Rings, Hopf Algebras and Brauer Groups, Ed. by S. Caenepeel and A. Verschoren, Lec. Notes in Pure and Appl. Math., 197, pp. 43-54, M. Dekker (1998), New York

Topological Methods in Algebra

Anthony Bak

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

1 Introduction

This article discusses the foundations for recent developments introducing a general concept of deformation and homotopy theory into pure algebra. Homotopy theory has a long tradition in analysis and geometry and is grounded in the concepts of topological space, continuous map, path, and deformation of continuous maps. Wherever these notions occur in a natural and meaningful way, one can endevour to apply homotopical concepts and methodology to formulate and solve problems.

The algebraic counterpart of a topological space is a global action. It has a very natural and intuitive notion of path which is easy to formalize and it has a good concept of deformation for morphisms, based on the notion of path. In terms of these concepts, one can develop in a purely algebraic setting the entire spectrum of homotopy theory. This article describes the notion of a global action, several kinds of morphisms between global actions, the notion of path in a global action and the notion of deformation of a morphism.

The body of the paper is organized as follows. We begin by defining a global action and giving a few examples including that of the line L. Next various concepts of morphism between global actions are introduced. A natural, intuitive notion of path in a global action A is described and then formalized as a certain kind of morphism from L to A. The next goal is defining a global action structure on the set of all morphisms from one global action to another and establishing the exponential law. The global action on morphism spaces plays the role of the compact open topology on function spaces in topology. Finally we define when two morphisms are homotopic.

2 Global actions

A group action consists by definition of a group G, a set X and a map G×X → X, (σ, x) ↦ σx, of the Cartesian product G × X to X such that for all elements σ, ρ ∈ G and x ∈ X, (σρ)x = σ(ρx) and 1x = x. If G acts on X then we write G ↷ X. A morphism G ↷ X → H ↷ Y of group actions conists of a group homomorphism f : G → H and a function f ′ : X → Y such that for all σ ∈ G and x ∈ X, f ′(σx) = f(σ)f ′(x).

A global action is formed by fitting together a set of group actions, according to a few simple principles. Each of the group actions making up the global action is then called a local action.

Definition 2.1 A global action A consists of a set X_{A} together with a set {(G_{A})_{α} ↷ X_{A} | α ∈ Φ_{A}} of group actions (G_{A})_{α} ↷ (X_{A})_{α} such that each (X_{A})_{α} ⫅ X_{A}. The set of group actions is structured by equipping the index set Φ_{A} with a reflexive relation ≦ and imposing the condition that if α ≦ β then (G_{A})_{α} leaves (X_{A})_{α} ∩ (X_{A})_{β} invariant and there is a group homomorphism (G_{A})_{α≦β} : (G_{A})_{α} → (G_{A})_{β} such that if σ ∈ (G_{A})_{α} and x ∈ (X_{A})_{α} ∩ (X_{A})_{β} then σx = (G_{A})_{α≦β}(σ)x.

The index set Φ_{A} of a global action A is called the coordinate system of A and each element of Φ_{A} is called a coordinate. It is possible that for distinct coordinates α and β, (X_{A})_{α} = (X_{A})_{β}, but (G_{A})_{α} ≠ (G_{A})_{β}. This allows one to have distinct groups acting on the same subset of X_{A}. The function G_{A} : Φ_{A} → ((groups)), α ↦ (G_{A})_{α}, is called the global group of A and each group (G_{A})_{α} is called a local group of A. Thus a global group is a group valued function on a coordinate system. Each set (X_{A})_{α} is called a local set of A and each group action (G_{A})_{α} ↷ (X_{A})_{α} is called a local action of A. The set X_{A} is called the enveloping set of A. The function Φ_{A} → subsets (X_{A}), α ↦ (X_{A})_{α}, is denoted also by X_{A} : Φ_{A} → subsets (X_{A}). When confusion might arise, we write |X_{A}| for the enveloping set X_{A}. The notation |A| will also be used for the enveloping set X_{A}.

A global action A is called covariant if the relation ≦ on Φ_{A} is transitive and if the global group G_{A} : Φ_{A} → ((groups)) is a covariant functor. In this case, the global group is called covariant also. A global action A is called contravariant if the relation ≦ on Φ_{A} is transitive and if α ≦ β ⇒ (X_{A})_{α} ⫆ (X_{A})_{β}. This condition is equivalent to requiring that the function X_{A} : Φ_{A} → subsets (X_{A}), α ↦ (X_{A})_{α}, is a contravariant functor. A global action is called bivariant of functorial if it is both covariant and contravariant.

An important example of a functorial action, which arises in several contexts, is the following.

Example 2.2 Let G ↷ X be a group action. Let Φ be a set which indexes a set {G_{α}|α ∈ Φ} of subgroups G_{α} of G. Assume that G_{α} = G_{β} ⇔ α = β. Partially order the set{G_{α}|α ∈ Φ} by inclusion and give Φ the induced partial ordering. Clearly the rule α ↦ G_{α} defines a functor G : Φ → ((groups)). Set |X| = X and define the function X : Φ → subsets |X|, α ↦ X_{α}, by X_{α} = |X| for all α ∈ Φ. Then one obtains a functorial global action (Φ, G, X).

If U is a set, let

Perm(U) =Group of all bijections of U onto itself,
fPerm(U) ={σ ∈ Perm(U)|σ fixes all but a finite number of elements of U}.

If U is a well ordered nonempty finite set, let

cPerm(U) = cyclic subgroups of Perm(U) generated by the cyclic

permutation which sends each element of U, except for the last, to its successor and sends the last element to the first.

The next example is important for the homotopy theory of global actions.

Example 2.3 This example is called the line action and is denoted by L. Let Φ_{L} = ℤ∪{∗}. Give Φ_{L} the partial ordering such that there is no relation between elements of ℤ and such that ∗ ≦ n for all n ∈ ℤ. Let |X_{L}| = ℤ and define the function X_{L} : Φ → subsets |X_{L}|, α = n ↦ {n, n + 1} and α = ∗ ↦ |X_{L}|. Define G_{L} : Φ_{L} ↦ ((groups)),α = n ↦ (G_{L})_{α} = Perm({n, n + 1}) and α = ∗ ↦ (G_{L})_{α} = {1}. Then the triple L = (Φ_{L}, G_{L}, X_{L}) is a functorial action.

The next example extends in several ways the one above, to arbitrary abstract simplicial complexes.

Example 2.4 Let S denote an abstract simplicial complex and let |X_{S}| denote the set of vertices of S. If α is a subcomplex of S, let (X_{S})_{α} denote the set of its vertices. Call a subcomplex α simple, if (X_{S})_{α} has a partition into subsets U such that any finite subset of U is a simplex in α and such that any simplex of α is a subset of some U. Clearly if α is simple then the partition above of (X_{S})_{α} is unique; let Part(X_{S})_{α} denote this partition. Let Φ_{S} denote the set of all simple subcomplexes of S. Partially order Φ_{S} by defining α ≦ β ⇔ (X_{S})_{α} ⫆ (X_{S})_{β} and every member of Part(X_{S})_{β} is a union of members of the Part(X_{S})_{α}. Clearly the subcomplex whose vertices are |X_{S}| and whose simplices are the singleton subsets of |X_{S}| is the smallest element of Φ_{S}. For α ∈ Φ_{S}, define

(G_{S})_{α} = ∏_{U∈Part(X_{S})_{α}} Perm(U)
(fG_{S})_{α} = ∏_{U∈Part(X_{S})_{α}} fPerm(U).

There is a canonical action of (G_{S})_{α} (resp. (fG_{S})_{α}) on (X_{S})_{α} defined by the action of each permutation group Perm(U) (resp. fPerm (U)) on U. Define

gl(S) = (Φ_{S}, G_{S}, X_{S})
fgl(S) = (Φ_{S}, fG_{S}, X_{S}).

Then gl(S) and fgl(S) are global actions called simplicial actions. They are not in general functorial.

Well order now the vertices |X_{S}| of S and let cΦ_{S} denote the subset of Φ_{S} of all simple subcomplexes α such that Part(X_{S})_{α} contains only finite sets. The smallest element of Φ_{S}, say ∗, clearly lies in cΦ_{S}. Give cΦ_{S} a new partial ordering such that α ≦ β ⇔ α = ∗. Thus if α ≠ ∗ ≠ β then either α = β or there is no relation between α and β. For α ∈ cΦ_{S}, define (cX_{S})_{α} = (X_{S})_{α} and

(cG_{S})_{α} = ∏_{U∈P art(X_{S})_{α}} cP erm(U).

Define

cgl(S) = (cΦ_{S}, cG_{S}, cX_{S}).

Then cgl(S) is a global action called a cyclic simplicial action. It is not in general functorial.

There are three kinds of morphisms for global actions. There is first of all a general concept. Then there is a structural concept which gives rise to so-called regular morphisms. Global actions which are regularly isomorphic are essentially the same. Finally there is a concept lying between the two above, which leads to the definition of normal morphism. The set Mor(A, B) of all morphisms from a global action A to a global action B will be given the structure of a global action such that the assignment (A, B) ↦ Mor(A, B) is functorial in A over all morphisms and functorial in B over precisely normal morphisms.

We define first the general concept of morphism. This concept is geometric in flavor and depends on the notion of local frame. A local frame at a point p ∈ |X_{A}| signifies a range of possible moves available at p . It is defined as follows.

Definition 2.5 Let A be a global action.Let x ∈ (X_{A})_{α}. A local frame at x in α or simply an α-frame at x is a sequence x = x_{o}, ⋯, x_{p} of points in (X_{A})_{α}, with the property that for each i (1 ≦ i ≦ p) there is a g_{i} ∈ (G_{A})_{α} such that g_{i}x = x_{i}. (Clearly x_{0}, x_{1}, ⋯, x_{p} is an α-frame at x_{0} ⇔ x_{0}, ⋯, x_{i}, ⋯, x_{p} is an α-frame at x_{i}.) A morphism f :A → B of global actions is a function f :|A| → |B| which preserves local frames. Specifically if x_{0}, ⋯, x_{p} is an α-frame at x_{0} then f(x_{0}), ⋯, f(x_{p}) is an β-frame at f(x_{0}) for some β ∈ Φ_{B}.

A path in a global action A is intuitively a sequence x_{0}, ⋯, x_{p} of points in |A| such that there are coordinates α_{0}, ⋯, α_{p} ∈ Φ_{A} and group elements σ_{i} ∈ (G_{A})_{x_{i}}(i = 0, ⋯, p − 1) such that σ_{i}x_{i} = x_{i+1}. The figure below illustrates a path when p = 3.

|A|  (X_{A})_{α_{2}} (X_{A})_{α_{1}} (X_{A})_{α_{0}} x_{0}  x_{1}  x_{2}  x_{3} •  •  • • σ_{0}  σ_{1}  σ_{2}

The notion is formalized as follows.

Definition 2.6 A path in a global action A is a morphism ω : L → A, where L is the line action in (2.3), such that there are integers n ≤ N ∈ ℤ = |L| with the property that ω(l) = ω(n) for all l ≤ n and ω(l) = ω(N) for all N ≦ l. Thus a path is stably constant on the left and on the right.

Definition 2.7 A regular morphism η : A → B of global actions is a triple (η_{Φ}, η_{G}, η_{X}) satisfying the following conditions.

(2.7.1) η_{Φ} : Φ_{A} → Φ_{B} is a relation preserving function.

(2.7.2) η_{G} : G_{A} → (G_{B})_{η_{Φ}( )} is a natural transformation of group valued functions on Φ_{A} where (G_{B})_{η_{Φ}( )} denotes the composition of η_{Φ} with G_{B}. This means by definition that if α ≦ α′ ∈ Φ_{A} then the diagram

\xymatrix{(G_{A})_{α} \ar[d] \ar[r]^{(η_{G})(α)} & (G_{B})_{η_{Φ}(α)} \ar[d] \\ (G_{A})_{α′} \ar[r]_{η_{G}(α′)} & (G_{B})_{η_{Φ}(α′)}}

commutes (2.7.3) η_{X} : |A| → |B| is a function such that η_{X}((X_{A})_{α}) ⫅ (X_{B})_{η_{Φ}(α)} for all α ∈ Φ_{A}. (2.7.4) For each α ∈ Φ_{A}, the pair (η_{G}, η_{X}) : (G_{A})_{α} ↷ (X_{A})_{α} → (G_{B})_{η_{Φ}(α)} ↷ (X_{B})_{η_{Φ}(α)} is a morphism of group actions. Clearly a regular morphism is one in the general sense of (2.7). A regular isomorphism η : A → B is a regular morphism such that there is a regular morphism η′ : B → A called the regular inverse of η with the property that η′_{Φ} is inverse to η_{Φ}, η′_{X} is inverse to η_{X}, and for each α ∈ Φ_{A}, η′_{G}(η_{Φ}(α)) is inverse to η_{G}(α). It is of course not true in general that a regular morphism which is an isomorphism in the general sense is a regular isomorphism. Example 2.8 Let G ↷ X and G′ ↷ X′ be group actions. Let {G_{α} | α ∈ Φ} be a set of subgroups of G such that G_{α} = G_{β} ⇔ α = β. Let {G′_{α′} | α′ ∈ Φ′} be a set of subgroups of G′ sucht that G′_{α′} = G′_{β′} ⇔ α′ = β′. Let (Φ, G, X) and (Φ′, G′.X′) denote the global actions constructed in (2.2). Let f : G → G′ be a group homomorphism and g : X → X′ be a function such that (f, g) defines a morphism (f, g) : G ↷ X → G′ ↷ X′ of ordinary group actions. Let φ : subgroups (G) → subgroups (G′) be a function such that if K ⊆ L ⊆ G are subgroups of G then K ⊆ φ(K) ⊆ φ(L). Suppose that {φ(f(G_{α})) | α ∈ Φ} = {G′_{α′} | α′ ∈ Φ′}. Then (f, g) defines in an obvious way a regular morphism (Φ, G, X) → (Φ′, G′, X′). Example 2.9 Let f : S → T be a morphism of abstract simplicial complexes. Then f defines morphisms gl(f) : gl(S) → gl(T), fgl(f) : fgl(S) → fgl(T), and clg(f) : cgl(S) → cgl(T) of global actions which do not in general have a regular structure. Moreover each of the assignments S ↦ gl(S), S ↦ fgl(S), and S ↦ cgl(S)’ defines a functor ((abstract simplicial complexes)) → ((global actions)). Next we define the concepts of chart and frame and then use them to define the notion of normal morphism. Definition 2.10 Let A and B be global actions. Let f : A → B be a morphism of global actions. A framing of f is a function β : |A| → Φ_{B} such that the following conditions hold.

(2.10.1) f(x) ∈ (X_{B})_{β(x)} for all x ∈ |A|.

(2.10.2) If x, x_{1}, ⋯, x_{p} is a local frame in A then f(x), f(x_{1}), ⋯, f(x_{p}) is a local frame in b ∈ Φ_{B} for some b ≧ β(x), β(x_{p}), ⋯, β(x_{p}).

An A-chart in B is a pair (f, β) consisting of a morphism f : A → B of global actions and a framing β : |A| → Φ_{B} of f.

Definition-Lemma 2.11 Let (f, β) be an A-chart in B.

If

σ = (σ_{x}) ∈ ∏_{x∈|A|}(G_{B})_{β(x)}

define

σf : |A| → |B|.
x ↦ σ_{x}f(x)

Then σf is a morphism A → B of global actions and (σf, β) is an A-chart in B.

PROOF Since σ_{x} ∈ (G_{B})_{β(x)}, it follows that σf(x) ∈ (X_{B})_{β(x)}. Thus the pair (σf, β) satisfies (2.10.1). To show that σf is a morphism of global actions and that (σf, β) is an A-frame in B, it suffices to show that (2.10.2) is satisfied. Let x_{o}, ⋯, x_{p} be a local frame at x_{o} ∈ |A|. By definition f(x_{o}), ⋯, f(x_{p}) is a b-frame at f(x_{o}) for some b ≥ β(x_{o}), ⋯, β(x_{p}). Let ρ_{x_{o}}, ⋯, ρ_{x_{p}} denote respectively the images of σ_{x_{o}}, ⋯, σ_{x_{p}} in (G_{B})_{b} under the canonical homomorphisms (G_{B})_{β(x_{i})} → (G_{B})_{b} (0 ≦ i ≦ p). Clearly ρ_{x_{o}}f(x_{o}), ⋯, ρ_{x_{p}}f(x_{p}) is a b-frame at ρ_{x_{o}}f(x_{o}). But ρ_{x_{i}}f(x_{i}) = σ_{x_{i}}f(x_{i}). Thus σf(x_{o}), ⋯, σf(x_{p}) is a b-frame at σf(x_{o}) and b ≧ β(x_{o}), ⋯, β(x_{p}). □

Definition 2.12 Let (f, β) be an A-chart in B. An A-frame at f on (f, β) is a set f = f_{o}, f_{1}, ⋯, f_{p} : A → B of morphisms for which there are elements σ_{1}, ⋯, σ_{p} ∈ ∏_{x∈|A|}

(G_{B})_{β(x)} such that σ_{i}f = f_{i} (1 ≦ i ≦ p). (In view of Lemma (2.11), f = f_{o}, f_{1}, ⋯, f_{p} is also an A-frame at f_{i} on (f_{i}, β) for any i (0 ≦ i ≦ p).)

The next lemma is very useful.

Local-Global Lemma 2.13 Let (f, β) be an A-chart in B. Then f = f_{o}, f_{1}, ⋯, f_{p} is an A-frame at f on (f, β) ⇔ for each x ∈ |A|, f(x), f_{1}(x), ⋯, f_{p}(x) is a local frame at f(x) in β(x).

PROOF The assertions are trivial consequences of Lemma (2.11).

Definition 2.14 Let A, B and C be global actions. An A-normal morphism g : B → C of global actions is one which preserves A-frames, i.e. if f, f_{1}, ⋯, f_{p} is an A-frame at f on (f, β) then there is a framing γ : |A| → Φ_{C} of gf such that gf,gf_{1}, ⋯, gf_{p} is an A-frame on the A-chart (gf, γ). A normal morphism g : B → C is one which preserves A-frames for any global action A. An A-normal (resp. normal) isomorphism is an A-normal (resp. normal) morphism which has an A-normal (resp. normal) inverse.

It is not true in general that an A-normal (resp. normal) morphism which is an isomorphism in the usual sense is an A-normal (resp. normal) isomorphism.

Lemma 2.15 A regular morphism is normal.

PROOF Let η : B → C be a regular morphism. If (f, β) is an A-chart in B then it follows straightforward that (η_{X}f, η_{Φ}β) is an A-chart in C. Let f, f_{1}, ⋯, f_{p} be an Aframe at f on (f, β) and let σ_{1}, ⋯, σ_{p} ∈ ∏_{x∈|A|}(G_{B})_{β(x)} such that σ_{i}f = f_{i} (1 ≦ i ≦ p).

If σ = (σ_{x}) ∈ ∏_{x∈|A|}(G_{B})_{β(x)}, define η_{G}(σ) = (η_{G}(β(x))(σ_{x})) ∈ ∏_{x∈|A|}(G_{C})_{η_{Φ}(β(x))}. Then

η_{G}(σ_{i})(η_{X}f) = η_{X}f_{i} (1 ≦ i ≦ p), by (2.7.4). Thus η_{X}f, η_{X}f_{1}, ⋯, η_{X}f_{p} is an A-frame at η_{X}f on (η_{X}f, η_{Φ}β). □

Next we use the concept of framing to give the set of all morphisms from a global action A to a global action B the structure of a global action.

Definition 2.16 Let A and B be global actions. Let |Mor(A, B)| denote the set of all morphisms from A to B. Define a global action

Mor(A, B) = (Φ_{(A,B)}, G_{(A,B)}, X_{(A,B)})

as follows. Its enveloping set is |Mor(A, B)|. Define

Φ_{(A,B)} = {β : |A| → Φ_{B}}.

Give Φ_{(A,B)} the reflexive relation defined by β ≦ β′ ⇔ β(x) ≦ β′(x) ∀x ∈ |A|. For β ∈ Φ_{(A,B)}, define

(G_{(A,B)})_{β} = ∏_{x∈|A|}(G_{B})_{β(x)}.

If β ≦ β′, there is for each x ∈ |A| a canonically defined homomorphism (G_{B})_{β(x)} → (G_{B})_{β′(x)} and therefore a homomorphism (G_{(A,B)})_{β} ⟶ (G_{(A,B)})_{β′}. For β ∈ Φ_{(A,B)}, define

(X_{(A,B)})_{β} = {f : A → B | f ∈ |Mor(A, B)|, β is a framing of f}.

By (2.11), if σ ∈ (G_{(A,B)})_{β} and f ∈ (X_{(A,B)})_{β} then σf ∈ (X_{(A,B)})_{β} and so there is an action of (G_{(A,B)})_{β} on (X_{(A,B)})_{β}. All the conditions for a global action are obviously satisfied. Moreover the global action Mor(A, B) is covariant, contravariant, or functorial whenever the same holds for B.

Proposition 2.17 As a functor in two variables with values in global actions, Mor(,) is contravariant and regular over all morphisms in the first variable and covariant over all normal morphisms in the second variable. More precisely the following holds.

(2.17.1) Let C be a global action and let f : A → B be a morphism of global actions. Then f defines a regular morphism

η = Mor(f, 1_{C}) : Mor(B, C) → Mor(A, C)

as follows. Define the relation preserving morphism

η_{Φ} : Φ_{(B,C)} → Φ_{(A,C)}.
β ↦ βf

Define the natural transformation

η_{G} : G_{(B,C)} → G_{(A,C)}

by

[figure: η_{G}(β) : (G_{(B,C)})_{β}; (G_{(A,C)})_{η_{Φ}(β)}; ∏_{y∈|B|}(G_{C})_{β(y)}; ∏_{x∈|A|}(G_{C})_{βf(x)}]

where

η_{G}(β)|_{(G_{C})_{β(y)}} is the diagonal homomorphism
(G_{C})_{β(y)} → ∏_{x∈|A|, f(x)=y}(G_{C})_{βf(x)},

under the convention that the empty product of groups, which can occur on the right hand side of the arrow above, is the trivial group. Define

η_{X} : |Mor(B, C)| → |Mor(A, C)|.
g ↦ gf

Then η = (η_{Φ}, η_{G}, η_{X}) is a regular morphism of global actions.

(2.17.2) Let A be a global action and let g : B → C be a morphism of global actions. Then the function

Mor(1_{A}, g) : |Mor(A, B)| → |Mor(A, C)|

is a morphism Mor(A, B) → Mor(A, C) of global actions ⇔ g is A-normal.

PROOF (2.17.1) Straightforward and routine. Details are left to the reader.

(2.17.2) Let (f, β) be an A-chart in B and let f = f_{0}, f_{1}, ⋯, f_{p} be an A-frame on (f, β). By definition of the term local frame, f_{0}, ⋯, f_{p} is also a local β-frame in the global action Mor(A, B) and conversely, any local frame in Mor(A, B) is an A-frame on some A-chart in B. Thus the function Mor(1_{A}, g) : |Mor(A, B)| → |Mor(A, C)| is a morphism of global actions ⇔ it preserves A-frames ⇔ g is A-normal.□

The next lemma is needed to show that the exponential map on global actions is regular.

Lemma 2.18 If g : B → C is a regular morphism then for any global action A, the morphism Mor(1_{A}, g) : Mor(A, B) → Mor(A, C) is regular.

PROOF By (2.15) and (2.17.2), the morphism Mor(1_{A}, g) : Mor(A, B) → Mor(A, C) exists. Let (η_{Φ}, η_{G}, η_{X} = g) be the regular structure of g. We define a regular structure (μ_{Φ}, μ_{g}, μ_{X} = Mor(1_{A}, g)) for Mor(1_{A}, g) as follows.

Define the coordinate morphism

μ_{Φ} : Φ_{(A,B)} → Φ_{(A,C)}.
β ↦ η_{Φ}β

Define the natural transformation

μ_{G} : G_{(A,B)} → G_{(A,C)}

by the commutative diagram

[figure: (G_{(A,B)})_{β}; μ_{G}(β); (G_{(A,C)})_{μ_{Φ}(β)}; ∏_{x∈|A|}(G_{(A,B)})_{β(x)}; ∏_{x∈|A|} η_{G}(β(x)); ∏_{x∈|A|}(G_{(A,C)})_{η_{Φ}(β(x))}]

One checks straightforward that (μ_{Φ}, μ_{G}, Mor(1_{A}, g)) is a regular morphism.□

For the results below on the exponential law, the notion of product is needed. We construct this next.

Definition-Lemma 2.19 Let A and B be global actions. Their product A × B is constructed as follows.

Φ_{A×B} = Φ_{A} × Φ_{B}

and (α, β) ≦ (α′, β′) ⇔ α ≦ α′andβ ≦ β′.

G_{A×B} = G_{A} × G_{B}
|A × B| = |A| × |B|
X_{A×B} = X_{A} × X_{B}.

For any coordinate (α, β) ∈ Φ_{A×B}, there is an obvious action of (G_{A×B})_{(α,β)} on (X_{A×B})_{(α,β)}, namely the one defined coordinatewise. One checks easily that A×B satisfies the universal property of a product.

The following notation will be used below. If S and T are sets, let

(S, T) = Mor_{((sets))}(S, T).

If U is also a set then there is a canonical isomorphism

(2.20)  E : (U, (S, T)) ⟶^{≅} (U × S, T)
f ⟼ Ef

of sets such that Ef(u, s) = f(u)(s). Its inverse is obviously the function

E′ : (U × S, T) ⟶ (U, (S, T))
f ⟼ E′f

where (E′f(u))(s) = f(u, s).

Definition 2.21 Let A, B and C be global actions. Define a regular morphism

E : Mor(A, Mor(B, C)) → Mor(A × B, C)

as follows. Denote the structural components of the global action Mor(A, Mor(B, C)) by (Φ_{(A,(B,C))}, G_{(A,(B,C))}, X_{(A,(B,C))}). Define

[figure: E_{Φ} : Φ_{(A,(B,C))}; Φ_{(A×B,C)}; (|A|, (|B|, Φ_{C})); (|A| × |B|, Φ_{C})]

to be the set theoretic exponential isomorphism (2.20). Clearly E_{Φ} preserves the reflexive relation. Define the natural transformation

E_{G} : G_{(A,(B,C))} → (G_{(A×B,C)})_{E_{Φ}( )}

such that

[figure: E_{G}(α) : (G_{(A,(B,C))})_{α}; (G_{(A×B,C)})_{E_{Φ}(α)}; ∏_{x∈|A|}( ∏_{y∈|B|}(G_{C})_{α(x)(y)}); ∏_{(x,y)∈|A|×|B|}(G_{C})_{(E_{Φ}α)(x,y)}]

maps the factor (G_{C})_{α(x)(y)} via the identity map onto the factor (G_{C})_{(E_{Φ}α)(x,y)} = (G_{C})_{α(x)(y)}. One verifies easily that the composite mapping |Mor(A, Mor(B, C))| → (|A|, (|B|, |C|)) @ > (2.20) >> (|A| × |B|, |C|) takes its image in |Mor(A × B, C)| and we define

E_{X} : |Mor(A, Mor(B, C))| → |Mor(A × B, C)|

to be the resulting mapping. One checks straightforward that

E = (E_{Φ}, E_{G}, E_{X})

is a regular morphism. (It fails in general to be an isomorphism (resp. regular isomorphism) because E_{X} is not necessarily surjective (resp. E_{X}((X_{(A,(B,C))})_{α}) is not necessarily all of (X_{(A×B,C)})_{E_{Φ}(α)}).

Let A_{n}, ⋯, A_{1} be an arbitrary sequence of global actions. Iterating the procedure above, one defines for any n ≥ 2 a regular morphism

E_{n} : Mor(A_{n}, Mor(A_{n−1}, ⋯, Mor(A_{1}, C)) ⋯) → Mor(A_{n} × ⋯ × A_{1}, C)

as follows. For n = 2, the morphism is defined above. Suppose n > 2 and that the morphism has been defined for every natural number N where 2 ≦ N ≦ n − 1. Let E_{n−1} denote the morphism for the sequence A_{n−1}, ⋯, A_{1}. Define E_{n} for the sequence A_{n}, A_{n−1}, ⋯, A_{1} as the composite of the regular morphism Mor(1_{A_{n}}, E_{n−1}) (see (2.18)) and the regular morphism E_{2} : Mor(A_{n}, Mor(A_{n−1} × ⋯ × A_{1}, B)) → Mor(A_{n} × ⋯ × A_{1}, B).

Many global actions arising in nature satisfy the following condition.

Definition 2.22 Let A be a global action. If Δ ⫅ Φ_{A}, let Φ_{A} ^{≧Δ} = {α ∈ Φ_{A}|α ≧ β ∀ β ∈ Δ}. A is called an strong infimum action if for any finite subset Δ ⫅ Φ_{A} and any finite nonempty set U ⫅ |A| such that (X_{A})_{β} ∩ U ≠ ∅ for each β ∈ Δ, the set {α ∈ Φ_{A} ^{≧Δ}|U an α − frame} is either empty or contains an initial element. A is called an infimum action if it satisfies the condition above at least for Δ = ∅ (empty set).

Theorem 2.23 The exponential map

E_{n} : Mor(A_{n}, Mor(A_{n−1}, ⋯, Mor(A_{1}, C)) ⋯) → Mor(A_{n} × ⋯ × A_{1}, C)

in (2.21) has a normal (resp. regular) inverse if C satisfies the infimum (resp. strong infimum) condition.

PROOF See [1] Theorem 3.23.

We close by defining when two morphisms are homotopic.

Definition 2.24 Let f, g : A → B be morphisms of global actions. Then f is homotopic to g if there is a morphism F : A×L → B of global actions and integers n ≦ N ∈ ℤ = |L| such that F |_{A×{l}} = f for all l ≦ n and F |_{A×{l}} = g for all N ≦ l.

References

1.A. Bak, Global actions: The algebraic counterpart of a topological space, Invited article for the 100’th anniversary of P. S. Alexandroff, Uspekhi Mat.Nauk, to appear (in Russian). English translation: Russian Math Surveys, to appear.

Anthony Bak Department of Mathematics University of Bielefeld 33501 Bielefeld Germany

e-mail: bak@@ mathematik.uni-bielefeld.de