Theory and Applications of Categories, Vol. 10, No. 15, 2002, pp. 392–409.

THE CYCLIC SPECTRUM OF A BOOLEAN FLOW

JOHN F. KENNISON

ABSTRACT. This paper defines flows (or discrete dynamical systems) and cyclic flows in a category and investigates how the trajectories of a point might approach a cycle. The paper considers cyclic flows in the categories of Sets and of Boolean algebras and their duals and characterizes the Stone representation of a cyclic flow in Boolean algebras. A cyclic spectrum is constructed for Boolean flows. Examples include attractive fixpoints, repulsive fixpoints, strange attractors and the logistic equation.

Introduction

Suppose we have a flow in Sets by which we mean a set S with a function t : S → S. We might think of S as the set of all possible states of a system and t(s) as the state the system will be in one time unit after being in state s. We are interested in the long-term future of a system that starts out in state s. If S has no further structure, then the successive states, {t^{n}(s)}, (called the “trajectory of s”) either repeat so that the system eventually cycles or the system takes on a sequence of distinct states.

But suppose S has more structure, such as a topological structure. In this case, the trajectory of s might approach a limiting state, or a limiting cycle, or a strange attractor or become chaotic. To account for this possible structure on S, we define a flow in a category and define when a flow is cyclic.

After some general observations, we focus on flows in Sets, the category of Sets, and Bool, the category of Boolean algebras, and their duals. We show that cyclic flows in Bool have special properties and show how any flow in this category can be broken down into its cyclic parts by constructing the cyclic spectrum (which is an example of a Cole spectrum, see [2]).

There is a one-to-one correspondence between flows in Bool and in the dual category Stone of compact, Hausdorff, totally disconnected spaces. But the notion of a cyclic flow in a category is not self-dual, so there are two different ways a flow in these categories can be cyclic. For example, in Stone we define what is meant by a nearly cyclic flow and, if not finite, such a flow is cyclic as a flow in Bool but not in Stone. It essentially corresponds to a “strange attractor”.

For applications we would like to examine flows on a set S which has a Stone topology. In many well-known examples, S has a topology but it is usually connected. However, as shown in Example 10 we can often find an associated Stone topology.

Received by the editors 2002 August 5.

Transmitted by Michael Barr. Published on 2002 August 28.

2000 Mathematics Subject Classification: 18B25, 37B99.

Key words and phrases: flow, discrete dynamical system, topos, Cole spectrum, strange attractor.

\begin{matrix}c \\ ◯ \end{matrix} John F. Kennison, 2002. Permission to copy for private use granted.

Much of this paper should be of interest to dynamical system theorists who know a little category theory as found in [5] or even the first chapter of [1]. Further concepts are in [1, 2, 3, 6], while [7, 8] give information about dynamical systems.

1. Flows in a Category

In this section we define flows and cyclic flows in a category.

Definition 1. Let C be a category. Then (X, t) is a flow (or discrete dynamical system) in C if X is an object of C and t : X → X is a morphism.

If (A, s) and (B, t) are flows in C, then h : A → B is a flow homomorphism from (A, s) to (B, t) if hs = th.

Definition 2. A category C is well-behaved if it is complete and co-complete and if every morphism f can be written as f = me where m is a mono and e is an extremal epi (meaning that whenever e = gh with g mono then g is an isomorphism.)

About well-behaved categories. The following notation and elementary properties for well-behaved categories is either fairly standard or well-known or readily proven:

(WB1) If A and B are objects of C, then A ⊆ B indicates that A is a subobject of B, meaning that there is an implicit mono from A to B.

(WB2) For f, g ∈ Hom(X, Y) we let Equ(f, g) ⊆ X denote the equalizer of f, g.

(WB3) If {A_{i} ⊆ B} for all i, then ⋁ A_{i} denotes the supremum of {A_{i}} in the partially ordered family of all subobjects of B.

(WB4) If f : X → Y and if A ⊆ X then then f(A) denotes the smallest subobject of Y through which A → X → Y factors.

(WB5) (Diagonal lemma) If se = mr with m mono and e extremal epi, then there exists a “diagonal” map d with de = r and md = s.

(WB6) Given f : X → Y and B ⊆ Y and {A_{i} ⊆ X} a set of subobjects of X with f(A_{i}) ⊆ B for all i, then f(⋁ A_{i}) ⊆ B.

Definition 3. Let (X, t) be a flow in a well-behaved category C. We say that (X, t) is cyclic if ⋁ Equ(t^{n}, Id_{X}) = X, the maximum subobject of X.

1.1. Lemma. Let h : (A, s) → (B, t) be a flow homomorphism and let E(n) = Equ(s^{n}, Id_{A}) and F (n) = Equ(t^{n}, Id_{B}). Then h(E(n)) ⊆ F (n).

Proof. Straightforward.

1.2. Lemma. Let C be a well-behaved category and let X = Colim {X_{i}} be a colimit in C with coprojections c_{i} : X_{i} → X. Let B ⊆ X be such that c_{i}(X_{i}) ⊆ B for all i. Then B = X, the maximum subobject of X.

Proof. Let m : B → X be the mono which represents B ⊆ X. Let d_{i} : X_{i} → B be such that md_{i} = c_{i} (which exist as c_{i}(X_{i}) ⊆ B). Because m is mono it is readily seen that the family {d_{i}} has the compatibility property which implies the existence of d : X → B with dc_{i} = d_{i} for all i.

It follows (using the colimit property) that md = Id_{X} and then (as m in mono) that dm = Id_{B} which shows that m is an isomorphism and so B represents the maximum subobject of X.

1.3. Proposition. In a well-behaved category, colimits of cyclic flows are cyclic.

Proof. Let {(X_{i}, t_{i})} be a small diagram of cyclic flows in the well-behaved category C. Let X = Colim {X_{i}} with coprojections c_{i} : X_{i} → X. Then it is readily shown that there is a unique t : X → X for which each c_{i} is a flow homomorphism. Let E_{i}(n) = Equ(t_{i}^{n}, Id_{X_{i}}) ⊆ X_{i} and let E(n) = Equ(t^{n}, Id_{X}) ⊆ X. Let B = ⋁ E(n), It suffices to show that B = X.

By the Lemma 1.1, we see that c_{i}(E_{i}(n)) ⊆ E(n) so c_{i}(E_{i}(n)) ⊆ B for all i. By (WB6) and Lemma 1.2, c_{i}(⋁{E_{i}(n)}) ⊆ B so c_{i}(X_{i}) ⊆ B for all i and so B = X.

1.4. Proposition. In a well-behaved category, extremal quotients of cyclic flows are cyclic.

Proof. Let (A, s) be a cyclic flow and let the extremal epi h : A → B be a flow homomorphism. We must show that (B, t) is cyclic. Let E(n) = Equ(s^{n}, Id_{A}) and F (n) = Equ(t^{n}, Id_{B}). It suffices to show that ⋁ F (n) = B. But since ⋁ E(n) = A it readily follows, using 1.1 and (WB6), that h factors through ⋁ F (n) and an extremal epi can only factor through the maximum subobject.

Example 1. In Sets, the category of Sets, (X, t) is a cyclic flow iff X is a disjoint union of finite cycles, where a finite cycle is a set {x_{0}, x_{1}, …, x_{n−1}} where t(x_{i}) = x_{i+1} where i + 1 is computed mod n. (We then say that x is a periodic point with period n.)

Example 2. Let t : X → X be a function on a set X. Then, regarded as a flow in Sets^{op}, (X, t) is cyclic iff t is one-to-one.

Proof. Let Q(n) be the equivalence relation on X generated by all pairs of the form (x, t^{n}(x)). Then X is cyclic iff (x, y) ∈ ⋂ Q_{n} implies x = y. Assume X is cyclic and that t(x) = t(y). Then (x, y) ∈ Q(n), for all n, as (x, t^{n}(x)), (y, t^{n}(y)) ∈ Q(n) and t^{n}(x) = t^{n}(y). So x = y and t is one-to-one.

Conversely, assume the t is one-to-one and that (x, y) ∈ Q(n) for all n. Since t is one-to-one, it can readily be shown that (x, y) ∈ Q(n) iff x = y or x = t^{k}(y) with k a multiple of n or y = t^{m}(x) with m a multiple of n. Now if x = t^{m}(y) for a unique m. then (x, y) cannot be in Q(n) once n > m. On the other hand, if x = t^{m}(y) for more than one m, then x, y are in a cyclic orbit of some period p (as t is one-to-one) and (x, y) ∈ Q(p) implies x = y.

Notation. The categories Bool and Stone were defined in the introduction. It is readily shown that both categories are well-behaved with the monos being the one-to-one maps and the extremal epis being just the epis which are the onto maps.

It is well-known that Bool and Stone are dual to each other, see [3]. We denote the contravariant equivalences between these categories by:

Clop : Stone → Bool and Pts : Bool → Stone So Clop(X) is the Boolean algebra of clopens of X and Pts(B) is the Stone space of

all points of B and is equivalent to the prime ideal spectrum of B (the Zariski and Patch topologies coincide).

If (X, t) is a flow in Stone then (B, τ) is a flow in Bool where B = Clop(X) and τ = Clop(t). We will often write (B, τ) = Clop(X, t)

Definition 4. (X, t) is a Boolean cyclic flow in Stone if (X, t) is a flow in Stone such that (B, τ) = Clop(X, t) is a cyclic flow in Bool.

Example 3. Let (X, t) be a flow in Stone. Then (X, t) is cyclic in Stone iff the set of periodic points of X is dense, where x ∈ X is periodic iff t^{n}(x) = x for some n ∈ N.

On the other hand, (X, t) is Boolean cyclic iff every b ∈ B is periodic under τ where (B, τ) = Clop(X, t).

Example 4. The group of p-adic integers, Ẑ_{p} = Lim Z_{p^{n}}, with t(z) = z + 1, is Boolean cyclic but is not cyclic in Stone.

Proof. It is not cyclic as a flow in Stone because it has no periodic points. In Stone, Ẑ_{p} is the limit of cyclic flows. So Clop(Ẑ) is the corresponding colimit in Bool which is cyclic by Proposition 1.3.

Remark. It is instructive to see why Ẑ_{p} of the above example is Boolean cyclic. Let b be a clopen of Ẑ_{p}. Note that finite intersections of sets of the form π_{p^{n}}^{−1}(a) for n ∈ N, a ∈ Z_{p^{n}} form a base for the topology on X. Since b is open, b is a union of such basic open sets and since b is closed, and therefore compact, b is a finite such union. So b depends only on finitely many coordinates, which means there exist k = p^{n} with b = π_{k}^{−1}(π_{k}(b)) and so τ^{k}(b) = b. Ẑ_{p} is an example of a “nearly cyclic” flow, see the remark following Definition 5 in the next section.

Example 5. Let X = ∏{Z_{n} : n ∈ N} where Z_{n} is the cyclic group of integers mod n. Then X has a compact product topology and (X, t) is a flow in Stone where t(x) = x + 1 (where 1 is the element of X which projects onto the image of 1 in each Z_{n}). Then X is not at all cyclic in Stone (it has no finite cycles) but X is Boolean cyclic.

Proof. By duality, Clop(X) is the coproduct of the cyclic flows Clop(Z_{n}). The result follows from Proposition 1.3.

Example 6. Let X be the coproduct in Stone of Z_{p} for p prime and let t(x) = x + 1 on each component. Then X is cyclic in Stone but not Boolean cyclic. (But see Example Proof. As a coproduct of Stone cyclic flows, X is Stone cyclic, by 1.3. Note that the coproduct is the Stone-Čech compactification of the discrete union of the Z_{p}’s. But the clopens of X coincide with the subsets of this discrete union (as both represent maps to the two-point Stone space). From this it is easy to see that X is not Boolean cyclic.

2. The Stone Space of a Cyclic Boolean Algebra.

In this section, we explore and characterize those flows (X, t) in Stone which are Boolean cyclic. Aside from the characterization result (Theorem 2.13) we show that, for a Boolean cyclic flow in Stone, the closure of the trajectory of a single point must be “nearly cyclic”.

Notation and Remark. In what follows: (X, t) will be a flow in Stone with (B, τ) = Clop(X, t). If B is (Boolean) cyclic, then τ is one-to-one and onto, so t : X → X must be one-to-one and onto also.

When (X, t) is a flow, we will often use just X, instead of (X, t), referring implicitly to the flow map t. Similarly, we may use B as short for (B, τ).

The Profinite Integers. We let Ẑ denote the profinite integers, or the free profinite group on one generator, 1. We regard Ẑ as a flow in Stone with t(ζ) = ζ +1 for all ζ ∈ Ẑ. (Most of the profinite groups we work with also have a ring structure and we let 1 denote the unit. When there are obvious maps from the integers, Z, then “1” denotes the image of 1 in Z.) Clearly the ring of integers Z is a subring of Ẑ and we will identify Z with its image in Ẑ.

By the Chinese Remainder Theorem, Ẑ = ∏{Ẑ_{p}: p prime}, the product of the p-adic integers as p varies over the primes of N. For every n ∈ N let q_{n} : Ẑ → Z_{n} be the continuous homomorphism which preserves the generator 1. It readily follows that if n divides m and if f : Z_{m} → Z_{n} is the homomorphism which preserves 1, then fq_{m} = q_{n}.

Note that Ẑ is Boolean cyclic by Proposition 1.3 because it is a limit of cyclic flows in Stone, so Clop(Ẑ) is a colimit of cyclic flows in Bool.

If ζ ∈ Ẑ is given, we will say that “ζ ≅ k (mod n)” if q_{n}(ζ) = q_{n}(k).

2.1. Theorem. Let (B, τ) be a cyclic Boolean flow. Then there is a continuous group action of Ẑ on B such that for n ∈ N and b ∈ B the action maps (n, b) to τ^{n}(b). This determines the action and τ^{ζ}(b) will denote the result of applying the action to (ζ, b).

Proof. Let ζ ∈ Ẑ and b ∈ B be given. Since B is cyclic, we can choose n ∈ N such that τ^{n}(b) = b and define τ^{ζ}(b) = τ^{k}(b) where ζ ≅ k (mod n). We claim that this is well-defined. Suppose τ^{n}(b) = τ^{m}(b) = b. Let s = lcm(n, m) and let f_{n} : Z_{s} → Z_{n} and f_{m} : Z_{s} → Z_{m} be the maps which preserve 1. Then f_{n}q_{s} = q_{n} and f_{m}q_{s} = q_{m}. Let ζ ≅ k (mod n) and ζ ≅ j (mod m).

We must show that τ^{k}(b) = τ^{j}(b). But let ζ ≅ h (mod s). Then (by considering the projections of ζ) we get h ≅ j (mod m) so τ^{h}(b) = τ^{j}(b) (as τ^{m}(b) = b) and, similarly, τ^{h}(b) = τ^{k}(b) so τ^{j}(b) = τ^{k}(b), proving that the action is well-defined.

We note that each map τ^{ζ} for ζ ∈ Ẑ is a Boolean homomorphism, because given any finite subset of B we can find n such that τ^{n}(b) = b for all b in the finite subset, so there is a k such that τ^{ζ} = τ^{k} on the entire finite subset. Since τ^{k} is a Boolean homomorphism, it follows that τ^{ζ} is too.

It is immediate that this is a group action on B so it remains to show that it is continuous (when B is given the discrete topology). Now suppose that τ^{ζ}(b) = c. Let n be chosen so that τ^{n}(b) = b. Then W = p_{n}^{−1}(p_{n}(ζ)) is open and τ^{ζ′}(b) = c for all ζ′ ∈ W. This proves continuity.

2.2. Theorem. Let B be a cyclic Boolean flow and let X be the corresponding flow in Stone. Then Ẑ acts continuously on X.

Proof. For each ζ ∈ Ẑ we have a Boolean homomorphism τ^{ζ} on B. By duality, there is a corresponding continuous map t^{ζ} on X. It also easily follows that this defines a group action of Ẑ on X. (A left group action on an object is the same as a right group action in the dual category, but since Ẑ is abelian there is no real difference between left and right actions.) It remains to show that this action is continuous, using the given Stone topology on X.

Let ζ ∈ Ẑ and x_{0} ∈ X be such that t^{ζ}(x_{0}) = y_{0}. Let b be a clopen neighborhood of y_{0}. We must find a neighborhood c of x_{0} and a neighborhood W of ζ such that the action maps W × c into b. Since t^{ζ} is continuous, (t^{ζ})^{−1}(b) = c is a clopen neighborhood of x_{0}. So τ^{ζ}(b) = c (recall that τ = t^{−1}). By the above proof, there exists a neighborhood W of ζ such that for ζ′ ∈ W we have τ^{ζ′}(b) = c. Then W and c have the required properties.

Definition 5. Let (X, t) be a flow in Stone. Then X is nearly cyclic if for every p, q ∈ X and every neighborhood U of q, there exists a clopen b with q ∈ b ⊆ U such that the sequence {t^{m}(p)} is periodically in b, meaning that for some n there is a non-empty subset {m_{1}, … m_{i} …} ⊆ {0, 1, … n − 1} with t^{m}(p) ∈ b iff m ≅ m_{i} (mod n) for some i.

Remark. Conceptually, a nearly cyclic flow is one which looks cyclic when we observe how often it visits a small neighborhood b. But when we observe more closely, using a smaller neighborhood, the period might change. Often it is doubled or multiplied by some prime p.

In effect, a nearly cyclic flow is a “strange attractor” because every trajectory comes close, infinitely often, to every point in the attractor and it is close to being cyclic (in the sense mentioned above) yet is not cyclic.

2.3. Lemma. A nearly cyclic flow in Stone is Boolean cyclic.

Proof. Let (X, t) be a nearly cyclic flow in Stone and let (B, τ) = Clop(X, t). We say that b has period n with respect to p ∈ X when t^{k}(p) ∈ b iff t^{k+n}(p) ∈ b. We claim that this implies that τ^{n}(b) = b. For let U = (t^{−1})^{n}(b) + b (where “+” denotes the Boolean addition, or symmetric difference). We need to show that U is empty, so suppose that q ∈ U. Then by the definition of nearly cyclic, there exists b′ ∈ B with q ∈ b′ ⊆ U such that the sequence of points t^{k}(p) frequently visits b′. But it is clear by the periodic property of b that no such power of p can be in U, let alone b′.

Now let W be any clopen of X. Clearly if q ∈ W there exists b with q ∈ b ⊆ W where b has the periodic property (with respect to any p) so τ^{n}(b) = b for some n. Thus W is a union of such periodic clopens b. But W is closed and therefore compact so W is a finite union of periodic elements and such a union is periodic, because the Boolean homomorphism τ, preserves finite unions.

Definition 6. If (X, t) is a flow in Stone and p ∈ X, we let T (p) denote the closure of the trajectory of p. We also call T (p) the closed trajectory of p.

We further say that p is a generating point of X if T (p) = X.

2.4. Lemma. Let (X, t) be a Boolean cyclic flow in Stone with a generating point p. If W is a clopen subset of X with t^{−1}(W) = W then W is either empty or all of X.

Proof. First we claim that t(u) ∈ W iff u ∈ W. Note that t(u) ∈ W iff u ∈ t^{−1}(W) iff u ∈ W (since W = t^{−1}(W)).

Now suppose W is non-empty. Since T (p) = X, the elements {t^{n}(p)} are dense in X so they meet every non-empty open set. Therefore there exists n with t^{n}(p) ∈ W. It follows from the above claim and induction that p ∈ W and then that every t^{k}(p) ∈ W. So W contains the trajectory of p and, since W is closed, the closed trajectory, T (p) of p. But T (p) = X as p is a generating point.

2.5. Proposition. A flow in Stone with a generating point is nearly cyclic iff it is Boolean cyclic.

Proof. Lemma 2.3 gives one direction. Conversely, let (X, t) be a Boolean cyclic flow in Stone with a generating point. Let q, r be in X and let U be a clopen neighborhood of r. We claim there exist k with t^{k}(q) ∈ U. Since Clop(X, t) is Boolean cyclic, there exists n ∈ N with τ^{n}(U) = (t^{−1})^{n}(U) = U. Let U_{k} = τ^{k}(U) for k = 1, …, n. Let W = ∪U_{k}. So W is clopen. Also τ(W) = W as the Boolean homomorphism τ preserves finite unions and permutes the U_{i}’s. Since r ∈ W we see, by the above lemma that W = X. So q ∈ W and so q ∈ τ^{k}(U) for some k and so t^{k}(q) ∈ U. Also q is periodically in U as τ^{n}(U) = U.

2.6. Lemma. Boolean cyclic flows in Stone are closed under the operations of taking subflows and quotient flows.

Proof. Cyclic flows in Bool are obviously closed under quotient flows and subflows so the result follows by duality.

Definition 7. Let (X, t) be a flow in Stone. By the t-action of N on (X, t) we mean the mapping N × X → X which maps (n, x) to t^{n}(x).

We say that (X, t) admits a continuous t-action by Ẑ if there is a continuous group action of Ẑ on X which extends the t-action of N. Clearly such an action is unique if it exists as N is dense in Ẑ. By Theorem 2.2, every Boolean cyclic flow in Stone has such a continuous action by Ẑ.

When (X, t) does admits a continuous t-action by Ẑ we let t^{ζ}(x) denote the result of applying the action to (ζ, x).

2.7. Lemma. Let (X, s) and (Y, t) be flows in Stone which admit continuous t-actions by Ẑ. If h : X → Y is a continuous homomorphism, then h preserves the actions of Ẑ on X and Y.

Proof. Given p ∈ X, we need to show that h(s^{ζ}(p)) = t^{ζ}(h(p)) for all ζ ∈ Ẑ. But, as h is a flow homomorphism, this is true for ζ ∈ N and the result follows as h is continuous and N is dense in Ẑ.

2.8. Lemma. Let (X, t), a flow in Stone, admit a continuous t-action by Ẑ. Then the orbit of a point under the action of Ẑ is the closed trajectory of that point under the action of t. Also, each closed trajectory T (p) is Boolean cyclic.

Proof. Let us say that the “orbit” of p under Ẑ is the set {t^{ζ}(p) : ζ ∈ Ẑ} while the “trajectory” of p under t is the set {t^{n}(p) : n ∈ N}. Then T (p) is the closure of the trajectory of p under t.

The orbit of p under Ẑ contains T (p) because it contains the trajectory and is closed (as a compact set).

Conversely, since N is dense in Ẑ we see that the orbit of p under Ẑ is contained in T (p), the closure of the trajectory under t.

Finally, Ẑ is Boolean cyclic by Proposition 1.3 as it is a limit (in Stone) of cyclic flows (hence a colimit in Bool). Therefore T (p) is Boolean cyclic by Proposition 1.4.

2.9. Corollary. Each trajectory T (p) of a Boolean cyclic flow in Stone is a quotient of Ẑ by a subgroup where the identity maps to p.

2.10. Lemma. Let (X, t), a flow in Stone, admit a continuous t-action by Ẑ. Then:

(1) X has a generating point iff every point of X is generating.

(2) Two closed trajectories, T (p) and T (q) overlap iff they coincide.

Proof.

(1) If X has a generating point, then X is Boolean cyclic by 2.8. The result now follows from the proof of Proposition 2.5, which showed that every neighborhood of r contains points of the form t^{k}(q) where q, r were arbitrary points of X.

(2) Suppose that r ∈ T (p) ∩ T (q). Then by (1), T (p) and T (q) are both generated by r so T (p) = T (r) = T (q).

Definition 8. Let (X, t) be a flow in Stone. We say that p ∼ q if T (p) ∩ T (q) is non-empty. We say that X is regular if ∼ is a closed equivalence relation on X (that is, an equivalence relation which is closed as subset of X × X.)

If X is regular, we let X/∼ denote the set of all ∼-equivalence classes and give it the quotient topology (induced by the obvious map X → X/∼).

Clearly X/∼ is the set of closed trajectories of X and, as noted below, it has a compact Hausdorff topology.

Remark. For compact Hausdorff spaces, the quotient topology induced by a closed equivalence relation is always compact Hausdorff. While this result is well-known (it is part of the theorem that compact Hausdorff spaces are equational over Sets) the proof is often abbreviated or omitted in standard references. So, for the record, here is a proof, supplied to me by Michael Barr:

2.11. Proposition. The quotient of a compact Hausdorff space by a closed equivalence relation is compact and Hausdorff.

Proof. Let X be a compact Hausdorff space and let E ⊆ X ×X be a closed equivalence relation. Let q : X → X/E be the quotient map and give X/E the quotient topology. Clearly X/E is compact, so it remains to show that it is Hausdorff.

We claim that q is a closed mapping. Let A ⊆ X be a closed subset. To prove that q(A) is closed in the quotient topology, we need to show that q^{−1}(q(A)) is closed. Regard A × X and E as closed subsets of X × X and let F = E ∩ (A × X). Then F is clearly closed and so is compact, so p_{2}(F) is a closed subset of X where p_{2} : X × X → X is the second projection. But p_{2}(F) = q^{−1}(q(A)).

Now let c, d be distinct points of X/E. Choose x, y in X with q(x) = c, q(y) = d. Then, since {x}, {y} are closed subsets so are {c}, {d} (as q is closed). We need to find disjoint open neighborhoods in X/E of c, d. But this is equivalent to finding closed subsets V, W of X/E such that c ∉ V and d ∉ W and V ∪W = X/E, as the complements of V, W are then the required neighborhoods. But q^{−1}(c) and q^{−1}(d) are disjoint closed subsets of X so, by normality, there exist disjoint open subsets E, F of X with q^{−1}(c) ⊆ E and q^{−1}(d) ⊆ F. Then V = q(X \ E) and W = q(X \ F) have the required properties.

2.12. Lemma. Let (X, t), a flow in Stone, admit a continuous t-action by Ẑ. Then X is regular.

Proof. That ∼ is an equivalence relation follows directly from Lemma 2.10. To show that ∼ is closed, let {p_{i} : i ∈ I} be an indexed family of members of X and let U be an ultrafilter on I with Lim_{U} p_{i} = p. Choose q_{i} with q_{i} ∼ p_{i} for all i and let Lim_{U} q_{i} = q. Then we must show that q ∼ p.

But by above results, if q_{i} ∼ p_{i} then there exists an ζ_{i} ∈ Ẑ such that q_{i} = t^{ζ_{i}}(p_{i}). Let Lim_{U} ζ_{i} = ζ (in the compact topology on Ẑ). Then, in X × Ẑ we see that Lim_{U} (ζ_{i}, p_{i}) = (ζ, p).

Since the action is continuous, it preserves this limit so q = t^{ζ}(p) which shows that q ∼ p.

2.13. Theorem. The following statements are equivalent for (X, t) a flow in Stone:

(1) (X, t) is Boolean cyclic

(2) (X, t) admits a continuous t-action by Ẑ.

(3) X is regular and every closed trajectory T (p) is nearly cyclic.

Proof.

(1) ⇒ (2) By Theorem 2.2.

(2) ⇒ (3) By Lemmas 2.8 and 2.12.

(3) ⇒ (1) Let X be regular and let every T (p) be nearly cyclic. Think of {T_{i} : i ∈ X/∼} as an indexing of the closed trajectories of X. (Strictly speaking, T_{i} = i but this might be notationally confusing.)

Let (B, τ) = Clop(X, t) be the corresponding Boolean flow. Let b ∈ B. Let b_{i} = b∩T_{i} and let n_{i} be the period of b_{i} in the cyclic Boolean algebra of all (relatively) clopen subsets of T_{i}. If the n_{i} are bounded, then let n be their least common multiple. It is clear that τ^{n}(b) = b.

On the other hand, assume that {n_{i}} is not bounded. Then we can find an ultrafilter U on the set X/∼ such that for every U ∈ U, the set {n_{i} : i ∈ U} is unbounded. Let T = Lim_{U} T_{i} in the quotient topology on X/∼.

Let b′ = b ∩ T. Then, since Clop(T) is Boolean cyclic, there exists n such that τ^{n}(b′) = b′. Now, by choice of U, we can choose q_{i} ∼ p_{i} such that q_{i} ∈ τ^{n}(b_{i}) + b_{i} for all i ∈ U where U ∈ U. (Note that + denotes the symmetric difference.) Then let q = Lim_{U} q_{i} so, by regularity, and because the map from X to X/∼ is continuous, q ∈ T. But q_{i} ∈ τ^{n}(b)+b so (as τ^{n}(b)+b is closed, q ∈ τ^{n}(b)+b) but this contradicts q ∈ T as T ∩ (τ^{n}(b) + b) is empty.

2.14. Proposition. Let (X, t) be a flow in Stone for which all closed trajectories are nearly cyclic and all but finitely many of them are actually cyclic. Further suppose there is a bound n such that every cyclic trajectory has period no bigger than n. Then (X, t) is Boolean cyclic.

Proof. The proof follows from the first two paragraphs of the above proof of (3) ⇒ (1).

Remark. The converse of 2.14 is false. Consider a product X = Ẑ^{m} where m is a large cardinal (say at least as large as the cardinal of Ẑ). Then X is clearly Boolean cyclic as X is a product (and so Clop(X) is a coproduct) of Boolean cyclic spaces. But X can have no finite cyclic trajectories, as they would have to project onto finite cycles on each component Ẑ. Nor can X have only finitely many closed trajectories (by cardinality considerations).

3. Constructing the Spectrum

The previous section showed that Boolean cyclic spaces are fairly special. We would like to construct a cyclic reflection for every Boolean flow. Unfortunately, the subcategory of cyclic Boolean flows is not reflective (as, for example, it is not closed under products, see Example 6). We can get a reflection if we work in a bigger category which might be called the “category of Boolean flows in Grothendieck toposes”.

The objects of this category are pairs (E, B) where E is a Grothendieck topos and B is a Boolean flow in E (that is, a flow in the category Bool(E) of Boolean algebras in E). By a morphism (λ^{∗}, f) : (E, B) → (F, C) we mean an inverse image functor λ^{∗} : E → F along with a flow homomorphism f : λ^{∗}(B) → C in F. Consider the subcategory of all (E, B) where B is cyclic and all morphisms (λ^{∗}, f) where f is mono. Then this category is reflective and the reflection is an example of a Cole spectrum. See [2, pages 205-210]. Remark. The category of Boolean cyclic flows is not a reflective subcategory of all Boolean flows, but it is coreflective. The coreflection seems not very interesting as it just gives the subalgebra of all b for which b = τ^{n}(b) for some n. However, by duality, this means that the Boolean cyclic flows in Stone are reflective in the category of all flows in Stone. If we start with N with the map t(n) = n + 1 and take its Stone-Čech compactification, we get a flow in Stone whose Boolean cyclic reflection is Ẑ.

Before describing the Cole spectrum, we will establish some properties of Boolean cyclic flows in a Grothendieck topos. 3.1. Lemma. If (C, τ) is a Boolean cyclic flow in a Grothendieck topos, then τ is an isomorphism. Proof. To prove that τ is one-to-one, let K be the kernel of τ. It suffices to prove that K = {0}. But let E(n) = Equ(Id_{C}, τ^{n}) then ⋁ E(n) = C. It is readily established that K ∩ E(n) = {0} for all n and so K = ⋁(E(n) ∩ K) = {0}.

A similar proof establishes that τ is onto as τ maps each E(n) onto itself. 3.2. Lemma. Boolean cyclic flows in any topos are closed under the formation of subflows. Proof. Let (C, τ) be a cyclic flow in the topos E. By the above lemma, τ is oneto-one and onto. Let A be a subflow. Then the maps Id_{C} and τ^{n} pull back along A to Id_{A} and τ_{A}^{n}. Since pullbacks preserve equalizers, E(n) = Equ(Id_{C}, τ^{n}) pulls back to E_{A}(n) = Equ(Id_{A}, τ_{A}^{n}) and the result follows because pulling back (or intersecting with the subobject A) preserves sups of subobjects. Remark. It can readily be seen that the above statement is not true for all categories. Even for the well-behaved category Stone, Example 6 shows a space that is Stone cyclic but with non-cyclic subobjects (for example, there are closed subsets of the Stone-Čech compactification which do not meet any Z_{p}). 3.3. Proposition. Let C be a Boolean flow in a Grothendieck topos E. If C is cyclic (in E) then there is a canonical action of Ẑ on C which extends the action of N.

Proof. Let E(n) = Equ(Id_{C}, τ^{n}) so that ⋁ E(n) = C. Given ζ ∈ Ẑ we define τ^{ζ} on E(n) as τ^{k} where ζ ≅ k (mod n). We claim that these definitions are compatible on the overlaps, E(n) ∩ E(m). The claim follows by showing that E(n) ∩ E(m) = E(gcd(n, m)) the proof of which uses negative powers of τ, as τ^{−1} exists.

By the claim, the above maps {τ^{k}} patch together to give us a morphism τ^{ζ} : C → C.

3.4. Corollary. Let C be a cyclic Boolean flow in a Grothendieck topos E. Then there is a canonical action of Ẑ on Γ(C) which extends the action of N. (where Γ is the global sections functor). Note that Ẑ need not act continuously on Γ(C).

We now start to construct the spectrum for B, a Boolean flow in Sets. This means finding a map (γ^{∗}, η) : (Sets, B) → (E, B^{#}) with the appropriate universal property (which is restated in Proposition 3.5 below) As a convenience, we will find the non-trivial cyclic spectrum, thus requiring that the cyclic object satisfy 0 ≠ 1. 3.5. Proposition. Let B be a flow in Boolean algebras (in Sets). Then the cyclic spectrum of B coincides with the topos which classifies non-trivial cyclic quotients of B. Proof. The classifying topos for cyclic quotients has a map (γ^{∗}, η) : (Sets, B) → (E, B^{#}) with η epi and B^{#} cyclic such that for any other map (λ^{∗}, f) : (Sets, B) → (F, C), with f epi, there is (to within equivalence) a unique inverse image functor δ^{∗} such that δ^{∗}γ^{∗} = λ^{∗} and δ^{∗}(B^{#}) = C. To show that this is the Cole spectrum, let (λ^{∗}, f) : (Sets, B) → (F, C) be given (without assuming that f is epi). Then use the mono/epi factorization of f to get a cyclic quotient of C and apply the above property (so that δ^{∗}(B^{#}) maps via a mono to C).

The details that the two concepts (Cole spectrum and quotient classifier) coincide are now straightforward. For example, if (γ^{∗}, η) : (Sets, B) → (E, B^{#}) is the Cole cyclic spectrum, then we need to prove that η is epi. But, if not, then η factors through a proper subobject of B^{#} and, by the spectral property, B^{#} must map monomorphically to that subobject which gives two ways to map B^{#} monomorphically to itself. Notation.

(1) We will use γ_{E}^{∗} or just γ^{∗} denote the unique inverse image functor from Sets to any given Grothendieck topos E. Its adjoint will be denoted by either γ_{∗} or by Γ which is the usual notation for the global sections functor which maps E to the set Hom_{E}(1, E).

(2) It follows that γ^{∗}(B) is just the coproduct of “B copies of 1” which we will denote by ∑ 1_{b} where the subscript b denotes the fact that the copies of 1 are indexed by B. We also let b : 1_{b} → γ^{∗}(B) denote the coprojections associated with this coproduct. We may then drop the subscript b and simply write b : 1 → γ^{∗}(B), particularly when we need to remember that 1_{a} = 1_{b} for all a, b ∈ B. The different sections labeled “b” are preserved by the functors and operations we are working with, so there is no danger of significant confusion.

First Step. We first construct the topos which classifies “Boolean monoflow quotients of B” (see below). Here the construction of [4] applies and it gives us a spatial topos which is a good starting point. Definition 9. By a Boolean monoflow we mean a non-trivial flow (B, τ) in Bool for which τ is one-to-one. Note that both Boolean monoflows and Boolean cyclic flows make sense in any Grothendieck topos and by Lemma 1.1, every Boolean cyclic flow is a monoflow.

Construction of the monoflow quotient classifier. Let (B, τ) be a flow in Bool (in Sets ). We follow the construction of [4].

Let {B_{v} : v ∈ V } be an indexing of the non-trivial monoflow quotients of B and let q_{v} : B → B_{v} be the quotient map. For every b ∈ B let N(b) = {v ∈ V : q_{v}(b) = 0}. Topologize V so that the sets {N(b)} form a base. (Note that N(a) ∩ N(b) = N(a ∨ b).) Define a set B^{o} and a function q : B^{o} → V so that B_{v} = q^{−1}(v). Regard each b ∈ B as a map b : V → B^{o} for which b(v) = q_{v}(b). Topologize B^{o} with the largest topology making each map a continuous. Then:

3.6. Proposition. With the notation of the above construction, B^{o} is a sheaf over V and it classifies non-trivial monoflow quotients of B in Grothendieck toposes.

Proof. We sketch the proof, most of which follows from [4]. It is straightforward to show that B^{o} is the etale space of a sheaf over V and that B^{o} is a monoflow because the stalks {B_{v}} are. It is clear that the canonical map from γ_{V}^{∗} (B) to B^{o} is an epi by considering its action on stalks.

Let E be a Grothendieck topos and let M be a Boolean monoflow in E along with an epi flow homomorphism f : γ_{E}^{∗}(B) → M.

For b ∈ B let E(b) be the equalizer of b, 0 : 1 → M. Let O(V) be the partially ordered set of open subsets of V and let Sub(1_{E}) be the partially ordered family of subobjects of 1 in E. Define λ : O(V) → Sub(1_{E}) so that λ(N(b)) = E(b) and extend to all of O(V) by taking unions. Then λ preserves all unions and finite intersections because each relevant property of O(V) follow from intuitionistic logic from a finite number of facts about B and axioms satisfied by non-trivial monoflows. As is well-known, this determines an inverse image functor λ^{∗} : Sh(V) → E.

Since M is a quotient of γ^{∗}(B), we see that M is γ^{∗}(B) modulo the ideal ⋁ b(E(b)). So M is determined by the subobjects E(b) and the coproduct and coequalizer struc-

ture on E. In the same way, B^{o} is determined by the equalizers N(b) and the coproduct and coequalizer structure on Sh(V) all of which is preserved by λ^{∗}. So λ^{∗}(B^{o}) = M.

Remark. In view of the above we need only to force the above generic monoflow quotient of B to be cyclic. This is done by using the smallest Lawvere-Tierney topology (or nucleus) j on Sh(V) for which the embedding of ⋁ E(n) to B^{o} becomes epi. To that end, we make the following definition:

Definition 10. Use the above notation and for each b ∈ B let b_{k} = b + τ^{k}(b). Then N(b_{k}) is the largest open subset of V on which b = τ^{k}(b) (as sections of B^{o}). Let O(V) be the Heyting algebra of open subsets of V and let j be the smallest Lawvere-Tierney topology on O(V) such that for each b ∈ B, the set {N(b_{k})} covers V. (These form a subbase for the covers). We must then consider finite intersections of the {N(b_{k})} and intersect them with U to form a cover of U for each U ∈ O(V).)

Notation.

(1) We let j-Sh(V) denote the resulting topos obtained from O(V) and let “j-sheaves” refer to sheaves over O(V) which are also sheaves with respect to the topology j.

(2) We let B^{#} denote the j-sheafification of B^{o} in the topos of j-Sh(V).

3.7. Proposition. The object B^{#} in the topos j-Sh(V) classifies cyclic quotients of B. Equivalently this gives us the Cole cyclic spectrum.

Proof. We must show that j is the smallest topology making B^{o} cyclic. Clearly, by the definition of j, each b ∈ B is in ⋁ E(n) and, as γ^{∗}(B) maps onto B^{o}, this means that ⋁ E(n) will cover B^{o}, and hence its j-sheafification, B^{#}.

Conversely, if ⋁ E(n) is the maximum subobject, then, pulling this back down to 1_{b}, we find that the covers defining the j-topology must be epi families.

3.8. Proposition. Let B, B^{#}, j-Sh(V) be as above. The points of j-Sh(V) correspond to cyclic quotients B → C in Sets and each such quotient extends to a flow homomorphism Γ(B^{#}) → C.

Proof. A point of j-Sh(V) is an inverse image functor to Sets. Such a functor preserves epis and cyclicity and so maps η to a cyclic quotient of B. Conversely, any such quotient arises in this way because of the universal property of the spectrum.

Let λ^{∗} : j-Sh(V) → Sets be such a point and let λ^{∗}(B^{#}) = C. By adjointness, there is a corresponding map Γ(B^{#}) → Γ(λ_{∗}(C)). But Γλ_{∗} is a geometric functor from Sets to Sets so is equivalent to the identity and the result follows.

Notation. Given a Boolean flow (B, τ), we let V and B^{#} be constructed as above. We let Γ(B^{#}) be the global sections of B^{#} and let η_{1} : B → Γ(B^{#}) be the adjunct of the canonical map η : γ^{∗}(B) → B^{#} in j-Sh(V).

We say that “B has enough maps into cyclic flows in toposes” if η_{1} is one-to-one. We note that for η_{1} to be one-to-one it is sufficient, but not presumably necessary that “B has enough maps into cyclic flows in Sets” meaning that if b ∈ B is non-zero then there exists a flow homomorphism h : B → C with C cyclic and h(b) non-zero.

Remark: Transfinite Time. Given a flow (X, t) in Stone, think of t(x) as the state a system is in one unit of time after being in state x. Then we would like to define t^{ζ}(x) and think of it as the state ζ units of time after state x where ζ could be transfinite (i.e. in Ẑ). But if X^{#} is defined as Γ(B^{#}) then, by 3.4, t^{ζ}(x) is defined only for x ∈ X^{#} not for x ∈ X. However the map η_{1} : B → Γ(B^{#}) gives us a map from X^{#} → X so in some sense, X^{#} is the space where transfinite time makes sense.

If B has “enough maps onto cyclic flows in Sets”, then η_{1} is one-to-one and X^{#} maps onto X. In this case, we might think of X^{#} as a sort of covering of X for which τ lifts to a one-to-one, onto map which comes from a cyclic flow in a topos.

Remark: Methods for computing the spectrum. The spectrum is further explored by the examples below. It is useful to first note the following:

Finding a Boolean quotient of B is equivalent to finding an ideal of B, by which we mean a subset I for which:

(1) 0 ∈ I.

(2) a, b ∈ I imply a ∨ b ∈ I.

(3) a ∈ I and b ≤ a imply b ∈ I.

If we want I to be a flow ideal (so that B/I has a natural flow) then we must add:

(4) a ∈ I implies τ(a) ∈ I.

If the quotient is to be non-trivial, then we require:

(5) 1 ∉ I

Finally, if we want the quotient to be a monoflow, then we need the kernel of τ to be {0}, so we add:

(6) τ(a) ∈ I implies a ∈ I.

Ideals satisfying (1)–(6) are the points of the space V. Another approach to finding quotients of B is to let (X, t) = Pts(B, τ). Then quotients of B correspond, by duality, to closed subsets of X. Given such a closed subset A, the corresponding ideal I is the set of all clopens b for which b ∩ A is empty. Point (4), above, is equivalent to the condition:

(4′) a ∈ A implies t(a) ∈ A. This makes sense as A is then a subflow in Stone.

Point (5) clearly translates into:

(5′) A is non-empty.

Point (6) is not so easy to restate as a condition about A but if t is one-to-one and onto, then point (6) is restated as:

(6′) (For t one-to-one and onto): t(x) ∈ A implies x ∈ A.

Also, if t is not onto, point (6) implies that A ⊆ t^{n}(X) for all n.

Example 7: An attractive fixpoint. Let X = N ∪ {∞} with t(n) = n + 1 and t(∞) = ∞. Let (B, τ) = Clop(X, t) (where X has the obvious Stone topology). We first find the monoflow quotients of B. They correspond to closed subsets A as discussed above. Since A ⊆ ⋂ t^{n}(X) we see that the only admissible A is {∞}. So V has a single point which contains those clopens b that do not contain ∞. The coverings add nothing as B^{o} is already cyclic (with a cyclic stalk of {0, 1} over I).

In effect, the spectrum eliminated all but the point at infinity and Γ(B^{#}) is just Clop{∞}.

Example 8: An attractive and a repulsive fixpoint. Let X = Z ∪ {−∞, ∞} and let t(z) = z + 1, with t(∞) = ∞ and t(−∞) = −∞. Give X the obvious Stone topology. Let (B, τ) = Clop(X, t) Then t is one-to-one, onto, so we need only consider quotients by closed subsets A which satisfy a ∈ A iff t(a) ∈ A.

This means that A ∩ Z must be either empty or all of Z. If it is all of Z then A must be X (as A is closed). If A ∩ Z is empty then A may also contain any non-empty subset of {−∞, ∞}. So V has four points. It is straightforward to verify that V has five open sets, namely:

∅ = N(X), N(∞), N(−∞), N(−∞, ∞), N(0) = X The j-coverings yield the fact that N(−∞, ∞) covers, so, in effect we get the two

points −∞ and ∞.

Remark. A goal for further work would be to extend this approach to try to distinguish repulsive from attractive fixpoints.

Example 9. Let X = ∑{Z_{p} : p ∈ P } where P is the set of primes, with t(z) = z + 1 on each component. Let B = Clop(X) We note that B is the algebra of clopens of X which is the Stone-Čech compactification of of S = ⋃{Z_{p}} (the disjoint union in Sets ). However B can more simply be viewed as the algebra of subsets of S (as subsets of S and clopens of X both correspond to maps from X to the two-point Stone space).

Since X is hard to work with, we will directly look for the right kind of ideals. We note that the zero ideal I_{0} = {0} is in V as t is onto (so τ is one-to-one). Also, the only open neighborhood in V of I_{0} is V itself, so the stalk at I_{0} will give us the set of global sections.

For any c the elements c_{k} = c + τ^{k}(c) will miss the Z_{p} ⊆ X once p divides k. So if a = b modulo any of these j-coverings, they must agree when intersected with each Z_{p} which implies a = b. This means that the separation and the sheafification of B^{o} with respect to the j-topology do not affect the stalk at I_{0}. (The stalks at ∞ are undoubtedly affected by closed subsets A which are disjoint from every Z_{p}.) But the upshot is that the set of global sections (= stalk at I_{0}) remains the same, so Γ(B^{#}) = B. Thus while B is not Boolean cyclic (see Example 6) B can be represented as the global sections of a cyclic flow in a topos. In particular, Ẑ acts on B (although here the action is obvious). In view of Theorem 2.13, the corresponding action on X cannot be continuous.

Example 10: A brief look at the Logistic Equation. Consider the map t(x) = rx(1 − x) which maps [0, 1] to itself, where r is a parameter satisfying 0 ≤ r ≤ 4. Unfortunately [0, 1] is not a Stone space because it is connected, but we can readily disconnect it by replacing each x ∈ [0, 1] with x− and x+. We then discard 0− and 1+ which are not needed. The resulting set, St[0, 1], has an obvious linear ordering and is a Stone space in the order topology (which is generated by the subbase {L(b) : b ∈ St[0, 1]} ∪ {U(a) : a ∈ St[0, 1]} where L(b) = {x : x < b} and U(a) = {x : x > a}).

The clopens of St[0, 1] are essentially finite unions of intervals, with b ∨ c being the interior of the closure of b ∪ c as in the double negation topology. Note that there is a natural way to define t as a continuous flow on St[0, 1].

In examining the spectrum of St[0, 1] we note that for small values of r, the methods used above show that the spectrum simply picks out the periodic points, without regard for whether they are attractive or repulsive. As r approaches a value called “r_{∞}” (which is about 3.569946) we pick up new periodic points whose period is twice the largest previous period. Eventually, at r = r_{∞}, a strange attractor appears, see [8, pages 3-4]. This strange attractor is nearly cyclic and will therefore be a point in the spectrum.

Beyond r_{∞}, chaos begins.

For the case of r = 4, the set P of periodic points of [0, 1] (and so of St[0, 1]) is dense and two clopen subsets are the same iff they have the same intersection with P. Because of this, it can be shown that there are enough Boolean cyclic quotients in Sets for η_{1} to be one-to-one. It follows that the dual map from X^{#} to St[0, 1] is onto. (r=4 is the only value for which X^{#} maps onto St[0, 1] as can be readily seen by the fact that t is not even onto when r < 4.)

Finally, there is the flow on [0, 4] × [0, 1] given by t(r, x) = (r, rx(1 − x)). All I have on this is a rough beginning. We need to replace [0, 4] × [0, 1] by a Stone space. This is equivalent to describing a subalgebra of the algebra of double negation opens which is preserved by t^{−1}. One candidate is generated by the “vertical trapezoids” or sets of the form {(r, x) : a < r < b, L(r) < x < U(r)} where L and U are linear functions. The trapezoid shape is needed in order for them to be preserved by t^{−1}.

References

[1] M. Barr, C. Wells. Toposes, Triples and Theories Springer-Verlag, New York, 1985.

[2] P.T. Johnstone. Topos Theory LMS Monograph 10, Academic Press, London, 1977.

[3] P.T. Johnstone. Stone Spaces Cambridge University Press, Cambridge, 1982.

[4] J.F. Kennison. Integral domain type representations in sheaves and other topoi, Mathematische Zeitschrift, 151, 35-56 (1976).

[5] S. Mac Lane. Category Theory for the Working Mathematician Springer-Verlag, New York, 1971.

[6] S. Mac Lane, I. Moerdijk. Sheaves in Geometry and Logic Springer-Verlag, New York, 1992.

[7] E. Ott. Chaos in Dynamical Systems Cambridge University Press, 1993.

[8] C. Preston. Iterates of Maps on an Interval Lecture Notes in Mathematics 999, Springer-Verlag, Berlin, 1983.

Clark University, Worcester, MA, 01610, USA

Email: jkennison@clarku.edu

This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anony-

mous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/10/15/10-15.{dvi,ps}

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods.

Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication.

The method of distribution of the journal is via the Internet tools WWW/ftp. The journal is archived electronically and in printed paper format.

Subscription information. Individual subscribers receive (by e-mail) abstracts of articles as they are published. Full text of published articles is available in .dvi, Postscript and PDF. Details will be e-mailed to new subscribers. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh,

rrosebrugh@mta.ca.

Information for authors. The typesetting language of the journal is TEX, and L^{A}TEX is the preferred flavour. TEX source of articles for publication should be submitted by e-mail directly to an appropriate Editor. They are listed below. Please obtain detailed information on submission format and style files from the journal’s WWW server at http://www.tac.mta.ca/tac/. You may also write to tac@mta.ca to receive details by e-mail.

Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org, Associate Managing Editor Lawrence Breen, Université Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Palo Alto Research Center: paiva@parc.xerox.com Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, University of Sydney: stevel@maths.usyd.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca, Managing Editor Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca