:Set_{Pc∗}, Set_{∗}_{PcS^{0}}, Gps_{PcZ}, Ab_{PcZ_{a}}.

To shorten notation, the composition of forgetful functors C_{PcG}⟶^{U}C⟶^{u}Set will be denoted by v = uU: C_{PcG} ⟶ Set and the composition of free functors

Set⟶^{f}C⟶^{F}C_{PcG} by g = F f: Set ⟶ C_{PcG}.

The main result of this section is the following:

6.3. Theorem. The functor P: proC ⟶ C_{PcG} has a left adjoint L: C_{PcG} ⟶ proC.

Proof. We are going to check that the conditions of Proposition 6.1 are satisfied. Take A = proC and B = C_{PcG}.

a) By Proposition 4.8, the object H = cG is admissible. Then for any X in proC,

proC(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 fibre product

(gvB ×_{B} gvB) ⟶^{pr_{2}} gvB

pr_{1} ↓  ↓ p_{B}
gvB  ⟶_{p_{B}} B

then

gvB ×_{B} gvB⟶^{pr_{1}} ; ⟶_{pr_{2}}gvB⟶_{p_{B}}B

is a difference cokernel. Since gv(gvB ×_{B} gvB) ⟶ gvB ×_{B} gvB is an epimorphism, it also follows that

gv(gvB ×_{B} gvB)⟶^{pr_{1}p} ; ⟶_{pr_{2}p}gvB⟶_{p_{B}}B

is also a difference cokernel.

Then we can define 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: proC ⟶ C_{PcG} has a left adjoint L: C_{PcG} ⟶ proC. 6.4. Corollary. The functors P: proSet ⟶ Set_{Pc∗}, P: proSet_{∗} ⟶ Set_{∗PcS^{0}}, P: proGps ⟶ Gps_{PcZ} and P: proAb ⟶ Ab_{PcZa} have left adjoints. 6.5. Remark. M.I.C. Beattie [Be] has constructed an equivalence between the category of finitely presented towers of abelian groups and finitely presented PcZ_{a}–abelian groups or PcZ_{a}–modules. This equivalence is also given by the restrictions of the functor P: proAb ⟶ Ab_{PcZ_{a}} and its left adjoint L: Ab_{PcZ_{a}} ⟶ proAb 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 T Ab denote the category of topological abelian groups and ST Ab the full subcategory determined by locally structured topological abelian groups.

Next we define two functors L: (proAb, Ab) ⟶ ST Ab and N: ST Ab ⟶ (proAb, Ab) such that L is left adjoint to N.

Recall that an object X of (proAb, Ab) is a morphism X = (^{∞}X ⟶ X_{0}) where ^{∞}X is an object of proAb and X_{0} is an object of the category Ab which can be considered as a full subcategory of proAb. A morphism f: X ⟶ Y in (proAb, Ab) consists of a pair of morphisms f = (^{∞}f, f_{0}) such that the following diagram

∞X ⟶^{^{∞}f} ^{∞}Y
↓  ↓
X_{0} ⟶_{f_{0}} Y_{0}

is commutative in proAb.

Any object of (proAb, Ab) can be represented up to isomorphism by a functor X: Λ ⟶ Ab where Λ is a directed set with a final element 0 (λ ≥ 0, ∀λ). If λ ≥ μ, let X_{μ}^{λ}: X_{λ} ⟶ X_{μ} denote the corresponding bonding morphism. Associated with X we have ^{∞}X = X: Λ ⟶ Ab which is an object of proAb, X_{0} which is an object of Ab (or a constant pro–abelian group) and the natural morphism ^{∞}X ⟶ X_{0} given by the identity ∞X_{0} ⟶ X_{0}.

Next we use this notation to define a functor L: (proAb, Ab) ⟶ ST Ab. Given an object X of (proAb, Ab), LX is defined to be the abelian group X_{0} together with the locally structured topology defined by the subgroups ImX_{0}^{λ}, where X_{0}^{λ}: X_{λ} ⟶ X_{0} are bonding maps of X. Notice that given λ, μ there exists γ such that ImX_{0}^{γ} ⊂ (ImX_{0}^{λ}) ∩ (ImX_{o}^{μ}). This implies that the neighbourhood local base {ImX_{0}^{λ}} defines a topology on X_{0}.

If f: X ⟶ Y is a morphism in (proAb, Ab), then the functor L is defined by Lf = f_{0}: X_{0} ⟶ Y_{0}. We must check that f_{0} is continuous. Assume that f: X ⟶ Y is given by a map φ: Λ_{Y} ⟶ Λ_{X} (φ0 = 0) and homomorphisms f_{μ}: X_{φ(μ)} ⟶ Y_{μ}. If ImY_{0}^{μ} is a neighbourhood at 0 ∈ Y_{0}, there is a λ ∈ Λ_{X} such that the following diagram is commutative

X_{φ(μ)} ⟶^{f_{μ}} Y_{μ}
X_{λ} _{↘}^{↗}  ↓ Y_{0}^{μ}
X_{0}  ⟶_{f_{0}} Y_{0}

This implies that f_{0}(ImX_{0}^{λ}) ⊂ ImY_{0}^{μ}. Therefore Lf: LX ⟶ LY is a continuous homomorphism.

To define a functor N: ST Ab ⟶ (proAb, Ab), for a given object B of ST Ab consider the directed set Λ = {S | S is a subgroup of B and S is a nbh at 0 } which has a final element S = B. Now define NB: Λ ⟶ Ab by NB_{S} = S, S ∈ Λ. Notice that NB_{0} = B.

7.2. Proposition. Consider the functors L and N defined above, then 1) L: (proAb, Ab) ⟶ ST Ab is left adjoint to N: ST Ab ⟶ (proAb, Ab)

2) The unit, B ⟶ LNB, induced by the pair of adjoint functors, is a natural equivalence. Then ST Ab can be considered as a full subcategory of (proAb, Ab).

Proof. Let X be an object of (proAb, Ab) and Y an object of ST Ab. If f: LX ⟶ Y is a continuous homomorphism, for each structured neighbourhood S of Y, there exists a structured neighbourhood ImX_{0}^{φS} at 0 such that f(ImX_{0}^{φS}) ⊂ S; for S = Y we take φS = 0. Define f^{b}: X ⟶ NY by f^{b} = (φ, f_{S}^{b}) where f_{S}^{b}: X_{φ(S)} ⟶ S is the composition f_{S}^{b} = (f|ImX_{0}^{φS})X_{0}^{φS}.

For a given g: X ⟶ NY, define g^{#}: LX ⟶ Y by g^{#} = g_{0}. Now it is easy to check that (f^{b})^{#} = f and (g^{#})^{b} = g.

If ST Ab/fc denotes the full subcategory of ST Ab determined by first countable topological abelian groups, we also have:

7.3. Proposition. The restriction functors L: (towAb, Ab) ⟶ ST Ab/fc and N: ST Ab/fc ⟶ (towAb, Ab) satisfy

1) L is left adjoint to N

2) The unit B ⟶ LNB is a natural equivalence and ST Ab/fc can be considered as a full subcategory of (towAb, Ab).

We also consider the following functors g: ST Ab ⟶ Ab that forgets the topology and the functor t: Ab ⟶ ST Ab defined as follows: If A is an abelian group, tA is the abelian group A together with the trivial topology. Notice that t is also a functor of the form t: Ab ⟶ ST Ab/fc. We have the following properties:

7.4. Proposition. The functors above satisfy 1) g: ST Ab ⟶ Ab is left adjoint to t: Ab ⟶ ST Ab

2) g: ST Ab/fc ⟶ Ab is left adjoint to t: Ab ⟶ ST Ab/fc.

Next we prove that the category ST Ab/fc does not have countable sums. To do this, we take X = LcZ_{a} that is given by the free abelian group

X = X(0) = f_{a}{x_{0}, x_{1}, x_{2},…}

and the local neighbourhood base at 0 given by

X(n) = f_{a}{x_{n}, x_{n+1}, x_{n+2},…}

In the category ST Ab, we can consider the countable sum S = ⊕_{i=0}^{∞} X_{i} together the

topology given by the following local base. For each sequence n = (n_{0}, n_{1}, n_{2},…) ∈ ∈ N × N × N × ⋯ we consider

S(n) = X(n_{0}) ⊕ X(n_{1}) ⊕ X(n_{2}) ⊕ ⋯

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

7.5. Lemma. S = ⊕_{i=0}^{∞} X_{i} is a non first countable topological abelian group.

Proof. Assume that we have a countable neighbourhood base at 0. This implies the existence of a sequence … m^{2} > m^{1} > m^{0} in N × N × N × ⋯, where

m^{0} = (m_{0}^{0}, m_{1}^{0}, m_{2}^{0},…)

m^{1} = (m_{0}^{1}, m_{1}^{1}, m_{2}^{1},…)

m^{2} = (m_{0}^{2}, m_{1}^{2}, m_{2}^{2},…)

⋯ such that S(m^{0}) ⊃ S(m^{1}) ⊃ S(m^{2}) ⊃ ⋯ is a countable neighbourhood base at 0, where

S(m^{0}) = X(m_{0}^{0}) ⊕ X(m_{1}^{0}) ⊕ X(m_{2}^{0}) ⊕ ⋯ S(m^{1}) = X(m_{0}^{1}) ⊕ X(m_{1}^{1}) ⊕ X(m_{2}^{1}) ⊕ ⋯ S(m^{2}) = X(m_{0}^{2}) ⊕ X(m_{1}^{2}) ⊕ X(m_{2}^{2}) ⊕ ⋯ ⋯ Now consider m = (m_{0}, m_{1}, m_{2},…) such that m_{0} > m_{0}^{0} m_{1} > m_{1}^{1} = max{m_{1}^{0}, m_{1}^{1}} m_{2} > m_{2}^{2} = max{m_{2}^{0}, m_{2}^{1}, m_{2}^{2}} ⋯

Then we have that

S(m) = X(m_{0}) ⊕ X(m_{1}) ⊕ X(m_{2}) ⊕ ⋯ is a neighbourhood at 0 that does not contain S(m^{i}) for i ≥ 0. This contradiction comes from the assumption that S was first countable. Therefore S is non first countable.

7.6. Corollary. The full subcategory ST Ab/fc of ST Ab is not closed under countable sums.

7.7. Lemma. The category ST Ab/fc does not have the countable sum Σ_{i=0}^{∞} X, where X = LcZ_{a}.

Proof. Suppose that we have a first countable topological abelian group Σ_{i=0}^{∞} X_{i}, where

X_{i} = X for i ≥ 0. Let in_{i}: X_{i} ⟶ Σ_{i=0}^{∞} X_{i} be the canonical “inclusions”. Now since g: ST Ab/fc ⟶ Ab is a left adjoint by Proposition 7.3, it follows that g preserves sums. Therefore there exists an isomorphism θ: g( Σ_{i=0}^{∞} X_{i}) ⟶ ⊕_{i=0}^{∞} gX_{i} such that for each i ≥ 0, the following diagram is commutative

g( Σ_{i=0}^{∞} X_{i}) ⟶^{θ} ⊕_{i=0}^{∞} gX_{i}
gin_{i} ↖  ↗ in_{gX}_{i}
gX_{i}

where gX_{i} = X for i ≥ 0. The isomorphism θ induces a topology τ_{f} on ⊕_{i=0}^{∞} X_{i} such

that in_{X_{i}}: X_{i} ⟶ ⊕_{i=0}^{∞} X_{i} is continuous for each i ≥ 0. Then ( ⊕_{i=0}^{∞} X_{i}, τ_{f}) together with the “inclusions” is the sum in the category ST Ab/fc.

Let ( ⊕_{i=0}^{∞} X_{i}, τ_{n}) be the sum in the category ST Ab. Since in_{i}: X_{i} ⟶ ⊕_{i=0}^{∞} X_{i} is continuous

for the topology τ_{f}, it follows that id: ( ⊕_{i=0}^{∞} X_{i}, τ_{n}) → ( ⊕_{i=0}^{∞} X_{i}, τ_{f}) is continuous. Therefore τ_{n} is finer than τ_{f}.

For each n = (n_{0}, n_{1}, n_{2},…), consider the abelian group ⊕_{i=0}^{∞} X_{i}/X(n_{i}) provided with

the discrete topology, dis, which is first countable. As in_{i}: X_{i} ⟶ ⊕_{i=0}^{∞} X_{i}/X(n_{i}) is contin-

uous for i ≥ 0, the natural projection p: ( ⊕_{i=0}^{∞} X_{i}, τ_{f}) → ( ⊕_{i=0}^{∞} X_{i}/X(n_{i}), dis) is continuous.

Then p^{−1}(0) = ⊕_{i=0}^{∞} X(n_{i}) is an open neighbourhood of 0 for τ_{f}. This implies that τ_{f} is finer than τ_{n}. As τ_{f} = τ_{n}, we have τ_{n} is a first countable topology. This fact contradicts Lemma 7.5.

7.8. Corollary. ST Ab/fc does not have countable sums.

7.9. Corollary. The category (towAb, Ab) does not have the countable sum Σ_{i=0}^{∞} cZ.

7.10. Corollary. The category (towAb, Ab) does not have countable sums.

7.11. Corollary. The category (towAb, Ab) is not equivalent to a category of modules.

Proof. Assume that the object Σ_{i=0}^{∞} cZ exists. Since L: (towAb, Ab) ⟶ ST Ab/fc is a

left adjoint functor, it follows that L preserves sums. Then L( Σ_{i=0}^{∞} cZ) ≅ Σ_{i=0}^{∞} LcZ. This contradicts Lemma 7.7.

7.12. Corollary. The category towAb does not have the countable sum Σ_{i=0}^{∞}(^{∞}cZ).

7.13. Corollary. The category towAb does not have countable sums.

7.14. Corollary. The category towAb is not equivalent to a category of modules.

Proof. If there is a countable sum Σ_{i=0}^{∞}(^{∞}cZ) in towAb, Σ_{i=0}^{∞}(^{∞}cZ) ⟶ Σ_{i=0}^{∞} cZ_{0} would be

isomorphic to a countable sum Σ_{i=0}^{∞} cZ in (towAb, Ab). This is not possible by Corollary 7.9.

Next we use topological abelian groups to prove that the extended functor P: proAb ⟶ Ab_{PcZa} is not faithful. This implies that cZ_{a} is not a generator of all the category proAb.

Let S be an infinite set and let f_{a}S be the free abelian group generated by S. Consider the topology defined on f_{a}S by the family of subgroups of the form f_{a}T where T ⊂S and S −T is a countable set. We are going to see that if a sequence y_{k} in f_{a}T converges to zero, then there exists k_{0} such that y_{k} = 0 for every k≥0. Assume that there is a subsequence x_{i} = y_{k_{i}} with x_{i} ≠ 0 for every i. Since y_{k} → 0, it follows that x_{i} → 0. Each x_{i} can be written as a linear combination of finitely many elements of S. Therefore the sequence x_{i} determines a countable set S′ of generators such that x_{i} ∉ f_{a}(S − S′) for every i≥0. However this contradicts the fact that x_{i} → 0.

Using the functor N: ST Ab ⟶ (proAb, Ab) we have the global proobject Nf_{a}S that also defines a proobject denoted in the same way in proAb.

7.15. Proposition. proAb(cZ_{a}, Nf_{a}S) ≅ 0.

Proof. The hom-set proAb(cZ_{a}, Nf_{a}S) is a quotient of (proAb, Ab)(cZ_{a}, Nf_{a}S). By Proposition 7.2, (proAb, Ab)(cZ_{a}, Nf_{a}S) ≅ ST Ab(LcZ_{a}, f_{a}S). Notice that ST Ab(LcZ_{a}, f_{a}S) is the set of sequences in f_{a}S converging to zero. Two converging sequences define the same morphism in proAb(cZ_{a}, Nf_{a}S) if and only if they have the same germ as the zero sequence. Therefore it follows that proAb(cZ_{a}, Nf_{a}S) ≅ 0.

7.16. Corollary. The functor P: proAb ⟶ Ab_{PcZ_{a}} is not faithful.

Proof. Since the bonding morphisms of Nf_{a}S are non trivial, we have that Nf_{a}S is not isomorphic to the zero object. This implies that the identity id of Nf_{a}S is not equal to the zero map 0: Nf_{a}S → Nf_{a}S. By Proposition 7.15 we have that P(id) = P(0) = 0.

7.17. Remark. Grossman’s result that P reflects isomorphisms does not work for the extended functor P: proAb ⟶ Ab_{PcZ_{a}}.

7.18. Corollary. The object cZ does not generate the whole of the category proAb.

Proof. It is easy to prove that if cZ were a generator for proAb, then P: proAb ⟶ Ab_{PcZ_{a}} would be a faithful functor.

7.19. Remark. At present, the author [He.2] is writing a paper that contains some topological applications of the embeddings given in this paper. It considers an extension of the P functor to categories whose objects are towers of simplicial sets or towers of simplicial groups. One of the main results of the new paper is the construction of a simplicial set hoPX associated with a tower of simplicial sets X. This space is constructed by considering a right-derived functor hoP of the version of Brown’s P functor defined for the category of towers of simplicial sets. Recall that the homotopy limit functor, holim, can be defined as the right-derived functor of the lim functor. The simplicial set holimX is a simplicial subset of hoPX. It is well known that the Hurewicz homotopy groups of holimX are the strong (or Steenrod) homotopy groups of the tower X, we obtain that the Hurewicz homotopy groups of the larger space hoPX are the Brown-Grossman homotopy groups of the tower X.

Acknowledgements. There are many mathematicians that helped me to develop this paper. I am very grateful to T. Porter for explaining to me the relations between proper homotopy theory and procategories. The Chipman’s preprints contain the idea of considering the Brown-Grossman groups with more structure. I thank M.I.C. Beattie for his questions and suggestions about towers of abelian groups. H. Baues and J. Zobel helped me with the construction of infinite sums and products to solve the problem of construction of the left adjoint of the P functor. I have also had very interesting discussions with Cabeza, Elvira, Extremiana, Navarro, Rivas y Quintero. I also thank Jose Antonio for helping me to type the paper.

The author acknowledges the financial help given by “DGICYT, PB93-0581-C02-01” and “Programa Acciones Integradas Hispano-Germanas, 1991-127A”.

Theory and Applications of Categories, Vol. 1, No. 2 52

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