arXiv:1907.10365v2 [math.CT] 2 Aug 2021 Introduction 0 space aifigtesefproperty: sheaf the satisfying in oemrhssfrom emnsagopi Homeo groupoid a termines on where h aeoyo h eaesae over ´etale spaces the of category the n h as presheaf A maps. the and pncovering open ha hoei hrceiaino ´etale groupoids of characterization theoretic Sheaf X X • Let o w pnsets open two For . w entoso ha:a n´tl pc n sacontravarian a as and ´etale space th an of as equivalence the sheaf: of a functor. analog of an definitions is two correspondence This ´etale any dogroup. that a show to We corresponds ´etale groupoid. an as pseudogroup o n homeomorphism any For restrictions fadol fteeeit noe covering open an exists there if only and if safunctor a is X X U X c.[]o eto 1.) Section or [1] (cf. . λµ top h td fHeir[,3 ugssta ti aua orgr a regard to natural is it that suggests 3] [2, Haeflier of study The eatplgclsae n let and space, topological a be = sa rddst therefore, set, oreded an is U { λ U E λ ∩ f X } | U → U top op of α F µ suoru sheaf pseudogroup h aeoyo h hae on sheaves the of category The . X U ( eogto belong U → U U h olwn iga sa equalizer: an is diagram following the , rmatplgclspace topological a from and ) to Set → F X V A . uut3 2021 3, August sa is where , oiYamazaki Koji Y h orcino h esHomeo sets the of correction The . V λ f pseudogroup in H Abstract F sheaf ∈ ( ( X U U Homeo 1 α λ e Homeo let , Set X f , e eeaiaino pseu- a of generalization new a , ) X f o n pnset open any for if, ⇒ An . ( top U stectgr fte(ml)sets (small) the of category the is X Y λ,µ α X sasalctgr.A category. small a is )). top sasubgroupoid a is ( ´etale space ,V U, F etesto h pnsets open the of set the be ( U X E { λµ ), ( U ,V U, otegvntopological given the to ) α f , } eog to belongs of etesto the of set the be ) X over U U seuvln to equivalent is uhta the that such H ⊂ X X fHomeo of X ( salocal a is ,V U, oid H n any and presheaf e t ( ,V U, de- ) X ) According to Haeflier [2, 3], it is natural to regard pseudogroups as ´etale groupoids. A topological groupoid (over X) is a groupoid object in the cate- gory of the topological spaces (such that the object space is X). A topolog- ical groupoid is ´etale if the source map is a local homeomorphism (cf. [5] or Section 1.1). Resende [6] introduced an abstract pseuedogroup as a complete and infinitely distributive , and he gave equivalent corre- spondence among the ´etale groupoids, the abstract pseuedogroups and the quantales, by locale theory. We define a pseudogroup sheaf (cf. Definition 3.2), a new generalization of a pseudogroup, and prove the equivalence of the ´etale groupoids over X and the pseudogroup sheaves on X, as an analogue of the equivalence of the ´etale spaces over X and the sheaves on X. The main result in this paper is the following theorem.

Theorem 4.3. Let X be a T1 space. Then, there exists an equivalence ´ Pse(X) ≃ EtGrpdX .

Pse(X) is the category of the pseudogroup sheaves on X, and EtGrpd´ X is the category of the ´etale groupoids over X. The requirement that the base space is a T1 space in Theorem 4.3 is contrasted with the requirement that the base space is a sober space in Resende’s result. We review sheaves and ´etale spaces in Section 1. We see the correspon- dence from ´etale groupoids to pseudogroup sheaves in Section 2. We define a pseudogroup sheaf, and see the correspondence from pseudogroup sheaves to ´etale groupoids in Section 3. The main result is obtained by summarizing the contents of the previous sections in categorical viewpoints in Section 4. As an added note, the interpretation of a pseudogroup in classical sence as a pseudogroup sheaf is given in Section 5. In this paper, we write the operation of a given category in three different manners. (−) ◦ (−) is the composition of two maps. (−) · (−) is the operation of a given ´etale groupoid. (−) • (−) is the operation of a given pre-pseudogroupoid. We often call a morphism “an arrow” in a given ´etale groupoid or a given pre- pseudogroupoid. The purpose of these manners is to eliminate confusion.

1 Sheaves

Let X be a . A presheaf on X is a contravariant functor on the ordered set of the open sets in X with valued in the category of

2 the (small) sets and the maps. A sheaf on X is a presheaf satisfying the sheaf condition (cf. Definition 1.3). The category of the sheaves on X is equivalent to the category of the ´etale spaces over X (cf. Section1.5).

1.1 Preliminaries Let X be a topological space. First, we mention some notations for topological spaces in this paper. Definition 1.1.

• Xtop is the set of the open sets in X. • N (x) is the set of the open neighborhoods of a point x ∈ X.

• For two points x,y ∈ X, we denote x → y if N (y) ⊂N (x). (In other words, we denote x → y if a sequence {x} converges to y.)

• A topological space X is a T1 space if a relation x → y implies the equality x = y for any two points x,y ∈ X.

Xtop is an oreded set, therefore, Xtop is a small category. We also mention some notations for categories. Definition 1.2. • Ob(C) is the class of the objects of a given category C.

• Cop is the opposite category of a category C.

• A set is small if it is in the given universe (cf. [4]). A category is small if the set of the objects and the set of the mor- phisms are small.

• Set is the category of the (small) sets and the maps. • A filtered set Λ is an ordered set Λ such that, for any two element λ, µ ∈ Λ, there exists an element ν ∈ Λ such that it satisfies the relations λ, µ ≤ ν.

• A direct system (resp. a inverse system) indexed by a filtered set Λ is a functor Λ → Set (resp. Λop → Set). • A direct limit limF (resp. a inverse limit limG) of a direct system F −→ ←− (resp. a inverse system G) indexed by a filtered set Λ is a set defined by the follows:

3 ◦ limF = Fλ/∼, −→ λ where ∼ais a equivalent relation defined by the follows:

∃ x ∼ y (x ∈ Fλ,y ∈ Fµ) ⇔ ν ≥ λ, µ s.t. Fλ≤ν (x)= Fµ≤ν (y).

◦ limG = {(xλ) ∈ Gλ | Gµ≥λ(xλ)= xµ}. ←− λ Y 1.2 Sheaves We define sheaves on a topological space. Definition 1.3 (cf. [1]). A presheaf on a topological space X is a functor op Xtop → Set. A presheaf F is a sheaf (or a sheaf of sections) if, for any U ⊂ X and any open covering {Uλ} of U, the following diagram is an equalizer: F(U) → F(Uλ) ⇒ F(Uλµ), Yλ Yλ,µ where Uλµ = Uλ ∩ Uµ. A section of a (pre)sheaf F on an open set U is an element s ∈ F(U). A morphism between (pre)sheaves is a natural transformation between functors.

1.3 Etale´ spaces The concept of ´etale spaces is equivalent to sheaves. Definition 1.4 (cf. [1]). An ´etale space (or a sheaf of germs) over a topo- logical space X is a local homeomorphism E → X from a topological space E to X. Let p : E → X and q : F → X be ´etale spaces over X. A morphism f : p → q over X is a continuous map f : E → F such that the following diagram is commutative: f E / F p q   X X Let p : E → X be a ´etale space over X. The sheaf Γ(−; p) associated with p is defined by the follows:

Γ(U; p)= {s : U → E | p(s(x)) = x (∀x ∈ U)}

4 1.4 Etale´ spaces associated with presheaves We will consider the inverse construction of the previous subsection. Let F be a presheaf on X. Let x ∈ X be a point. The set N (x) of open neighborhoods of x is an oreded set, therefore, N (x) is a small category. The opposite category N (x)op of N (x) is a filtered set.

Definition 1.5 (cf. [1]). The stalk Fx of F at x is the direct limit of the direct system F|N (x)op . For a section s ∈ F(U) and a point x ∈ U, the germ sx of s at x is the image of s for the natural map F(U) → Fx.

We define a set EF and a map pF : EF → X as the follows:

EF = Fx, x a pF (sx) = x (sx ∈ Fx).

For an open set U ⊂ X and a section s ∈ F(U), a subset [s, U] ⊂ EF is defined as [s, U]= {sx | x ∈ U}. Proposition 1.6 (cf. [1]). The correction of subsets {[s, U] | s, U} deter- mines a unique topology on EF such that {[s, U] | s, U} is a basis for the topology. Moreover, the map pF : EF → X is a local homeomorphism.

pF : EF → X is the ´etale space associated with the presheaf F.

1.5 Categorical viewpoints Some concrete categories are defined as follows. PSh(X) the category of the presheaves on X. Sh(X) the category of the sheaves on X. Et´ X the category of the ´etale spaces over X.

The construction p 7→ Γ(−; p) defines a functor Γ : Et´ X → Sh(X) ⊂ PSh(X). The construction F 7→ pF defines a functor p : PSh(X) → Et´ X . Then, there exists an adjunction p ⊣ Γ : PSh(X) → Et´ X . The unit map µ : Id → Γ ◦ p of the adjunction p ⊣ Γ is called the sheafification. For a presheaf F, the component µF : F → Γ(−,pF ) of µ is defined as follows:

µF,U : F(U) → Γ(U, pF ); s 7→ [x 7→ sx] The counit map p ◦ Γ → Id of the adjunction p ⊣ Γ is isomorphic. A component of the unit map µF : F → Γ(−; pF ) is an isomorphism if and only if the presheaf F is a sheaf. Therefore, the restruction of the adjunction p ⊣ Γ induces an equivalence Sh(X) ≃ Et´ X .

5 2 Etale´ groupoids

A topological groupoid (over X) is a groupoid object in the category of the topological spaces (such that the object space is X). A topological groupoid is ´etale if the source map is a local homeomorphism (cf. Definition 2.1). Any ´etale groupoid has a sheaf associated with the ´etale groupoid (cf. Section 2.2).

2.1 Etale´ groupoids Recall, a (small) groupoid G is a (small) category G such that any mor- phism in G is an isomorphism. (In this paper, we assume that a groupoid is small unless it is confusing.) A topological groupoid is a groupoid G = (G0, G1,s,t,i,inv,comp) such that the set of objects G0 and the set of ar- rows G1 are topological spaces, and that the structure maps (i.e. source map s, target map t, identities map i, inversion map inv and composition map comp) are continuous.

Definition 2.1. A topological groupoid is ´etale if the source map is a local homeomorphism.

Remark 2.2. A topological groupoid is ´etale if and only if the all structure maps are local homeomorphisms.

In this paper, we fix the object space G0. Definition 2.3. Let X be a topological space. A topological groupoid over X is a topological groupoid such that the object space is X.

Morphisms between topological groupoids are continuous functors.

Definition 2.4. Let G and H be topological groupoids over a topological space X. A morphism φ : G→H over X is a functor φ : G→H such that the object map of φ is the identity map of X and that the arrow map of φ is continuous.

2.2 Sheaves associated with ´etale groupoids Let G be an ´etale groupoid over X. We define G(U, V ) as the set {f : −1 U → t (V )|s ◦ f = id} for each open sets U, V ∈ Xtop. The correction of these sets G(U, V ) define a small category G such thatb the objects are the open sets in X. (The composition (−) • (−) of the category G is defined as (g • f)(x)=bg(t(f(x))) · f(x) for x ∈ U, whereb the operation (−) · (−) is the b 6 composition of the groupoid G.) The category G includes the category Xtop as a subcategory. For each open set V ∈ Xtop and each pointb x ∈ X, let Gx(V ) = y limG(U, V ). For each points x,y ∈ X, let Gx = limGx(V ). −→ ←− U∋x V ∋y b b b b Proposition 2.5. The above G satisfies the following:

(1) Ob(Xtop)= Ob(G). b y (2.1) The natural projectionsb Gx → Gx(V ) are injections. y y (This identifies the set Gx as the set Im[Gx → Gx(V )].) b b y (2.2) The map Gx → Gx(Vb) is surjection. b b y∈V a b b (2.3) For three points x,y,z ∈ X, if we have a relation y → z (cf. Definition y z 1.1), then we obtain the inclusion Gx ⊂ Gx. y (2.4) For a germ fx ∈ Gx(V ), fx belongsb to Gbx if and only if we have the relation f(x) → y. b b (3) The presheaf G(−,V ) is a sheaf for each V ∈ Xtop.

Proof. The claims (1)b and (3) is obvious by definition. We will prove the claim (2.1). The map G(U, V1) → G(U, V2) is an injection by definition for any open sets U, V1 and V2 (V1 ⊂ V2). The map Gx(V1) → Gx(V2) is an injection for any pointb x and anyb open sets y V1 and V2 (V1 ⊂ V2). Therefore, the set Gx(= limGx(V )) is identified with ←− b b V ∋y y the set Gx(V ). In other words, the naturalb projectionsb Gx → Gx(V ) are V ∋y \ injections. b b b y y We identify the set Gx as the set Im[Gx → Gx(V )] by the claim (2.1). y For a germ fx ∈ Gx(V ), fx belongs to Gx if and only if, for any open neighborhood V ′ of y, thereb exist an open neighborhoodb b U of x and a section ′ f ∈ G(U, V ) such thatb the germ of f at x coincidesb with fx. We obtain the f(x) claim (2.2) because fx belongs to the set Gx for any fx ∈ Gx(V ). Web will prove the claim (2.3). Suppose that we have the relation y → z. y Take any germ fx ∈ Gx. For any open neighborhoodb V of y,b there exist an open neighborhood U of x and a section f ∈ G(U, V ) such that the germ of f at x coincides with fxb. Any open neighborhood of z is an open neighborhood b

7 of y because of the relation y → z. Therefore, the germ fx belongs to the z set Gx. We will prove the claim (2.4). A proof of the implication f(x) → y ⇒ y fx ∈b Gx is the same as the proof of the claim (2.3). We will prove the y inverse. Suppose that a germ fx ∈ Gx(V ) belongs to Gx. Take any open neighborhoodb V ′ of y. There exist an open neighborhood U of x and a ′ section f ∈ G(U, V ) such that the germb of f at x coincidesb with fx. The point f(x) belongs to the open set V ′. Then, V ′ is an open neighborhood of f(x). Therefore,b we obtain the relation f(x) → y.

Remark 2.6. Suppose that X is a T1 space (cf. Definition 1.1). If there exists y z a germ fx belonging to Gx ∩ Gx, then we obtain the equality y = f(x) = z because of the relation y ← f(x) → z. Therefore, if X is a T1 space, all together the above conditionsb b(2.1), (2.2), (2.3) and (2.4) equivalents the following: y (2) The cannonical map Gx → Gx(V ) induced by the natural projec- y∈V y a tions Gx → Gx(V ) is an isomorphismb b for each open set V ∈ Xtop and each point x ∈ X. b b 3 Pseudogroup sheaves

A pseudogroup is a subgroupoid H of HomeoX satisfying the sheaf prop- erty. We define a pseudogroup sheaf (cf. Definition 3.2), a new generalization of a pseudogroup. The category of the pseudogroup sheaves on X is equiv- alent to the category of the ´etale groupoids over X (cf. Section 4). The interpretation of a pseudogroup in classical sence as a pseudogroup sheaf is given in Section 5.

3.1 Pseudogroup sheaves Let X be a topological space. Suppose that C is a small category with including Xtop as a subcategory, and Ob(Xtop)= Ob(C). Then, the functor op op C(−,V ) : Xtop ⊂C → Set is a presheaf on X for each V ∈ Xtop. For each open set V ∈ Xtop and each point x ∈ X, let Cx(V ) = y limC(−,V ). For each points x,y ∈ X, let Cx = limCx(V ). −→ ←− U∋x V ∋y y Suppose that the cannonical map Cx →Cx(V ) induced by the natural y∈V y a projections Cx → Cx(V ) is an isomorphism for each open set V ∈ Xtop and

8 y y each point x ∈ X. We identify the set Gx as the set Im[Gx → Gx(V )]. Then, y the correction of these sets Cx define a small category G such that the sets y of the objects is X and that the sets ofb the arrows fromb x toby is Cx. (The ⋆ composition (−) · (−) of the category C is defined as (gby · fx) = (g • f)x for x ∈ U, where the operation (−) • (−) is the composition of the ctegory C.)

Lemma 3.1. The above composition (−) · (−) of the category C⋆ is well- defined.

z y Proof. Take a composable pair (gy,fx) ∈ Cy × Cx. For any open neigh- borhood W of z, there exist an open neighborhood V of y and a section g ∈C(V,W ) such that the germ of g at y coincides with gy. There exist an open neighborhood U of x and a section f ∈ C(U, V ) such that the germ of f at x coincides with fx. The composition gy · fx is defined as the germ (g • f)x. Take any sections g′ ∈ C(V ′,W ) and f ′ ∈ C(U ′,V ′) such that the germ ′ ′ of g at y coincides with gy and that the germ of f at x coincides with ′′ ′ fx. There exist an open neighborhood V (⊂ V ∩ V ) of y and a section ′′ ′′ ′ ′′ g ∈ C(V ,W ) such that the restrictions g|V ′′ and g |V ′′ coinside with g . There exist an open neighborhood U ′′ of x and a section f ′′ ∈ C(U ′′,V ′′) ′ ′′ such that the restrictions f|U ′ and f |U ′′ coinside with f . Then, we obtain ′′ ′′ ′ ′ the equality (g • f)x = (g • f )x = (g • f )x. Therefore, the operation (−) · (−) is well-defined.

Definition 3.2. Let X be a T1 space. A pre-pseudogroup on X is a small category C with an embedding functor Xtop ⊂C satisfying the following:

(1) Ob(Xtop)= Ob(C).

y (2) The cannonical map Cx → Cx(V ) induced by the natural projec- y∈V y a tions Cx → Cx(V ) is an isomorphism for each open set V ∈ Xtop and each point x ∈ X.

(3) the category C⋆ is a groupoid.

A pseudogroup sheaf on X is a pre-pseudogroup on X satisfying the following:

(4) the presheaf C(−,V ) is a sheaf for any V ∈ Xtop.

Remark 3.3. When X is not a T1 space, this definition is possible, but could not be reasonable (cf. Proposition 2.5).

9 The first example is the psudogroup sheaf associated with an ´etale groupoid. Example 3.4. Let G be a ´etale groupoid, and let G be the category defined in Section 2.2. If X is a T1 space, then the category G satisfies the conditions (1), (2) and (4) in Definition 3.2. In fact, the categoryb G is a pseudogroup sheaf on X because of Proposition 3.5. b b Proposition 3.5. The groupoid G is isomorphic to the category G⋆. In particular, the ctegory G⋆ is a groupoid. b ⋆ Proof. Define a functorbφ : G → G as the follows:

b φ(fx)= f(x).

In fact, the map φ is compatible with the operations by the follows:

φ(gy · fx) = φ(g • f)x) = (g • f)(x) = g(y) · f(x) (cf. Section 2.2) = φ(gy) · φ(fx).

(Remark that the pair (gy,fx) is a composable pair, therefore, we obtain the equality y = t(f(x)).) The functor φ is a bijection because the source map of the groupoid G is a local homeomorphism. Therefore, we obtain an isomorphism φ : G⋆ → G.

We give otherb examples. Example 3.6. The category Xtop is a pseudogroup sheaf on X.

Example 3.7. Let LoHomeoX (U, V )= {f : U → V : local homeomorphisms} for each open sets U, V ∈ Xtop. The correction of these sets LoHomeoX (U, V ) define a small category LoHomeoX such that the objects are the open sets in X. The category LoHomeoX satisfies the conditions (1), (3) and (4) in Definition 3.2. If X is a T1 space, then LoHomeoX is a pseudogroup sheaf on X. Example 3.8. Let F be a sheaf of group on X. We define C as

F(U) (U ⊂ V ) C(U, V )= (∅ (U 6⊂ V ).

Then C is a pseudogroup sheaf on X.

10 3.2 Etale´ groupoids associated with pre-pseudogroups

Let C be a pre-pseudogroup on a T1 space X. Let s : EC → X be an ´etale space associated with the presheaf C(−, X). Then, EC = Cx(X) =∼ x∈X y y a Cx and s(fx) = x for fx ∈ Cx (cf. Section 1.4). In other words, EC is x,y∈X thea set of arrows of the groupoid C⋆, and s is the source map of C⋆. Proposition 3.9. C⋆ is an ´etale groupoid.

Proof. We have to proof that the identities map i : X → EC , the inversion map inv : EC → EC and the composition map comp : EC ×X EC → EC are continuous. (Then, the target map t : EC → X is continuous because of the equality t = s ◦ inv.) For an open set U ⊂ X and a section f ∈C(U, X), a subset [s, U] ⊂ EC is defined as [s, U] = {fx | x ∈ U}. Recall that the collection of these [f, U] generates the open sets of EC (cf. Section 1.4). Claim.1 i is continuous.

The map i coincides with the inverse of the homeomorphism s|[idX ,X] : [idX , X] → X. Therefore, i is continuous. Claim.2 inv is continuous.

−1 Take any point gy ∈ inv ([f, U]). Denote inv(gy) = fx. This satisfies the equality fx · gy = idy. U is an open neighborhood of x(= g(y)). There exist an open neighborhood V of y and a section g ∈C(V,U) such that the germ of g at y coincides with gx. We obtain the equality (f • g)y = idy. ′ There exists an open neighborhood V ⊂ V of y such that f • g|V ′ = idV ′ . ′ −1 We obtain the inclusion gy ∈ [g, V ] ⊂ inv ([f, U]). Therefore, inv is continuous. Claim.3 comp is continuous.

−1 Take any point (gy,fx) ∈ comp ([h, U]). Denote gy · fx = hx. There exist an open neighborhood V of y and a section g ∈ C(V, X) such that the germ of g at y coincides with gy. There exist an open neighborhood U ′ ⊂ U of x and a section f ∈ C(U ′,V ) such that the germ of f at x coincides with fx. Because of the equality gy · fx = hx, there exists an ′′ ′ open neighborhood U ⊂ U of x such that g • f|U ′′ = hU ′′ . We obtain the ′ −1 inclusion (gy,fx) ∈ [g, V ] ×X [f, U ] ⊂ comp ([h, U]). Therefore, comp is continuous.

11 4 Main result

Let X be a T1 space. Some concrete categories are defined as follows. PPse(X) the category of the pre-pseudogroups on X. Pse(X) the category of the pseudogroup sheaves on X. EtGrpd´ X the category of the ´etale groupoids over X. ´ The construction G 7→ G (cf. Section 2.2) defines a functor G : EtGrpdX → Pse(X) ⊂ PPse(X). The construction C 7→ C⋆ (cf. Section 3.2) defines a ´ functor F : PPse(X) b→ EtGrpdX . We will prove that there exists an ´ adjunction F ⊣ G : PPse(X) → EtGrpdX . We will construct the unit map µ : Id → GF of the adjunction F ⊣ G. ⋆ Let C be a pre-pseudogroup on X. Define a morphism µC : C → C over X as follows: µC(f)(x)= fx (for f ∈C(U, V ) and x ∈ U). c Remark that the morphism C(−,V ) → C⋆(−,V ) between presheaves induced by µC is the sheafification (cf. Section 1.5). c Proposition 4.1. Let C and µC be as above. For any ´etale groupoid G on X and any morphism φ : C → G between pre-pseudogroups, there exists a unique morphism φ¯ : C⋆ → G between ´etale groupoids such that the following diagram is commutative: b µC / ⋆ C ❂ C ❂❂ ❂❂ G(φ¯) φ ❂❂  ❂ b G

Proof. The morphism C(−, X) → C⋆(−,b X) between presheaves induced by ⋆ µC is the sheafification. sC⋆ : C1 → X is the ´etale space associated with the presheaf C(−, X). The sheafcC⋆(−, X) (resp. G(−, X)) is the sheaf associated with the ´etale space sC⋆ (resp. sG : G1 → X, where sG is the source map of the groupoid G). Thec morphism φ :bC(−, X) → G(−, X) between presheaves induces a unique morphism φ¯1 : sC⋆ → sG such that the following diagram is commutative: b

µ C(−, X) C / C⋆(−, X) ❑❑❑ ❑❑ Γ ¯ ❑❑❑ (φ1) φ ❑❑b%  G(−, X),

b 12 (where the functor Γ is the same as in Section 1.5.) Then, the map φ¯1 : ⋆ C1 → G1 can be written concretely as follows:

φ¯1(fx)= φ(f)(x).

Write the target maps as t, and the following equality holds:

t(φ¯1(fx)) = t(φ(f)(x)) = t(φ(f)x) = t(fx).

For any composable pair (gy,fx), the following equality holds:

φ¯1(gy · fx) = φ¯1((g • f)x) = φ(g • f)(x) = (φ(g) • φ(f))(x) = φ(g)(y) · φ(f)(x) = φ¯1(gy) · φ¯1(fx).

⋆ Therefore, the map φ¯1 determines a morphism φ¯ : C → G between ´etale groupoids. Then, the following diagram is commutative:

µC / ⋆ C ❂ C ❂❂ ❂❂ G(φ¯) φ ❂❂  ❂ b G

⋆ Conversely, suppose there exists a morphismb ψ : C → G that makes the above diagram commutative. Then, the following diagram is commutative:

µC C(−, X) / C⋆(−, X) ❑❑❑ ❑❑ Γ ❑❑❑ (ψ1) φ ❑❑b%  G(−, X)

We obtain the equality ψ1 = φ¯1 becauseb of the universality of the sheafi- fication. This implies the equality ψ = φ¯. Therefore, the morphism φ¯ is unique.

Proposition 4.2. The counit map ǫ : F G → Id of the adjunction F ⊣ G is isomorphic.

13 ⋆ Proof. For a ´etale groupoid G, the morphism ǫG : G → G is the unique morphism such that the following diagram is commutative: b µGb G / G⋆ ❁❁ ❁❁ ❁❁ cG(ǫG ) b id ❁❁ b G. Then, the following diagram is commutative: b µGb G(−,V ) / G⋆(−,V ) ❏❏ ❏❏ ❏❏ Γ ǫ′ ❏❏ c ( ) b id ❏❏b%  G(−,V ), where ǫ′ is the morphism between ´etale spaces determined by the arrow map ⋆ ′ b (ǫG)1 : G1 → G1. These morphisms ǫ coincide with the components of the counit map of the adjunction p ⊣ Γ : PSh(X) → Et´ X (cf. Section 1.5) becauseb of the universality of the sheafification. These are isomorphisms. Therefore, ǫ is isomorphic.

⋆ A component of the unit map µC : C → C is an isomorphism if and only if the pre-pseudogroup C is a pseudogroup sheaf. Therefore, the restriction ´ of the adjunction F ⊣ G induces an equivalencec Pse(X) ≃ EtGrpdX . We obtain the following theorem.

Theorem 4.3. Let X be a T1 space. Then, there exists an equivalence ´ Pse(X) ≃ EtGrpdX .

5 Classical pseudogroups

Let X be a T1 space, and let C be a pre-pseudogroup on X. Any section f ∈C(U, V ) has the underlying map f : U → V in the following way: Take ∼ y any point x ∈ U. The germ fx of f at x belongs to Cx(V ) = Cx. Then, y∈V a y there exists exactly one point y ∈ V such that the germ fxbelongs to Cx. We define f(x)= y. The map f : U → V is a local homeomorphism because of the equality −1 f = t ◦ (s|[f,U]) , where t is the target map and s is the source map. Moreover, the correspondence f 7→ f define a morphism over X from the pre-pseudogroup C to the pseudogroup sheaf LoHomeoX (cf. Example 3.7).

14 Lemma 5.1.

• id = id.

• g • f = g ◦ f.

Proof. We will prove the equality id = id. Take any identity id ∈ C(U, U) and any point x ∈ U. For any open neighborhood V ⊂ U of x, the restriction id|V of id belongs to the set C(V,V ). Therefore, the germ idx belongs to the x set Cx . We obtain the equality id(x)= x. We will prove the equality g • f = g ◦ f. Take any composable pair (g,f) ∈C(V,W )×C(U, V ) and any point x ∈ U. Let y = f(x). The germ gy g(y) ′ beongs to the set Cy . For any open neighborhood W ⊂ W of g(y), there exist an open neighborhood V ′ ⊂ V of y and a section g ∈ C(V ′,W ′) such that the germ of g at y coincides with gy. There exist an open neighborhood U ′ ⊂ U of x and a section f ∈C(U ′,V ′) such that the germ of f at x coincides g(y) with fx. Therefore, the germ (g • f)x belongs to the set Cx . We obtain the equality (g • f)(x)= g(y) = (g ◦ f)(x).

A pseudogroup in classical sence corresponds with a concrete pseudogroup sheaf.

Definition 5.2. A pseudogroup sheaf C on X is concrete if the functor C → LoHomeoX ; f 7→ f is faithful.

Let C be a concrete pseudogroup on X. Then the subcategory of invert- ible morphisms of C is a pseudogroup in a classical sence. Conversely, let Q be a pseudogroup in a classical sence. For two open sets, define a set C′(U, V ) by the follows:

.{ C′(U, V )= {i ◦ f | f ∈ Q(U, V ′), i is the inclusion map V ′ ֒→ V

The correction of these sets C′(U, V ) determines a pre-pseudogroup C′. We can obtain a concrete pseudogroup sheaf C by the sheafification of the pre- pseudogroup C′.

References

[1] Otto Forster. Lectures on Riemann surfaces, volume 81. Springer Science & Business Media, 2012.

15 [2] Andr´eHaefliger. Structures feuillet´ees et cohomologie `avaleur dans un faisceau de groupo¨ıdes. Commentarii Mathematici Helvetici, 32:248–329, 01 1958.

[3] Andr´eHaefliger. Homotopy and integrability, volume 197, pages 133–163. 11 2006.

[4] Saunders Mac Lane. Categories for the working mathematician, vol- ume 5. Springer Science & Business Media, 2013.

[5] Ieke Moerdijk and Janez Mrcun. Introduction to foliations and Lie groupoids, volume 91. Cambridge University Press, 2003.

[6] Pedro Resende. Etale´ groupoids and their quantales. Advances in Math- ematics, 208(1):147–209, 2007.

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