Renault has defined in [7] the cohomology of the inverse semigroup G of the G-sets of a given groupoid G as a functor from the category of G-presheaves to that of abelian groups. We show in our paper that G-presheaves is isomorphic to AbD(G) where D(G) is the division category defined from Loganathan in [6] and used there to give another description of the Lausch cohomology of inverse semigroups. This isomorphism allows us in turn to prove that Renault and Lausch cohomology groups of G are isomorphic

Krakulli, Anjeza

Teliti, Xhezair

Epoka University

122On the cohomology of the inverse semigroup Gof the G-sets of a groupoid GAnjeza Pasku∗, Xhezair Teliti †∗Universiteti Aleksande¨r MoisiuFakulteti i Teknologjise¨ dhe Informacionitanjezakrakulli@uamd.edu.al†Universiteti i Tirane¨sFakulteti i Shkencave te¨ Natyre¨sDepartamenti i Matematike¨sxhezair.teliti@fshn.edu.alAbstract—Renault has defined in [7] the cohomology of theinverse semigroup G of the G-sets of a given groupoid G asa functor from the category of G-presheaves to that of abeliangroups. We show in our paper that G-presheaves is isomorphicto AbD(G) where D(G) is the division category defined fromLoganathan in [6] and used there to give another description ofthe Lausch cohomology of inverse semigroups. This isomorphismallows us in turn to prove that Renault and Lausch cohomologygroups of G are isomorphic.Key words: Groupoid, cohomology, presheaves, inverse semi-group.I. INTRODUCTION AND PRELIMINARIESWe give in this section a few basic notions from groupoidsand inverse semigroups associated to them and show howcohomology groups of a groupoid are defined. All these can befound in [7]. By definition, a groupoid G is a set endowed witha product map (x, y) 7→ xy: G2 → G where G2 is a subsetof G × G called the set of composable pairs, and an inversemap x × x−1 : G → G such that the following relations aresatisfied:(i) (x−1)−1 = x;(ii) (x, y), (y, z) ∈ G2, then (xy, z), (x, yz) ∈ G2 and(xy)z = x(yz);(iii) (x−1, x) ∈ G2 and if (x, y) ∈ G2, then x−1(xy) =y;(iv) (x, x−1) ∈ G2 and if (z, x) ∈ G2, then (zx)x−1 =z.For every x ∈ G, we define d(x) = x−1x as the domain of xand r(x) = xx−1 as the range of x. Note that a pair (x, y) iscomposable only if r(y) = d(x). Also the relations xd(x) =x = r(x)x, suggest that we call the set G0 = r(G) = d(G) theunit space of G. Here is a non trivial example of a groupoid.Example I.1 Let U be a set and S a group which acts on Uon the right. The action of s on u is denoted by u ·s. We let Gbe U × S and define the following groupoid structure: (u, s)and (v, t) are composable only if v = u · s; (u, s)(u · s, t) =(u, st), and (u, s)−1 = (u ·s, s−1). Then, r(u, s) = (u, e) andd(u, s) = (u · s, e). The map (u, e) 7→ u identifies G0 with U .An important notion in the theory of groupoids is that of aG-set. Let G be a groupoid and S a subset of G. We callS a G-set if the restriction of r and d to it is one-to-one, orequivalently if SS−1, S−1S ⊆ G0. The set G of all G-sets ofG can be made into an inverse semigroup, for if S and T areG-sets, then their product ST is again a G-set, and for anyS ∈ G, G−1 ∈ G.To define the cohomology of G we need to define first thepresheaves. For this, let C be any category and A0 a set. The set2A0 of all subsets of A0 when ordered by inclusion becomes acategory: there is an arrow U → V if V ⊆ U . By definition aC-presheaf A from 2A0 to C is a contravariant functor whoseobject map is denoted by U → AU and its morphism mapby AU → AV whenever V ⊆ U . A partial isomorphism,or a partial symmetry φ of A is a bijection φ : V → Uwhere V and U are subsets of A0 together with isomorphismsφ : AV ′ → Aφ(V ′) for any V ′ ⊆ V , which are compatiblewith the restriction morphism. The latter means that for everyV ′′ ⊆ V ′, the following diagram commutesAV ′// Aφ(V ′)AV ′′ // Aφ(V ′′)Two partial isomorphisms φ and φ′ can be composed: if φ :V → U and φ′ : V ′ → U ′, we let V ′′ be φ−1(U ′ ∩ V ) andU ′′ be φ(U ′ ∩ V ); φ′′ = φ ◦ φ′ is the bijection V ′′ → U ′′obtained by composing φ and φ′; and for W ⊆ V ′′ we defineφ′′ : AW → Aφ′′(W ) by composingAW φ” // Aφ′(W ) φ // Aφ◦φ′(W ).The inverse of a partial isomorphism is defined in the obviousway. In this way the set T (A) of partial isomorphisms of Abecomes an inverse semigroup which we call the isomorphisminverse semigroup of the given C-presheaf A.For a given inverse semigroup G, we define a G-presheaf(A,L) to be a C-presheaf A together with a homomorphismL : G → T (A) such that L0 : G0 → 2A0 is an injection.Given a G-presheaf (A,L) of abelian groups one can formISCIM 2013, pp. 122-125 c© 2013 Authors123the following cochain complex. A n-cochain is a functionf : Gn → A which satisfies the following conditions:(i) f(s0, s1, ..., sn−1) ∈ Ar(s0s1...sn−1) ;(ii) f is compatible with the restriction map, that isif U = r(s0s1...sn−1) and V = r(t0t1...tn−1)where ti = esi for some idempotent ei,then f(t0, t1, ..., tn−1) is the restriction off(s0, s1, ..., sn−1) ∈ AU to V ; and(iii) for n > 0, f(s0, ..., si, ...sn−1) ∈ 2A0 whenever siis an idempotentThe set Cn(G,A) of n-cochains is an abelian group underpointwise addition. The sequence0 // C0(G,A) // C1(G,A) // · · ·// Cn(G,A) δn // Cn+1(G,A) // · · ·whereδ0(f(s)) = L(s)f ◦ d(s)− f ◦ r(s) andδnf(s0, ..., sn) = L(s0)f(s1, ..., sn)+n∑i=1(−1)if(s0, ...si−1si, ..., sn)+ (−1)n+1f(s0, ..., sn−1)is a cochain complex. We denote by Zn(G,A) and Bn(G,A)the groups of n-cocycles and that of n-coboundaries. The n-th cohomology group Zn(G,A)/Bn(G,A) will be denoted byHn(G,A).In the next section we will show that for any inverse semigroupS, S-presheaves form a category and that this category isisomorphic to the functor category AbD(S) where D(S) hasobjects all the idempotents of S and morphisms e → f aretriples (e, x, x′) where x′ is the inverse of x and e ≥ xx′,x′x = f . The main result of [2] states that there is only onecohomology functor from a given category to Ab, thereforethe Lausch cohomology defined on AbD(S) has to coincideto that of renault define on S-presheaves.II. S-PRESHEAVES AS FUNCTORSLet S be an inverse semigroup, X a presheaf of abeliangroups over E(S) and α : S → T (X) be a representation ofS by partial symmetries of X .Lemma II.1 Representation α gives rise to an S-module inthe sense of Lausch.Proof. Theorem 5.8 ((i) ⇔ (ii)) of [5] states that α can beregarded as an action of S on the right of the presheaf X withvalues in Ab. Then as shown in p. 33 of [5] one can constructa clifford semigroup (X,⊗) with semilattice of idempotentsE(S) and with a right action of S on X given bya ◦ s = α(es)ρeess−1(a).which satisfies all the properties of an S-module.Let S be a fixed inverse semigroup, we form the categoryof S-presheaves with objects representations of S by partialsymmetries of presheaves of abelian groups over E(S) andmorphisms between two representations α : S → T (X) andβ : S → T (Y ) are S-module morphisms τ : X→ Y betweenthe corresponding S-modules of Lemma II.1 such that ∀s ∈ S,τ(α(s)(x)) = β(s)(τ(x)). (1)Here α(s) is meant to be be one of the components of thecorresponding family and x ∈ X(e) where X(e) is the domainof that component of α(s). We have to show that S-presheavesis indeed a category. The only thing we have to check is thatif α : S → T (X), β : S → T (Y ) and γ : S → T (Z) areobjects from S-presheaves and τ1 : α → β, τ2 : β → γ aremorphisms, then for every s ∈ S and x from some domain ofsome component of α(s) we haveτ2τ1(α(s)(x)) = γ(s)(τ2τ1(x)). (2)From the definitions of τ1 and τ2 we haveτ1(α(s)(x)) = β(s)(τ1(x)) (3)andτ2(β(s)(y)) = γ(s)(τ2(y)). (4)Then replacing in (4) y by τ1(x) we getτ2(β(s)(τ1(x))) = γ(s)(τ2τ1(x)). (5)Now (3) and (5) imply (2).Given an inverse semigroup S with semilattice of idempo-tents E we define a category P(S) with objects the idem-potents E of S and morphisms e → f are pairs (e, s) ∈E × S such that f = s−1es. Composition is given by(s−1es, t)(e, s) = (e, st). Let P (S) the quotient of P(S) bythe congruence on the hom-sets of P(S) generated by the pairs(e, s) ∼ (e, es) and (e, e) ∼ ide.We will write morphisms of P (S) by the same symbols astheir representatives in P(S). Note that the semilattice E(S)is a subcategory of P (S).The next two lemmas show two properties of functors fromAbP (S).Lemma II.2 Every X ∈ AbP (S) gives rise to a right actionof S on the Ab-bundle X = ∪e∈EX(e).Proof. Define a function ◦ : X× S → X bya ◦ s = X(e, s)(a) whenever a ∈ X(e) .Let us check the three properties for the right action of S onX.(Act 3) If a ∈ X(e), then from the definition a◦s ∈ X(s−1es)and the map a 7→ a ◦ s is a morphism in Ab since X(e, s) issuch.(Act 1) If a ∈ X(e), then a◦e = X(e, e)(a) = idX(e)(a) = a.(Act 2) (a ◦ s) ◦ t = X(s−1es, t)X(e, s)(a) = X(e, st)(a) =a ◦ (st).124Lemma II.3 Every X ∈ AbP (S) gives rise to an S-moduleX = ∪e∈EX|E(e) where X|E is the restriction of X in E(S).Proof. We will show that the clifford semigroup X has thestructure of an S-module. From (iii) ⇒ (ii) of Theorem 5.8of [5] we have that the Ab-bundle X of Lemma II.2 can beregarded as a representation of S by partial symmetries of apresheaf with values in Ab in the following way. First, as in theproof of Theorem 5.6 of [5] we form a semilattice of groupsX(e) (though we have one already) by defining for e ≥ f , ρef :X(e)→ X(f) by ρef (a) = a◦f . But a◦f = X(e, f)(a) = a+f . This shows that the clifford semigroup arising by restrictingX in E(S) is the same as the one described in Theorem 5.6of [5]. Then define a partial functiona · s ={a ◦ s if a ∈ X(e) and ss−1 = eundefined elseThis is a right action of S on the presheaf X|E which satisfies(Rep 1)-(Rep 5) of Proposition 5.7 of [5] therefore fromExample 3 of [5] X becomes an S-module with the S actiondefined bya ? s = ρeess−1 · (es) = ρeess−1(a) ◦ (es). (6)On the other hand we see thatρeess−1(a) ◦ (es) = X(ess−1, es)X(e, ss−1)(a)= X(e, es)(a)= X(e, s)(a)= a ◦ s.Comparing with (6) we see that actions ? and ◦ are equal,therefore X is an S-module.Define G : S-presheaves→ AbP (S) on objects by sendingeach representation α : S → T (X) to G(α) : P (S) → Abwhich sends each idempotent e to X(e) and each morphism(e, s) : e→ s−1es to the compositeG(α)((e, s)) = α(es)ρeess−1 . (7)The functorial properties of G(α) are easy to proof if we recallthat (7) defines a right action on the presheaf X and that fora ∈ X(e), G(α)((e, s))(a) is the same as a ◦ s of Example 3of [5].Let τ : α → β is a morphism in S-presheaves where α :S → T (X) and β : S → T (Y ). DefineG(τ) : G(α)→ G(β)as the family{τe : X(e)→ Y (e)|e ∈ E}.To show that G(τ) is natural we have to show that for eache ∈ E, every morphism (e, s) : e → s−1es and every a ∈X(e), we haveτs−1esG(α)(e, s)(a) = G(β)(e, s)τe(a),which from (7) is equivalent toτ(a ◦ s) = τ(a) ◦ s.This is true since from Lemma II.1 X and Y are S-moduleswith action ◦ and τ : X → Y is an S-module morphism.Define G′ : AbP (S) → S-presheaves on objects X inthe following way. From Lemma II.2 X gives rise to a rightaction of S on the Ab-bundle X = ∪e∈EX(e) and then asin the proof of (iii) ⇒ (ii) of Theorem 5.8 of [5] one candefine a representation G′(X) of S by partial symmetries ofthe presheaf X|E . It turns out that G′(X) : S → T (X|E) isdefined by s 7→ X(ss−1, s) where X(ss−1, s) : X(ss−1) →X(s−1s) is the map a 7→ a ◦ s.Lemma II.4 The module of Lemma II.1 arising from therepresentation G′(X) is the same as the module of LemmaII.3 arising from X .Proof. Theorem 5.8 ((ii) ⇒ (i)) and Example 3 of [5] showthat the module of Lemma II.1 arising from the representationG′(X) is the clifford semigroupX of Lemma II.3 consisting ofgroups X(e) together with structure morphisms ρef = X(e, f),and the action of S on X is given bya ? s = ρeess−1 · (es)= X(ess−1, es)X(e, ss−1)(a)= X(e, es)(a)= X(e, s)(a)= a ◦ s.This proves the lemma.Define G′ on morphisms. If τ : X → Y is a naturaltransformation of functors in AbP (S) then τ induces an S-module morphism τ∗ : X → Y of the corresponding S-modules X and Y of Lemma II.3. But Lemma II.4 claims thatX matches to the module arising from G′(X) and so does Yto G′(Y ). Also the fact that τ∗ is a module morphism impliesτ∗X(ss−1, s) = Y (ss−1, s)τ∗,which shows that τ∗ : X→ Y can be regarded as a morphismbetween the respective representations G′(X) and G′(Y ). WedefineG′(τ) = τ∗.The functorial properties are now clear.Theorem II.1 Categories AbP (S) and S-presheaves are iso-morphic.Proof. Let us first show that for every α ∈ S-presheaves wehave G′Gα = α. From the definition of G′ we have that G′Gαis the homomorphismG′Gα : S → T (X)defined bys 7→ Gα(ss−1, s)where from (7), G(α)(ss−1, s) is the morphismGα(ss−1, s) :X(ss−1)→ X(s−1s) = X(s−1(ss−1)s).125defined byGα(ss−1, s) = α((ss−1)s)ρss−1(ss−1)ss−1 = α(s),therefore G′Gα = α. Secondly we show that for every X ∈AbP (S), GG′X = X . For this we have to show that GG′Xsends every morphism (e, s) : e→ s−1es of P (S) to X(e, s).From (7) we haveGG′X(e, s) = G′X(es)ρeess−1 (8)and from the definition of G′ we haveG′X(es) = X((es)(es)−1, es) = X(ess−1, es). (9)But ρeess−1 = X(e, ss−1) and then from (8) and (9) we haveGG′X(e, s) = X(ess−1, es)X(e, ss−1)= X(e, (ss−1)(es))= X(e, es) = X(e, s)as desired.Proposition II.1 For an inverse semigroup S, categoriesP (S) and D(S) of [6] coincide.Proof. First notice that P(S) coincides with C(S) of [6].Let (e, x) : x → x−1ex be a morphism in P(S). We canwrite x−1ex as (ex)−1(ex) and observe that e ≥ (ex)(ex)−1,therefore (e, x) coincides with (e, (ex), (ex)−1) : e → f =(ex)−1(ex) of C(S). Conversely, let (e, x, x−1) : e → f bea morphism in C(S). Since e ≥ xx−1, we have e(xx−1) =xx−1 and then x−1e(xx−1)x = x−1xx−1x which is equiva-lent to x−1ex = x−1x. But f = x−1x, then x−1ex = f andas a consequence (e, x, x−1) matches with (e, x) : e→ x−1exof P(S). Lastly observe that our ∼ is the same as ∼ of p. 379of [6], hence P (S) = D(S).Corollary II.1 Cohomology groups of an inverse semigroupdefined by Lausch are isomorphic to those defined by Renault.Proof. The cohomology of an inverse semigroup S afterRenault is defined in S-presheaves which from Theorem II.1and Proposition II.1 is isomorphic to AbD(S). But AbD(S)is an abelian category with enough injectives, therefore fromthe uniqueness theorem for cohomology functors [2] (see also[3]), there is only one cohomology functor on AbD(S). Sincethe cohomology after Lausch is defined in AbD(S), we havethat both cohomologies coincide.REFERENCES[1] Hilton, P.J., Stammbach, U., A Course in Homological Algebra, SecondEdition, Springer-Verlag, 1997[2] Lang, S., Rapport sur la Cohomologie des Groupes, W. A. Benjamin,New York, Amsterdam, 1966[3] Lausch, H., Cohomology of inverse semigroups, J. Algebra 35 (1975),273-303[4] Lawson, M. V., Inverse Semigroups. The Theory of Partial Symmetries,World Scientific, 1998[5] Lawson, M. V., Steinber, B., Ordered groupoids and Etendues, Cahiersde Topologie et Geometrie Differentielle Categoriques 45.2 (2004): 82-108.[6] Loganathan, M., Cohomology of Inverse Semigroups, J. Algebra 70(1981), 375-393[7] Renault, J., A groupoid approach to C∗-algebras, LNM 793, Springer-Verlag, 1980[8] Schubert. H., Categories, Springer Verlag, 1972

On the cohomology of the inverse semigroup G of the G-sets of a groupoid G

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On the cohomology of the inverse semigroup G of the G-sets of a groupoid G

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