Dimension Theory and Group Valued Functors

Anthony Bak Department of Mathematics University of Bielefeld 33501 Bielefeld, Germany e-mail: bak@mathematik.uni-bielefeld.de

March 5, 2002

Abstract

This article defines concepts of structure and dimension in arbitrary categories and illustrates them with well known examples. It shows that arbitrary group and coset valued functors define in a natural way notions of structure and dimension on their source categories and that this data predicts group theoretic properties of the functors over finite dimensional objects, such as solvability, nilpotence, or normality of subfunctors. The results are illustrated with applications.

1 Introduction

The dimension theory of categories provides valuable information concerning the properties of functors on categories. This article focuses on group and coset valued functors and examines group theoretic properties of these functors such as solvability, nilpotence, or normality of subfunctors. Further articles will treat functors with values in other categories and investigate other properties, such as splitting and cancellation for functors with values in additive categories.

The current article begins by defining a concept of structure in an arbitrary category and for a category with structure, it defines many kinds of dimension functions. It compares dimension functions and shows that for a given kind of dimension function, there is a unique smallest one which is called the universal one of its kind. Well known dimension functions are discussed in this context.

It is then shown how arbitrary group and coset valued functors lead naturally to specific structures and dimension functions on their source categories. Further it is shown that this data predicts group theoretic properties of the functors on finite dimensional objects, such as solvability, nilpotence, or normality of subfunctors. Applications to classical-like groups and K-theory are given.

A generalization of a coset valued functor is a natural transformation valued functor of group valued functors. It is shown that such functors lead also to structures and dimension functions on their source categories and that this information predicts group theoretic properties of the target of the natural transformation on finite dimensional objects. Applications are also given to classical-like groups and K-theory.

The current article has a sequel [ ] titled Dimension Theory and Linked Sequences of Group Valued Functors. A sequence ⋯ G_{i}, G_{i+1} ⋯ of group valued functors is called linked if each consecutive pair G_{i}, G_{i+1} is a linked pair. Such sequences occur very naturally in homotopical settings. The current article defines the concept of a linked pair and shows how to associate to a natural transformation of group valued functors a linked pair of functors. We then use linked pairs to recover results above and as a bridge to the next paper.

The rest of the article is organized as follows. In §2, concepts of structure and dimension in arbitrary categories are defined. In §3, some well known and not so well known examples are given. In §4, it is shown how coset valued functors lead to specific structures and dimension functions on their source categories and this data is used to determine group theoretic properties of the functors on finite dimensional objects. In §5 applications of the above are made to classical-like groups and K-theory. In §6, it is shown how natural transformation valued functors of group valued functors lead also to structures and dimension functions on their source categories and how this information determines group theoretic properties of the target of the natural transformation. In §8, the concepts of linking diagram and linked pair of functors are introduced. It is shown that group theoretic properties of the second functor in a linked pair influence those of the first. In §9, it is shown how to construct a linked pair of functors from a natural transformation valued functor of group valued functors. In §10, the results of §8 and §9 are applied to classical-like groups and K-theory.

It is a pleasure for me to acknowledge my hosts the late O. Villamayor and G. Cortinas at Buenos Aires and La Plata Universities and my host A. Martsinkovsky at Northeastern University where I lectured on materials presented here. Special thanks are given to L. D’Alfonso and G. Minian for taking notes which resulted in a preliminary version [ ] of materials presented here and to A. Mundkur and A. Stepanov for making their Bielefeld notes available to me for the current paper.

2 Categories with Structure and Dimension

This section defines the concept of a category with structure and of a dimension function on a category with structure. On a given category with structure, there are many kinds of dimension functions, corresponding to a notion called type. A general category 𝒞 becomes a category with structure when we fix in it four ingredients of structure. A function d : Ob(𝒞) → ZZ^{≥0} ∪ {∞} is a dimension function if it satisfies certain properties with respect to the structure on 𝒞. Different structures determine usually different classes of dimension functions. The four ingredients of structure are the following:

• A class 𝒱(𝒞) of morphisms in 𝒞, called virtual isomorphisms.

• A class ℐ(𝒞) of functors on directed preordered sets with values in 𝒞, called infrastructure functors.

• A class of commutative squares in 𝒞, called structure squares.

The dimension theory developed in this article is geared to show that a class of functors on a general category 𝒞 will have a given property, if 𝒞 has a structure which is tied to the functors themselves and the property as follows:

• The property is invariant under 𝒱(𝒞).

• The functors preserve direct limits of functors in ℐ(𝒞).

• The behavior of the functors in the upper left hand object of a structure square is suitably tied to its behavior on the rest of the square.

Under the relationship above of functors to structure, the results of the article will show that the property holds over all finite dimensional objects.

We define now in detail the ingredients of a category with structure.

2.1

Throughout let 𝒞 denote an arbitrary category.

Let Iso (𝒞) denote the class of all isomorphisms in 𝒞. A class 𝒱(𝒞) of morphisms in 𝒞 is called a class of virtual isomorphisms, if 𝒱(𝒞) contains Iso(𝒞) and if 𝒱(𝒞) is closed under composition.

Let x = s, t, or ϕ. The letter s stands for source, the letter t for target, and the symbol ϕ stands for the empty letter. A class 𝒞_{0} of objects of 𝒞 is called x-closed under virtual isomorphism, if one of the following holds:

(i) x = s and given B ∈ 𝒞_{0} and A → B ∈ 𝒱(𝒞), it follows that A ∈ 𝒞_{0}.

(ii) x = t and given A ∈ 𝒞_{0} and A → B ∈ 𝒱(𝒞), it follows that B ∈ 𝒞_{0}.

(iii) x = ϕ and 𝒞_{0} is both s-closed and t closed under virtual isomorphisms.

A class 𝒞_{0} is called closed under virtual isomorphism, if it is ϕ-closed.

Let A and A′ be objects of 𝒞. A virtual chain equivalence from A to A′ is by definition a chain of objects A = A_{0}, A_{1}, ⋯, A_{n} = A′ and morphisms f_{0}, f_{1}, ⋯, f_{n−1} ∈ 𝒱(𝒞) such that for each i (0 ≦ i ≦ n − 1) either source (f_{i}) = A_{i} and target (f_{i}) = A_{i+1} or source (f_{i}) = A_{i+1} and target (f_{i}) = A_{i}. Clearly virtual chain equivalence is an equivalence relation on Ob(𝒞).

Obviously a class 𝒞_{0} of objects of 𝒞 is closed under virtual isomorphisms ⇔ it is closed under virtual chain equivalence.

2.2

Recall that a preordered set I is a set together with a reflexive, transitive relation ≦. Equivalently I is a category whose objects form a set and for any pair i, j of objects of I, there is at most one morphism whose source is i and target is j. A partially ordered set I is a preordered set whose relation is also anitsymmetric, i.e. i ≦ j and j ≦ i implies i = j. A preordered set I is called directed, if given elements i, j ∈ I, there is an element k ∈ I such that i ≦ k and j ≦ k.

Let {∗} denote the directed preordered set with precisely one element ∗. If A ∈ Ob(𝒞), let F_{A} denote the functor

F_{A} : {∗} → 𝒞
∗ ↦ A
1_{∗} ↦ 1_{A}

where 1_{∗} and 1_{A} denote the identity morphisms on ∗ and A, respectively. F_{A} will be called a trivial infrastructure functor.

A class ℐ(𝒞) of functors with values in 𝒞 is called a class of infrastructure functors, if it contains all trivial infrastructure functors and is closed under natural isomorphisms and if each F ∈ ℐ(𝒞) has the property that its source category is a directed, preordered set I and the direct limit lim_{⟶_{I}} F exists in 𝒞. By definition, lim_{⟶_{I}} F is the colimit colim_{I}F.

2.3

If A ∈ Ob(𝒞) then the commutative square

\xymatrix{A \ar[d]_{1} \ar[r]^{1} & A \ar[d]^{1} \\ A \ar[r]_{1} & A}

will be called a trivial square.

A class 𝒮(𝒞) of commutative squares

\xymatrix{A \ar[d] \ar[r] & B \ar[d] \\ C \ar[r] & D}

in 𝒞 is called a class of structure squares, if it contains all trivial squares and is closed under isomorphism of commutative squares.

A class of 𝒢𝒮(𝒞) ⫅ 𝒮(𝒞) of commutative squares in 𝒞 is called a class of generating squares for 𝒮(𝒞) , if it is closed under isomorphism of commutative squares, and every square in 𝒮(𝒞) is a direct limit in a prescribed way of squares in 𝒢𝒮(𝒞). This means that we have a class ℱ𝒢𝒮(𝒞) of functors S whose source categories are directed, preordered sets J, whose values are in 𝒢𝒮(𝒞), and whose direct limits lim_{⟶_{J}} S exist and are in 𝒮(𝒞) and every square

in 𝒮(𝒞) is some direct limit lim_{⟶_{J}} S. The purpose for introducing ℱ𝒢𝒮(𝒞) rather

than allowing all direct limits of squares in 𝒢𝒮(𝒞), is that the latter would permit us to treat only functors on 𝒞 which commute with all direct limits of generating squares.

2.4

A category with structure is a quadruple (𝒞, 𝒱(𝒞), ℐ(𝒞), 𝒮(𝒞) ) consisting of a category 𝒞, a class 𝒱(𝒞) of virtual isomorphisms, a class ℐ(𝒞) of infrastructure functors, and a class 𝒮(𝒞) of structure squares. A functor or morphism ℱ : (𝒞, 𝒱(𝒞), ℐ(𝒞), 𝒮(𝒞)) → (𝒟, 𝒱(𝒟), ℐ(𝒟), 𝒮(𝒟)) of categories with structure is a functor ℱ : 𝒞 → 𝒟 which preserves virtual isomorphisms, infrastructure functors, and structure squares. This means that

• if f is a virtual isomorphism then so is ℱ(f),

• if F : I → 𝒞 is an infrastructure functor then so is the composition

ℱF : I → 𝒟 and the canonical morphism lim_{⟶_{I}} ℱF → ℱ (lim_{⟶_{I}} F ) is an

isomorphism, • if

\xymatrix{A \ar[d] \ar[r] & B \ar[d] \\ C \ar[r] & D}

is a structure square in 𝒞 then

\xymatrix{ℱ(A) \ar[d] \ar[r] & ℱ(B) \ar[d] \\ ℱ(C) \ar[r] & ℱ(D)}

is one in 𝒟.

2.5

Let \underline{𝒞} = (𝒞, 𝒱(𝒞), ℐ(𝒞), 𝒮(𝒞)) denote a category with structure. Let ∞ denote the first infinite ordinal. Let x = s, t, or ϕ be as in (2.1). An x−dimension function on \underline{𝒞} is a function

d : Ob(𝒞) → ℤ^{≥0} ∪ {∞}

such that one of the following holds:

(i) x = s and given A → B ∈ 𝒱(𝒞), it follows that d(A) ≦ d(B).

(ii) x = t and given A → B ∈ 𝒱(𝒞), it follows that d(A) ≧ d(B).

(iii) x = ϕ and given A → B ∈ 𝒱(𝒞), it follows that d(A) = d(B), i.e. d is both an s- and a t-dimension function.

A dimension function on \underline{𝒞} is by definition a ϕ-dimension function on \underline{𝒞}. If d and e are x-dimension functions on 𝒞 then d ≦ e means that d(A) ≦ e(A) for each A ∈ Ob(𝒞). A category with x-dimension is a pair (\underline{𝒞}, d) consisting of a category with structure \underline{𝒞} and an x-dimension function d on \underline{𝒞}. A functor or morphism ℱ : (\underline{𝒞}, d) → (\underline{𝒟}, e) of categories with x-dimension is a functor \underline{𝒞} → \underline{𝒟} of categories with structure such that d ≧ eℱ.

The most important concepts for an x-dimension function are the notions of type and reduction. The problems we can solve using dimension theory depend on these notions. They are introduced next.

2.6

A type t is a triple t_{B}, t_{C}, t_{D} : ℤ^{≥0} ∪ {∞} → ℕ ∪ {∞, ∞ + 1} of functions. It is usually written in the form

t = \binom{t_{B}}{t_{C} t_{D}} .

If t and t′ are types then t ≦ t′ means that t(x) ≦ t′(x) for each element x ∈ ℤ^{≥0} ∪ {∞}, i.e. if X = A, B, or C then t_{X}(x) ≦ t′_{X}(x).

Let \underline{𝒞} be a category with structure. Let A ∈ Ob(𝒞). A reduction of A consists of

(i)

\{ \begin{matrix}a morphism A → A′ ∈ 𝒱(𝒞), if x = s. \\ a morphism A′ → A ∈ 𝒱(𝒞), if x = t \\ a virtual chain equivalence from A to A′, if x = ϕ \end{matrix} \} and

(ii) an infrastructure functor B : I → 𝒞, i ↦ B_{i}, and

(iii) a set

\xymatrix{A′ \ar[d] \ar[r] & B_{i} \ar[d] \\ C_{i} \ar[r] & D_{i}} (i ∈ I)

of structure squares indexed by I (C and D are not necessarily functorial in I) with the property that for each relationship (i ≦ j) ∈ I, the triangle

\xymatrix{ & B_{j} \\ A′ \ar[ur] \ar[dr] \\  & B_{i} \ar[uu]}

commutes.

Let d be an x-dimension function on \underline{𝒞}. Let t be a type and n ∈ ℕ. A reduction of A is said to be a d-reduction of type t at n, if in addition to the above

(iv) d (lim_{⟶_{I}} B) < t_{B}(n)

and for each i ∈ I,

(v) d(C_{i}) < t_{C}(n), and

(vi) d(D_{i}) < t_{D}(n).

Suppose now that 0 < d(A) < ∞. Then a d-reduction of type t at d(A) for A will be simply called a d-reduction of type t for A. An xdimension function d on \underline{𝒞} is said to have type t, if each A ∈ Ob(𝒞) such that 0 < d(A) < ∞ has a d-reduction of type t. An x-dimension function can have many types and obviously if it has type t and t ≦ t′ then it also has type t′. The applicability of a given x-dimension function d to a given problem depends on the minimal types of d.

2.7

Let d be an x-dimension function of type t on the category with structure \underline{𝒞}. Let A ∈ Ob(𝒞) and n ∈ ℕ. Even if A has a d-reduction of type t at n, it does not follow that d(A) ≦ n. However if the conclusion above is true for any A and any n then we shall say that d is a tame x-dimension function of type t.

2.8

Let \underline{𝒞} be a category with structure. Define x − dim(\underline{𝒞}) to be the partially ordered class of all x-dimension functions on \underline{𝒞}, with partial ordering given by ≦. If t is a type, let x − dim(\underline{𝒞}, t) denote the partially ordered subclass of x − dim(\underline{𝒞}) consisting of all x-dimension functions of type t.

2.9

Theorem The partially ordered class x−dim(\underline{𝒞}) has arbitrary greatest lower bounds. If S ⫅ x − dim(\underline{𝒞}) is a nonempty subclass then its greatest lower bound glb_{S} is computed by the formula

glb_{S}(A) = inf{d(A) | d ∈ S}.

Moreover if S ⫅ x − dim(\underline{𝒞}, t) is a nonempty subclass then glb_{S} ∈ x − dim(\underline{𝒞}, t).

Proof The function Ob(𝒞) → ZZ^{≥0}∪{∞}, A ↦ inf {d(A) | d ∈ S}, obviously satisfies the conditions of definition 2.5 of an x-dimension function. Clearly this function is the greatest lower bound for S. Suppose S ⫅ x−dim(\underline{𝒞}, t). It will be shown that glb_{S} ∈ x − dim(\underline{𝒞}, t). Let A ∈ Ob(𝒞) such that glb_{S}(A) ∈ ℕ. We must show that A has a glb_{S}-reduction of type t at glb_{S}(A). Let d ∈ S such that glb_{S}(A) = d(A). By definition, A has a d-reduction of type t at d(A). Since glb_{S}(X) ≦ d(X) for all X ∈ Ob(𝒞), it follows that any dreduction above of A is also a glb_{S}-reduction of A of type t at d(A) = glb_{S}(A). Thus by definition, glb_{S} has type t.□

2.10

Theorem Let d be an x-dimension function of type t on the category with structure \underline{𝒞}. Then there is a tame x-dimension function d_{t} of type t, called the tame closure of d at t, which is the unique maximum among all tame xdimension functions d′ of type t such that d′ ≦ d. Moreover d_{t} is computed according to the following rule: Let T (d) = {d′ | d′ a tame x-dimension function of type t, d′ ≦ d} and S(d) = {d′′ | d′′ an x-dimension function of type t, d′′ ≦ d, d′ ≦ d′′ ∀ d′ ∈ T (d)}. Then

d_{t} = glb_{S(d)}.

Proof Clearly d′ ≦ glb_{S(d)} ≦ d for all d′ ∈ T (d) and by Theorem 2.9, glb_{S(d)} is an x-dimension function of type t. To complete the proof of the theorem, it suffices to show that glb_{S(d)} is tame of type t. Suppose glb_{S(d)} is not tame. Define f : Ob(𝒞) → ZZ^{≥0} ∪ {∞} by

f(A) = inf({glb_{S(d)}(A)}∪{n ∈ ℕ | A has a glb_{S(d)}-reduction of type t at n}).

Since glb_{S(d)} is not tame, f < glb_{S(d)}, i.e. f ≦ glb_{S(d)} and there is an B ∈ Ob(𝒞) such that f(B) < glb_{S(d)}(B). Thus f ∉ S(d). It will be shown next that f ≦ d. Since glb_{S(d)} is an x-dimension function, it follows by construction that f is one also. Let A ∈ Ob(𝒞) such that 0 < f(A) < ∞. By definition A has a glb_{S(d)}-reduction of type t at f(A). Since f(X) ≦ glb_{S(d)}(X) for all X ∈ Ob(𝒞), it follows that each glb_{S(d)}-reduction of A above is also an f-reduction of A of type t at f(A). Thus f has type t, by definition. Suppose d′ ∈ T (d). We must show that d′ ≦ f. From the definition of glb_{S(d)}, it follows that d′ ≦ glb_{S(d)}. It suffices to show that if f(A) < ∞, then d′(A) ≦ f(A).If f(A) = 0 then from the definition of f, it follows glb_{S(d)}(A) = 0 and so d′(A) = 0. Suppose f(A) > 0. From the definition of f, it follows that A has a glb_{S(d)}-reduction of type t at f(A). Since d′(X) ≦ glb_{S(d)}(X) for all X ∈ Ob(𝒞), it follows that any glb_{S(d)}- reduction above of A is also a d′-reduction of A of type t at f(A). Since d′ is tame, it follows that d′(A) ≦ f(A).□

2.11

Let \underline{𝒞} be a category with structure. Let t be a type. An x-dimension function δ of type t is called universal of type t, if δ ≦ d for any x-dimension function d of type t, whose 0-dimensional objects are also 0-dimensional under δ.

2.12

Corollary Let \underline{𝒞} be a category with structure. Let 𝒞^{0} be a class of objects of 𝒞, which is x-closed under virtual isomorphism. Then there is a unique universal x-dimension function of type t, whose 0-dimensional objects are precisely those of 𝒞^{0}. Furthermore it is tame. This function is often denoted by

d_{\underline{𝒞} ,𝒞^{0},t}.

Proof Let S denote the class of all x-dimension functions of type t, whose 0-dimensional objects are in 𝒞^{0}. This class is nonempty and contains an xdimension function d whose 0-dimensional objects are precisely 𝒞^{0}, namely the function d : Ob(𝒞) → ZZ^{≥0} ∪ {∞} defined by

d(A) = \{ \begin{matrix}0 & if A ∈ 𝒞^{0} \\ ∞ & if A ∉ 𝒞^{0}. \end{matrix}

By Theorem 2.10, glb_{S} exists and is an x-dimension function of type t. Obviously glb_{S} has 𝒞^{0} as its class of 0-dimensional objects, is universal of type t and unique. Furthermore by Theorem 2.8, it must be tame. □

2.13

Theorem Let \underline{𝒞} be a category with structure. Let t be a type such that given X ∈ {A, B, C} either t_{X} ≦ id or t_{X} = ∞ + 1. Let d be a tame x-dimension function of type t on \underline{𝒞}. Then d is universal of type t and is computed by the following rule: Define recursively disjoint, x-closed classes 𝒞^{n} as follows. Let 𝒞^{0} = {A ∈ Ob(𝒞) | d(A) = 0}. Let n ∈ ℕ and assume that disjoint, x-closed classes 𝒞^{m} have been defined for all 0 ≦ m < n. Define δ^{n−1} : Ob(𝒞) → ZZ^{≥0} ∪ {∞} by

δ^{n−1}(A) = \{ \begin{matrix}m & if & A ∈ 𝒞^{m} & where 0 ≦ m < n, \\ ∞ & if & A ∉ ∪_{m=0}^{n−1}𝒞^{m}. \end{matrix}

Obviously δ^{n−1} is an x-dimension function. Define 𝒞^{n} = {A ∈ Ob(𝒞) | A ∉

∪_{m=0}^{n−1}𝒞^{m}, A has a δ^{n−1}-reduction of type t at n}. Define 𝒞^{∞} = Ob(𝒞)\ ∪_{n∈ZZ^{≥0}}

𝒞^{n}. Then for any A ∈ Ob(𝒞) and any n ∈ ZZ^{≥0} ∪ {∞},

d(A) = n ⇔ A ∈ 𝒞^{n}.

Furthermore if n ∈ ZZ^{≥0} and 𝒞^{n} = ϕ then 𝒞^{n′} = ϕ for all n′ ≥ n, i.e. d = δ^{n−1}.

Proof We shall treat only the case t ≦ \binom{id}{id id}. The other cases are

done similarly.

Let 𝒞^{n} (n ∈ ZZ^{≥0} ∪ {∞}) be defined as in the theorem. Clearly these classes are disjoint and x-closed under virtual isomorphisms. Define δ : Ob(𝒞) → ZZ^{≥0} ∪ {∞} by δ(A) = n ⇔ A ∈ 𝒞^{n}. By construction, δ is a tame type t x-dimension function.

We shall show that δ is universal. Let δ′ = d_{\underline{𝒞},𝒞^{0},t} be the universal xdimension function of type t whose 0-dimensional objects form the class 𝒞^{0}. We must show that δ ≦ δ′. It suffices to show that if δ′(A) < ∞ then δ(A) ≦ δ′(A). We proceed by induction on δ′(A). If δ′(A) = 0 then δ(A) = 0 by definition of δ′. Let n ∈ ℕ such that δ′(A) < n implies δ(A) ≦ δ′(A). Suppose that δ′(A) = n. By definition, A has a δ′-reduction of type t at n.

Since t ≦ \binom{id}{id id} and since for any X ∈ Ob(𝒞), δ(X) ≦ δ′(X) whenever

δ′(X) < n, it follows that any δ′-reduction above of A is also a δ-reduction of A of type t at n. Since δ is tame, it follows that δ(A) ≦ n = δ′(A). Thus δ ≦ δ′.

We show next that d = δ. Since δ is universal, it suffices to show that if δ(A) < ∞ then d(A) ≦ δ(A). We proceed by induction on δ(A). If δ(A) = 0 then d(A) = 0, by definition of δ. Let n ∈ ℕ such that δ(A) = n implies d(A) ≦ δ(A). Suppose δ(A) = n. By definition, A has a δ-reduction of type

t at n. Since t ≦ \binom{id}{id id} and since for any X ∈ Ob(𝒞), d(X) ≦ δ(X)

whenever δ(X) < n, it follows that the reduction of A above is also a dreduction of type t at n. Since d is tame, this implies d(A) ≦ n = δ(A).

Finally suppose that 𝒞^{n} = ϕ. We shall show that 𝒞^{n+1} = ϕ. Suppose A ∈ 𝒞^{n+1}. Then A has a δ^{n}-reduction of type t at n + 1. Since 𝒞^{n} = ϕ and

t ≦ \binom{id}{id id}, this reduction must also be a δ^{n−1}-reduction of type t at n.

Since 𝒞^{n} = ϕ, it follows that A ∈ ∪_{m=0}^{n−1}𝒞^{m}, which contradicts the fact that the

𝒞^{k}’s are disjoint. Thus 𝒞^{n+1} = ϕ. By induction, we conclude that 𝒞^{n′} = ϕ for all n′ ∈ ZZ^{≥0} such that n′ ≥ n.□

2.14

Let (\underline{𝒞}, d) be a category with structure and d an x-dimension function on \underline{𝒞}. For n ∈ ZZ^{≥0} ∪ {∞}, define the classes

𝒞_{d}^{n} = {A ∈ Ob(𝒞) | d(A) = n}
𝒞_{d}^{≦n} = {A ∈ Ob(𝒞) | d(A) ≦ n}.

The next lemma is left as an easy exercise to the reader.

2.15

Lemma The classes above are x-closed under virtual isomorphisms and Ob(𝒞) =

∪_{n=0}^{∞}𝒞_{d}^{n}. Conversely any partition of Ob(𝒞) as a disjoint union Ob(𝒞) = ⋃_{n=0}^{∞}𝒞^{n}

of x-closed classes 𝒞^{n} defines an x-dimension function d on \underline{𝒞} such that 𝒞_{d}^{n} = 𝒞^{n}. Partially order the members of {x − closed classes in Ob(𝒞)} by inclusion. Then the function \{x-dimension functions of type t on \underline{𝒞}\} → {x-closed classes in Ob(𝒞)}, d ↦ 𝒞_{d}^{0}, preserves partial orderings and induces a bijection between universal x-dimension functions of type t and x-closed classes in Ob(𝒞). If (\underline{𝒟}, e) is a category with x-dimension and ℱ : \underline{𝒞} → \underline{𝒟} is a functor of categories with structure then ℱ defines a morphism (\underline{𝒞}, d) → (\underline{𝒟}, e) of categories with x-dimension ⇔ ℱ(𝒞_{d}^{≤n}) ⫅ 𝒟_{e}^{≦n} for all n ∈ ZZ^{≥0}. Furthermore if d = d_{\underline{𝒞},𝒞^{0},t} and e = d_{\underline{𝒟} ,𝒟^{0},t} for some x-closed classes 𝒞^{0} ⊆ Ob(𝒞) and 𝒟^{0} ⊆ Ob(𝒟) then ℱ induces a morphism of categories with x-dimension ⇔ ℱ(𝒞^{0}) ⫅ 𝒟^{0}.

2.16

Let \underline{𝒞} be a category with structure. If S ⫅ x − dim(\underline{𝒞}), define the partially ordered set type(S) = {types t | each d ∈ S has type t }, with partial ordering that of types.

Since the results we prove for group valued functors are formulated using the concept of type for x-dimension functions, it is useful to have some general guidelines how large the set type(S) above is, in particular when S has just one element. The following lemmas provide such information. They are given without proof.

2.17

Lemma The partially ordered set type(S) has the property that every element is bounded below by a minimal element. Furthermore if t ∈ type(S) and t′ is a type such that t′ ≥ t then t′ ∈ type(S). Thus the minimal elements of type(S) define it uniquely.

2.18

Lemma Let ℱ : \underline{𝒞} → \underline{𝒟} be a functor of categories with structure. Let T ⫅ x − dim(\underline{𝒟}). Define S = \{dℱ | d ∈ T \}.

(i) Then S ⫅ x − dim(\underline{𝒞}) and type(S) ⫅ type(T).

(ii) Let x ∈ {s, t, ϕ} as in 2.1. If x = s, suppose that for any (ℱ(A) → D) ∈ 𝒱(𝒟) where A ∈ Ob(𝒞) and D ∈ Ob(𝒟), there is a (A → B) ∈ 𝒱(𝒞) and an isomorphism ℱ(B) → D. If x = t, suppose that for any (D → ℱ(A)) ∈ 𝒱(𝒟) where A ∈ Ob(𝒞) and D ∈ Ob(𝒟), there is a (B → A) ∈ 𝒱(𝒞) and an isomorphism ℱ(B) → D. If x = ϕ, suppose that for any virtual chain equivalence in \underline{𝒟} from ℱ(A) to D where A ∈ Ob(𝒞) and D ∈ Ob(𝒟), there is a virtual chain equivalence in \underline{𝒞} from A to some B ∈ Ob(𝒞) and an isomorphism from ℱ(B) to D. Suppose that any data consisting of an infrastructure functor (B : I → 𝒟) ∈ ℐ(𝒟) and a set

\xymatrix{A \ar[d] \ar[r] & B_{i} \ar[d] \\ C_{i} \ar[r] & D_{i}} (i ∈ I)

of structure squares in 𝒮(𝒟), which together satisfy (2.6) (ii) and (iii) and have the property that all objects A, B_{i}, C_{i}, D_{i} (i ∈ I) lie in ℱ(Ob(𝒞)), must be the image under ℱ of analogous data in 𝒞. Then type(T) ⫅ type(S).

3 Categories with Global Reduction and Dimension

The previous section introduced concepts of structure and dimension in categories for the purpose of studying properties of functors, in subsequent sections and papers. The current section develops further these concepts in order to expand their range of application. Once the section is complete, all of the constructions and results of the previous one will be special cases of those of the current one. Proofs for the current results will be omitted, since they are obvious extensions of their counterparts in the previous section. Examples of concepts introduced in the current section are provided in section 5.

The key construction for extending the scope of the previous section is that of a global reduction. In the previous section, reductions were used only one at a time. In the current section, we want to specify for each object in a category, certain sets of reductions to be called global reductions. The concept of dimension will be formulated such that certain properties hold across an entire set of reductions instead of just for one reduction. The constructions and results of the previous section are recovered from those of the current, by letting each global reduction consist of just one ordinary reduction and by assigning to each object in the category all of its single member global reductions.

Throughout this section, let \underline{𝒞} = (𝒞, 𝒱(𝒞), ℐ(𝒞), 𝒮(𝒞)) denote a category with structure.

3.1

A category with global reduction consists of a category with structure \underline{𝒞} and for each A ∈ Ob(𝒞) a class glrd(A) of sets X such that each member of X is a reduction of A in the sense of (2.6) (i) - (iii). Each member X ∈ glrd(A) is called a global reduction of A. It will be assumed that if X ∈ glrd(A) and Y is a set of reductions of A such that each reduction in X is isomorphic (in the straightforward sense) to a reduction in Y and conversely then Y ∈ glrd(A). In this case, we say that X is isomorphic to Y. A category with global reduction will be denoted by a pair

(\underline{𝒞}, glrd)

where glrd assigns to each A ∈ Ob(𝒞) a class glrd(A) of global reductions of A, which is closed under isomorphism. We explicitly allow the possibility that for some A′s, glrd(A) = ϕ.

Let ℱ : \underline{𝒞} → \underline{𝒞}′ be a functor of categories with structure. It follows from the definition of such functors that ℱ takes any reduction of an object A of 𝒞 to a reduction of ℱ(A). If R is a reduction of A, let ℱ(R) denote the corresponding reduction of ℱ(A). If X is a global reduction of A in (\underline{𝒞}, glrd), let ℱ(X) = \{ℱ(R) | R ∈ X\}. We say that ℱ defines a functor or morphism (\underline{𝒞}, glrd) → (\underline{𝒞′}, glrd′) of categories with global reduction, if for each A ∈ Ob(𝒞) and X ∈ glrd(A), ℱ(X) ∈ glrd′(ℱ(A)).

3.2

Let x = s (source), t (target), or ϕ (no condition), be as in (2.1). An x - dimension function d on (\underline{𝒞}, glrd) is by definition an x-dimension function on \underline{𝒞}, as in (2.5). A category with x-dimension is a pair ((\underline{𝒞}, glrd), d) consisting of a category with global reduction (\underline{𝒞}, glrd) and an x-dimension function d on (\underline{𝒞}, glrd). A functor or morphism ℱ : ((\underline{𝒞}, glrd), d) → ((\underline{𝒞′}, glrd′), d′) of categories with x-dimension is a functor ℱ : (\underline{𝒞}, glrd) → (\underline{𝒞′}, grld′) of categories with global reduction such that d ≧ d′ℱ.

As in section 2, the most important properties of x-dimension functions are described using the notions of type and global reduction. This is done in the following paragraphs.

3.3

Let d be an x-dimension function on the category with global reduction (\underline{𝒞}, glrd). Let t denote a type, as in (2.6), and let n ∈ ℕ. A global reduction X ∈ glrd(A) is called a global d-reduction of type t at n, if X ≠ ϕ and each reduction in X is a d-reduction of type t at n, as in (2.6) (iv) - (vi). A global d-reduction of type t at A is by definition an X ∈ glrd(A), which is a global d-reduction of type t at d(A). Note that this definition makes sense only when d(A) > 0, because t is defined on ℕ ∪ {∞}. An x-dimension function d is said to have type t, if each A ∈ Ob(𝒞) such that 0 < d(A) < ∞ has a global d-reduction of type t at A.

An x-dimension function on (\underline{𝒞}, glrd) can have many types and obviously if it has type t and t ≦ t′ then it also has type t′. The applicability of a given x-dimension function d to a given problem depends on the minimal types of d.

For the rest of this section, let

\overline{𝒞} = (\underline{𝒞}, glrd)

denote a category with global reduction. We continue developing the properties of x-dimension functions.

3.4

Let d be an x-dimension function of type t on the category with global reduction \overline{𝒞}. Let A ∈ Ob (𝒞) and n ∈ ℕ. Even if A has a global d-reduction of type t at n, it does not follow that d(A) ≦ n. However if the conclusion above is true for any A and any n then we shall say that d is a tame xdimension function of type t.

3.5

Let \overline{𝒞} be a category with global reduction. Define x − dim(\overline{𝒞}) to be the partially ordered class of all x-dimension functions on \overline{𝒞}, with partial ordering given by ≦. If t is a type, let x − dim(\overline{𝒞}, t) denote the partially ordered subclass of x − dim(\overline{𝒞}) consisting of all x-dimension functions of type t.

3.6

The next result generalizes Theorem 2.9.

Theorem The partially ordered class x−dim(\overline{𝒞}) has arbitrary greatest lower bounds. If S ⫅ x − dim(\overline{𝒞}) is a nonempty subclass then its greatest lower bound glb_{S} is computed by the formula

glb_{S}(A) = inf{d(A) | d ∈ S}.

Moreover if S ⫅ x − dim(\overline{𝒞}, t) is a nonempty subclass then glb_{S} ∈ x − dim(\overline{𝒞}, t).

3.7

Let next result generalizes Theorem 2.10.

Theorem Let d be an x-dimension function of type t on the category with global reduction \overline{𝒞}. Then there is a tame x-dimension d_{t} of type t, called the tame closure of d at t, which is the unique maximum among all tame xdimension functions d′ of type t such that d′ ≦ d. Moreover d_{t} is computed according to the following rule: Let T (d) = {d′ | d′ a tame x-dimension function of type t, d′ ≦ d} and S(d) = {d′′ | d′′ an x-dimension function of type t, d′′ ≦ d, d′ ≦ d′′ ∀ d′ ∈ T (d)}. Then

d_{t} = glb_{S(d)}.

3.8

Let \overline{𝒞} be a category with global reduction. Let t be a type. An x-dimension function δ of type t is called universal of type t, if δ ≦ d for any x-dimension function d of type t, whose 0-dimensional objects are also 0-dimensional under δ.

3.9

The next result generalizes Corollary 2.12.

Corollary Let \overline{𝒞} be a category with global reduction. Let 𝒞^{0} be a class of objects of 𝒞, which is x-closed under virtual isomorphism. Then there is a unique universal x-dimension function of type t, whose 0-dimensional objects are precisely those of 𝒞^{0}. Furthermore it is tame. This function is often denoted by

d_{\overline{𝒞} ,𝒞^{0},t}.

3.10

The next result generalizes Theorem 2.13.

Theorem Let \overline{𝒞} be a category with global reduction. Let t be a type such that given X ∈ {A, B, C} either t_{X} ≦ id or t_{X} = ∞ + 1. Let d be a tame x-dimension function of type t on \overline{𝒞}. Then d is universal of type t and is computed by the following rule: Define recursively disjoint, x-closed classes 𝒞^{n} as follows. Let 𝒞^{0} = {A ∈ Ob(𝒞) | d(A) = 0}. Let n ∈ ℕ and assume that disjoint, x-closed classes 𝒞^{m} have been defined for all 0 ≦ m < n. Define δ^{n−1} : Ob(𝒞) → ZZ^{≥0} ∪ {∞} by

δ^{n−1}(A) = \{ \begin{matrix}m & if & A ∈ 𝒞^{m} & where 0 ≦ m < n, \\ ∞ & if & A ∉ ∪_{m=0}^{n−1}𝒞^{m}. \end{matrix}

Obviously δ^{n−1} is an x-dimension function. Define 𝒞^{n} = {A ∈ Ob(𝒞) | A ∉

∪_{m=0}^{n−1}𝒞^{m}, A has a δ^{n−1}-reduction of type t at n}. Define 𝒞^{∞} = Ob(𝒞)\ ∪_{n∈ZZ^{≥0}}

𝒞^{n}. Then for any A ∈ Ob(𝒞) and any n ∈ ZZ^{≥0} ∪ {∞},

d(A) = n ⇔ A ∈ 𝒞^{n}.

Furthermore if n ∈ ZZ^{≥0} and 𝒞^{n} = ϕ then 𝒞^{n′} = ϕ for all n′ ≥ n, i.e. d = δ^{n−1}.

3.11

Let (\overline{𝒞}, d) be a category with global reduction and d an x-dimension function on \overline{𝒞}. For n ∈ ZZ^{≥0} ∪ \{∞\}, define the classes

𝒞_{d}^{n} = {A ∈ Ob(𝒞) | d(A) = n}
𝒞_{d}^{≦n} = {A ∈ Ob(𝒞) | d(A) ≦ n}.

3.12

The next result generalizes Lemma 2.15.

Lemma The classes above are x-closed under virtual isomorphisms and Ob(𝒞) =

∪_{n=0}^{∞}𝒞_{d}^{n}. Conversely any partition of Ob(𝒞) as a disjoint union Ob(𝒞) = ⋃_{n=0}^{∞}𝒞^{n}

of x-closed classes 𝒞^{n} defines an x-dimension function d on \overline{𝒞} such that 𝒞_{d}^{n} = 𝒞^{n}. Partially order the members of {x − closed classes in Ob(𝒞)} by inclusion. Then the function \{x-dimension functions of type t on \overline{𝒞}\} → \{xclosed classes in Ob(𝒞)}, d ↦ 𝒞_{d}^{0}, preserves partial orderings and induces a bijection between universal x-dimension functions of type t and x-closed classes in Ob(𝒞). If (\overline{𝒟}, e) is a category with x-dimension and ℱ : \overline{𝒞} → \overline{𝒟} is a functor of categories with global reduction then ℱ defines a morphism (\overline{𝒞}, d) → (\overline{𝒟}, e) of categories with x-dimension ⇔ ℱ(𝒞_{d}^{≤n}) ⫅ 𝒟_{e}^{≦n} for all n ∈ ZZ^{≥0}. Furthermore if d = d_{\overline{𝒞},𝒞^{0},t} and e = d_{\overline{𝒟} ,𝒟^{0},t} for some x-closed classes 𝒞^{0} ⊆ Ob(𝒞) and 𝒟^{0} ⊆ Ob(𝒟) then ℱ induces a morphism of categories with x-dimension ⇔ ℱ(𝒞^{0}) ⫅ 𝒟^{0}.

3.13

Let \overline{𝒞} be a category with global reduction. If S ⫅ x − dim(\overline{𝒞}), define the partially ordered set type(S) = {types t | each d ∈ S has type t }, with partial ordering that of types.

Since the results we prove for group valued functors are formulated using the concept of type for x-dimension functions, it is useful to have some general guidelines how large the set type(S) above is, in particular when S has just one element. The following lemmas provide such information.

3.14

The next lemma generalizes Lemma 2.17.

Lemma The partially ordered set type(S) has the property that every element is bounded below by a minimal element. Furthermore if t ∈ type(S) and t′ is a type such that t′ ≥ t then t′ ∈ type(S). Thus the minimal elements of type(S) define it uniquely.

3.15

The next lemma generalizes Lemma 2.18.

Lemma Let ℱ : \overline{𝒞} → \overline{𝒟} be a functor of categories with global reduction. Let T ⫅ x − dim(\overline{𝒟}). Define S = \{dℱ | d ∈ T \}.

(i) Then S ⫅ x − dim(\underline{𝒞}) and type(S) ⫅ type(T).

Suppose that for each A ∈ Ob(𝒞), the function glrd\overline{𝒞}(A) → glrd\overline{𝒟} (A) induced by ℱ is surjective up to isomorphism of global reductions in glrd\overline{𝒟} (A). Then type(T) ⫅ type(S).

4 Categories with Costructure, Global Coreduction, and Dimension

This section dualizes the concepts and results of the previous two sections. Proofs are the duals of their predecessors and will be omitted. Examples of the concepts are provided in section 5.

The principle of dualization reverses arrows in categories and replaces each functor by its dual. I.e., if one starts with a functor F : 𝒞 → 𝒟 of categories then dualizing replaces each arrow f : A → B in either 𝒞 or 𝒟 by an arrow f^{∘} : B^{∘} → A^{∘} and the functor F by the functor F^{∘} : 𝒞^{∘} → 𝒟^{∘} where F^{∘} sends each arrow f^{∘} : B^{∘} → A^{∘} in 𝒞^{∘} to the arrow (F (f))^{∘} : F (B)^{∘} → F (A)^{∘} in 𝒟^{∘}. Thus if F is covariant (resp. contravariant) then so is F^{∘}. Note that whereas dualization reverses the orientation of arrows in categories, it does not reverse the orientation of functors between categories being dualized.

Applying dualization to a category with structure \underline{𝒞}, we get a category with costructure \underline{𝒞}^{∘} such that each coreduction in \underline{𝒞}^{∘} is the dual of a reduction in \underline{𝒞}. Applying dualization to a category with global reduction \overline{𝒞}, we get a category \overline{𝒞}^{∘} with global coreduction such that each global coreduction in \overline{𝒞}^{∘} is the dual of a global reduction in \overline{𝒞}. Dimension in a category with costructure or category with global coreduction is defined by reversing arrows and applying the existing definition of dimension in the corresponding category with structure or global reduction.

In order to fix terminology and avoid confusion later one, we carry out the details of the program above, but leave as mentioned already the proofs of the results as an exercise in dualization.

4.1

Throughout let 𝒞 denote an arbitrary category.

Let Iso (𝒞) denote the class of all isomorphisms in 𝒞. A class 𝒱(𝒞) of morphisms in 𝒞 is called a class of virtual isomorphism, if 𝒱(𝒞) contains Iso(𝒞) and if 𝒱(𝒞) is closed under composition.

Let x = s, t, or ϕ, as in (2.1). Recall that the letter s stands for source, the letter t for target, and the symbol ϕ stands for the empty letter. In dualizing, s and t get interchanged. Define the concepts x-closed under virtual isomorphism, closed under virtual isomorphism, and virtual chain equivalence, as in (2.1).

4.2

A preordered set I is called inverse directed, if given elements i, j ∈ I, there is an element k ∈ I such that i ≧ k and j ≧ k. In dualizing, directed preordered sets are replaced by inverse directed ones, and conversely. This implies that direct limits lim_{⟶} are replaced by inverse limits lim_{⟵}, and conversely.

Let {∗} denote the inverse directed preordered set with precisely one element ∗. If A ∈ Ob(𝒞), let F_{A} denote the functor

F_{A} : {∗} → 𝒞
∗ ↦ A
1_{∗} ↦ 1_{A}

where 1_{∗} and 1_{A} denote the identity morphisms on ∗ and A, respectively. F_{A} will be called a trivial coinfrastructure functor.

A class coℐ(𝒞) of functors with values in 𝒞 is called a class of coinfrastructure functors, if it contains all trivial coinfrastructure functors, is closed under natural isomorphism, and each F ∈ coℐ(𝒞) has the property that its source category is an inverse directed, preordered set I and the inverse limit lim_{⟵_{I}} F exists in 𝒞. By definition, lim_{⟵_{I}} F is the limit lim_{I}F.

4.3

If A ∈ Ob(𝒞) then the commutative square

\xymatrix{A & A \ar[l]_{1} \\ A \ar[u]^{1} & A \ar[l]^{1} \ar[u]_{1}}

will be called a trivial square.

A class 𝒮(𝒞) of commutative squares

\xymatrix{A & B \ar[l] \\ C \ar[u] & D \ar[l] \ar[u]}

in 𝒞 is called a class of structure squares, if it contains all trivial squares and is closed under isomorphism of commutative squares.

A class of 𝒢𝒮(𝒞) ⫅ 𝒮(𝒞) of commutative squares in 𝒞 is called a class of generating squares for 𝒮(𝒞) , if it is closed under isomorphism of commutative squares, and every square in 𝒮(𝒞) is an inverse limit in a prescribed way of squares in 𝒢𝒮(𝒞). This means that we have a class ℱ𝒢𝒮(𝒞) of functors S whose source categories are inverse directed, preordered sets J, whose values are in 𝒢𝒮(𝒞), and whose inverse limits lim_{⟵_{J}} S exist and are in 𝒮(𝒞) and

every square in 𝒮(𝒞) is some inverse limit lim_{⟵_{J}} S.

4.4

A category with costructure is a quadruple (𝒞, 𝒱(𝒞), co ℐ(𝒞), 𝒮(𝒞) ) consisting of a category 𝒞, a class 𝒱(𝒞) of virtual isomorphisms, a class coℐ(𝒞) of coinfrastructure functors, and a class 𝒮(𝒞) of structure squares. A functor or morphism ℱ : (𝒞, 𝒱(𝒞), coℐ(𝒞), 𝒮(𝒞)) → (𝒟, 𝒱(𝒟), coℐ(𝒟), 𝒮(𝒟)) of categories with costructure is a functor ℱ : 𝒞 → 𝒟 which preserves virtual isomorphisms, coinfrastructure functors, and structure squares. This means that • if f is a virtual isomorphism then so is ℱ(f),

• if F : I → 𝒞 is a coinfrastructure functor then so is the composition

ℱF : I → 𝒟 and the canonical morphism lim_{⟵_{I}} ℱF → ℱ (lim_{⟵_{I}} F ) is an

isomorphism,

• if

\xymatrix{A & B \ar[l] \\ C \ar[u] & D \ar[l] \ar[u]}

is a structure square in 𝒞 then

\xymatrix{ℱ(A) & ℱ(B) \ar[l] \\ ℱ(C) \ar[u] & ℱ(D) \ar[u] \ar[l]}

is one in 𝒟.

4.5

Let \underline{𝒞} = (𝒞, 𝒱(𝒞), coℐ(𝒞), 𝒮(𝒞)) denote a category with costructure. Let ∞ denote the first infinite ordinal. Let x = s, t, or ϕ be as in (4.1). The concepts of an x−dimension function on \underline{𝒞} and of a dimension function on \underline{𝒞} are defined exactly as in (2.5) for a category with structure. This means that dualizing interchanges s-dimension and t-dimension functions, whereas dimension functions remain dimension functions. A category with costructure and x-dimension is a pair (\underline{𝒞}, d) consisting of a category with costructure \underline{𝒞} and an x-dimension function d on \underline{𝒞}. A functor or morphism ℱ : (\underline{𝒞}, d) → (\underline{𝒟}, e) of categories with costructure and x-dimension is a functor \underline{𝒞} → \underline{𝒟} of categories with costructure such that d ≧ eℱ.

The most important concepts for an x-dimension function are the notions of type and coreduction. The problems we can solve using dimension theory depend on these notions. They are discussed next.

4.6

As in (2.6), a type t is a triple t_{B}, t_{C}, t_{D} : ℤ^{≥0} ∪ {∞} → ℕ ∪ {∞, ∞ + 1} of functions and is usually written in the form

t = \binom{t_{B}}{t_{C} t_{D}} .

Let \underline{𝒞} be a category with costructure. Let A ∈ Ob(𝒞). A coreduction of A consists of

(i)

\{ \begin{matrix}a morphism A → A′ ∈ 𝒱(𝒞), if x = s. \\ a morphism A′ → A ∈ 𝒱(𝒞), if x = t \\ a virtual chain equivalence from A to A′, if x = ϕ \end{matrix} \} and

(ii) an infrastructure functor B : I → 𝒞, i ↦ B_{i}, and

(iii) a set

\xymatrix{A′ & B_{i} \ar[l] \\ C_{i} \ar[u] & D_{i} \ar[l] \ar[u]} (i ∈ I)

of costructure squares indexed by I (C and D are not necessarily functorial in I) with the property that for each relationship (i ≦ j) ∈ I, the triangle

\xymatrix{ & B_{j} \\ A′ \ar[ur] \ar[dr] \\  & B_{i} \ar[uu]}

commutes.

Let d be an x-dimension function on \underline{𝒞}. Let t be a type and n ∈ ℕ. A coreduction of A is said to be a d-coreduction of type t at n, if in addition to the above

(iv) d (lim_{⟵_{I}} B) < t_{B}(n)

and for each i ∈ I,

(v) d(C_{i}) < t_{C}(n), and

(vi) d(D_{i}) < t_{D}(n).

Suppose now that 0 < d(A) < ∞. Then a d-coreduction of type t at d(A) for A will be simply called a d-coreduction of type t for A. An x-dimension function d on \underline{𝒞} is said to have type t, if each A ∈ Ob(𝒞) such that 0 < d(A) < ∞ has a d-coreduction of type t. An x-dimension function can have many types and obviously if it has type t and t ≦ t′ then it also has type t′. The applicability of a given x-dimension function d to a given problem depends on the minimal types of d.

4.7

Next we generalize the notion of a category with costructure to that of a category with global coreduction and define the notion of a dimension function on such categories, generalizing the existing notion on categories with costructure.

A category with global coreduction consists of a category with costructure \underline{𝒞} and for each A ∈ Ob(𝒞) a class gcrd(A) of sets X such that each member of X is a coreduction of A. Each member X ∈ gcrd(A) is called a global coreduction of A. It will be assumed that if X ∈ gcrd(A) and Y is a set of coreductions of A such that each coreduction in X is isomorphic (in the straightforward sense) to a coreduction in Y and conversely then Y ∈ gcrd(A). In this case, we say that X is isomorphic to Y. A category with global coreduction will be denoted by a pair

(\underline{𝒞}, gcrd)

where gcrd assigns to each A ∈ Ob(𝒞) a class gcrd(A) of global coreductions of A, which is closed under isomorphism. We explicitly allow the possibility that for some A′s, gcrd(A) = ϕ.

Let ℱ : \underline{𝒞} → \underline{𝒞}′ be a functor of categories with costructure. It follows from the definition of such functors that ℱ takes any coreduction of an object A of 𝒞 to a coreduction of ℱ(A). If R is a coreduction of A, let ℱ(R) denote the corresponding coreduction of ℱ(A). If X is a global coreduction of A in (\underline{𝒞}, gcrd), let ℱ(X) = \{ℱ(R) | R ∈ X\}. We say that ℱ defines a functor or morphism (\underline{𝒞}, gcrd) → (\underline{𝒞′}, gcrd′) of categories with global coreduction, if for each A ∈ Ob(𝒞) and X ∈ gcrd(A), ℱ(X) ∈ gcrd′(ℱ(A)).

4.8

Let x = s (source), t (target), or ϕ (no condition), be as in (2.1). An x -dimension function d on (\underline{𝒞}, gcrd) is by definition an x-dimension function on \underline{𝒞}, as in (2.5). A category with global coreduction and xdimension is a pair ((\underline{𝒞}, gcrd), d) consisting of a category with global coreduction (\underline{𝒞}, gcrd) and an x-dimension function d on (\underline{𝒞}, gcrd). A functor or morphism ℱ : ((\underline{𝒞}, gcrd), d) → ((\underline{𝒞′}, gcrd′), d′) of categories with global coreduction and x-dimension is a functor ℱ : (\underline{𝒞}, gcrd) → (\underline{𝒞′}, gcld′) of categories with global coreduction such that d ≧ d′ℱ.

As in sections 2 and 3, the most important properties of x-dimension functions are described using the notions of type and global coreduction. This is done in the following paragraphs.

4.9

Let d be an x-dimension function on the category with global coreduction (\underline{𝒞}, gcrd). Let t denote a type, as in (4.6), and let n ∈ ℕ. A global coreduction X ∈ gcrd(A) is called a global d-coreduction of type t at n, if X ≠ ϕ and each coreduction in X is a d-coreduction of type t at n, as in (4.6) (iv) - (vi). A global d-coreduction of type t at A is by definition an X ∈ gcrd(A), which is a global d-coreduction of type t at d(A). Note that this definition makes sense only when d(A) > 0, because t is defined on ℕ ∪ {∞}. An x-dimension function d is said to have type t, if each A ∈ Ob(𝒞) such that 0 < d(A) < ∞ has a global d-coreduction of type t at A.

An x-dimension function on (\underline{𝒞}, gcrd) can have many types and obviously if it has type t and t ≦ t′ then it also has type t′. The applicability of a given x-dimension function d to a given problem depends on the minimal types of d.

For the rest of this section, let

\overline{𝒞} = (\underline{𝒞}, gcrd)

denote a category with global coreduction. We continue developing the properties of x-dimension functions.

4.10

Let d be an x-dimension function of type t on the category with global coreduction \overline{𝒞}. Let A ∈ Ob (𝒞) and n ∈ ℕ. Even if A has a global dcoreduction of type t at n, it does not follow that d(A) ≦ n. However if the conclusion above is true for any A and any n then we shall say that d is a tame x-dimension function of type t.

4.11

Let \overline{𝒞} be a category with global coreduction. Define x − dim(\overline{𝒞}) to be the partially ordered class of all x-dimension functions on \overline{𝒞}, with partial ordering given by ≦. If t is a type, let x − dim(\overline{𝒞}, t) denote the partially ordered subclass of x − dim(\overline{𝒞}) consisting of all x-dimension functions of type t.

4.12

The next result is the dual of Theorem 3.6.

Theorem The partially ordered class x−dim(\overline{𝒞}) has arbitrary greatest lower bounds. If S ⫅ x − dim(\overline{𝒞}) is a nonempty subclass then its greatest lower bound glb_{S} is computed by the formula

glb_{S}(A) = inf{d(A) | d ∈ S}.

Moreover if S ⫅ x − dim(\overline{𝒞}, t) is a nonempty subclass then glb_{S} ∈ x − dim(\overline{𝒞}, t).

4.13

Let next result is the dual of Theorem 3.7.

Theorem Let d be an x-dimension function of type t on the category with global coreduction \overline{𝒞}. Then there is a tame x-dimension d_{t} of type t, called the tame closure of d at t, which is the unique maximum among all tame xdimension functions d′ of type t such that d′ ≦ d. Moreover d_{t} is computed according to the following rule: Let T (d) = {d′ | d′ a tame x-dimension function of type t, d′ ≦ d} and S(d) = {d′′ | d′′ an x-dimension function of type t, d′′ ≦ d, d′ ≦ d′′ ∀ d′ ∈ T (d)}. Then

d_{t} = glb_{S(d)}.

4.14

Let \overline{𝒞} be a category with global coreduction. Let t be a type. An x-dimension function δ of type t is called universal of type t, if δ ≦ d for any x-dimension function d of type t, whose 0-dimensional objects are also 0-dimensional under δ.

4.15

The next result is the dual of Corollary 3.9.

Corollary Let \overline{𝒞} be a category with global coreduction. Let 𝒞^{0} be a class of objects of 𝒞, which is x-closed under virtual isomorphism. Then there is a unique universal x-dimension function of type t, whose 0-dimensional objects are precisely those of 𝒞^{0}. Furthermore it is tame. This function is often denoted by

d_{\overline{𝒞} ,𝒞^{0},t}.

4.16

The next result is the dual of Theorem 3.10.

Theorem Let \overline{𝒞} be a category with global coreduction. Let t be a type such that given X ∈ {A, B, C} either t_{X} ≦ id or t_{X} = ∞ + 1. Let d be a tame x-dimension function of type t on \overline{𝒞}. Then d is universal of type t and is computed by the following rule: Define recursively disjoint, x-closed classes 𝒞^{n} as follows. Let 𝒞^{0} = {A ∈ Ob(𝒞) | d(A) = 0}. Let n ∈ ℕ and assume that disjoint, x-closed classes 𝒞^{m} have been defined for all 0 ≦ m < n. Define δ^{n−1} : Ob(𝒞) → ZZ^{≥0} ∪ {∞} by

δ^{n−1}(A) = \{ \begin{matrix}m & if & A ∈ 𝒞^{m} & where 0 ≦ m < n, \\ ∞ & if & A ∉ ∪_{m=0}^{n−1}𝒞^{m}. \end{matrix}

Obviously δ^{n−1} is an x-dimension function. Define 𝒞^{n} = {A ∈ Ob(𝒞) | A ∉

∪_{m=0}^{n−1}𝒞^{m}, A has a δ^{n−1}-reduction of type t at n}. Define 𝒞^{∞} = Ob(𝒞)\ ∪_{n∈ZZ^{≥0}}

𝒞^{n}. Then for any A ∈ Ob(𝒞) and any n ∈ ZZ^{≥0} ∪ {∞},

d(A) = n ⇔ A ∈ 𝒞^{n}.

Furthermore if n ∈ ZZ^{≥0} and 𝒞^{n} = ϕ then 𝒞^{n′} = ϕ for all n′ ≥ n, i.e. d = δ^{n−1}.

4.17

Let (\overline{𝒞}, d) be a category with global coreduction and d an x-dimension function on \overline{𝒞}. For n ∈ ZZ^{≥0} ∪ \{∞\}, define the classes

𝒞_{d}^{n} = {A ∈ Ob(𝒞) | d(A) = n}
𝒞_{d}^{≦n} = {A ∈ Ob(𝒞) | d(A) ≦ n}.

4.18

The next result is the dual of Lemma 3.12.

Lemma The classes above are x-closed under virtual isomorphisms and Ob(𝒞) =

∪_{n=0}^{∞}𝒞_{d}^{n}. Conversely any partition of Ob(𝒞) as a disjoint union Ob(𝒞) = ⋃_{n=0}^{∞_{•}}𝒞^{n}

of x-closed classes 𝒞^{n} defines an x-dimension function d on \overline{𝒞} such that 𝒞_{d}^{n} = 𝒞^{n}. Partially order the members of {x − closed classes in Ob(𝒞)} by inclusion. Then the function \{x-dimension functions of type t on \overline{𝒞}\} → {x-closed classes in Ob(𝒞)}, d ↦ 𝒞_{d}^{0}, preserves partial orderings and induces a bijection between universal x-dimension functions of type t and xclosed classes in Ob(𝒞). If (\overline{𝒟}, e) is a category with global coreduction and x-dimension and ℱ : \overline{𝒞} → \overline{𝒟} is a functor of categories with global coreduction then ℱ defines a morphism (\overline{𝒞}, d) → (\overline{𝒟}, e) of categories with global coreduction and x-dimension ⇔ ℱ(𝒞_{d}^{≤n}) ⫅ 𝒟_{e}^{≦n} for all n ∈ ZZ^{≥0}. Furthermore if d = d_{\overline{𝒞},𝒞^{0},t} and e = d_{\overline{𝒟} ,𝒟^{0},t} for some x-closed classes 𝒞^{0} ⊆ Ob(𝒞) and 𝒟^{0} ⊆ Ob(𝒟) then ℱ induces a morphism of categories with global coreduction and x-dimension ⇔ ℱ(𝒞^{0}) ⫅ 𝒟^{0}.

4.19

Let \overline{𝒞} be a category with global coreduction. If S ⫅ x − dim(\overline{𝒞}), define the partially ordered set type(S) = {types t | each d ∈ S has type t }, with partial ordering that of types.

Since the results we prove for group valued functors are formulated using the concept of type for x-dimension functions, it is useful to have some general guidelines how large the set type(S) above is, in particular when S has just one element. The following lemmas provide such information.

4.20

The next lemma is the dual of Lemma 3.14.

Lemma The partially ordered set type(S) has the property that every element is bounded below by a minimal element. Furthermore if t ∈ type(S) and t′ is a type such that t′ ≥ t then t′ ∈ type(S). Thus the minimal elements of type(S) define it uniquely.

4.21

The next lemma is the dual of Lemma 3.15.

Lemma Let ℱ : \overline{𝒞} → \overline{𝒟} be a functor of categories with global coreduction. Let T ⫅ x − dim(\overline{𝒟}). Define S = \{dℱ | d ∈ T \}.

(i) Then S ⫅ x − dim(\underline{𝒞}) and type(S) ⫅ type(T).

Suppose that for each A ∈ Ob(𝒞), the function gcrd\overline{𝒞}(A) → gcrd\overline{𝒟} (A) induced by ℱ is surjective up to isomorphism of global coreductions in glrd\overline{𝒟} (A). Then type(T) ⫅ type(S).