The classifying space of an inverse semigroup
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The classifying space of an inverse semigroup Ganna Kudryavtseva · Mark V. Lawson
Published online: 13 January 2015 © Akadémiai Kiadó, Budapest, Hungary 2015
Abstract We refine Funk’s description of the classifying space of an inverse semigroup by replacing his ∗-semigroups by right generalized inverse ∗-semigroups. Our proof uses the idea that presheaves of sets over meet semilattices may be characterized algebraically as right normal bands. Keywords
Generalized inverse semigroups · étale actions · Right normal bands
1 Statement of the theorem With each inverse semigroup S, we shall associate two categories, the aim of this paper being to prove that these two categories are equivalent. To define the first, we need the concept of an étale action of an inverse semigroup. These were first explicitly defined in [4], but their origins lie in [7,10] and they played an important role in [9]. Let X be a non-empty set. A left S -action of S on X is a function S × X → X , defined by (s, x) → s · x (or sx), such that (st)x = s(t x) for all s, t ∈ S and x ∈ X . If S acts on X we say that X is an S-set. In this paper, all actions will be assumed left actions. An étale action (S, X, p) of S on X is defined as follows [4,11]. Let E(S) denote the semilattice of idempotents of S. There is a function p : X → E(S) and an action S × X → X such that the following two conditions hold: (E1) p(x) · x = x; (E2) p(s · x) = sp(x)s −1 .
G. Kudryavtseva Faculty of Computer and Information Science, University of Ljubljana, Tržaška cesta 25, 1001 Ljubljana, Slovenia e-mail: [email protected] M. V. Lawson (B) Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK e-mail: [email protected]
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The classifying space of an inverse semigroup
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The set X is also partially ordered when we define x ≤ y when x = p(x) · y. A morphism ϕ : (S, X, p) → (S, Y, q) of étale actions is a map ϕ : X → Y such that q(ϕ(x)) = p(x) for any x ∈ X and ϕ(s · x) = s · ϕ(x) for any s ∈ S and x ∈ X . The category of all étale S-actions is called the classifying space or classifying topos of S and is denoted by B (S). This space is the subject of Funk’s paper [3]. In the last section, we shall need a more general notion of morphism. Let (S, X, p) and (T, Y, q) be étale actions where we do not assume that S and T are the same. Then (α, β) : (S, X, p) → (T, Y, q) is called a morphism if α : S → T is a semigroup homomorphism, β : X → Y is a function such that q(β(x)) = p(x), and β(s · x) = α(s) · β(x). To define our second category, we need some definitions from semigroup theory. An element s of a semigroup S is said to be (von Neumann) regular if there is an element t, called an inverse of s, such that s = sts and t = tst. The set of inverses of the element s is denoted by V (s). In an inverse semigroup S, we define d(s) = s −1 s and r(s) = ss −1 where s −1 is the unique inverse of s. A band is a semigroup in which every element is idempotent and a right normal band is
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