Some Results on Pseudomonoids

Journal of Advanced Studies in Topology 6:2 (2015), 43–55

Some results on pseudomonoids

A. Dehghan Nezhada,b,∗, S. Shahriyarib aDepartment of Mathematics, Yazd University, 89195-741, Yazd, Iran. bSchool of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran.

Abstract In this paper we follow the work of L´opez and Masa (The problem 31.10) given in [1]. The purpose of this work is to study the new concepts of pseudomonoids. We also obtain some interesting results. Keywords: Pseudogroup, pseudomonoid, diﬀeology space. 2010 MSC: 58H10, 20N20, 20N25, 08A72.

1. Preliminaries

In mathematics, a pseudogroup is an extension of the group. But one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra (quasigroup, for example). A theory of pseudogroups developed by E´lie Cartan in the early 1900’s. Recall that a transformation group G on X is a subgroup in a group Hom(X), so each g ∈ G is a diffeomorphism of X. A local homeomorphism is a homeomorphism f : U −→ V, where U and V are open subsets in X. Certainly, the set of local homeomorphisms is not a group of transformations because the composition is not well-defined. In studying a geometrical structure, it is fruitful to study its, group of automorphisms. Usually, these automorphisms are not globally defined. They therefore do not form a group in the present sense of the word but rather a pseudogroup, or the more general case, pseudomonoid. In general, pseudogroups were studied as a possible theory of infinite- dimensional Lie groups. The concept of a local Lie group, namely a pseudogroup of functions defined in neighborhoods of the origin of E, is actually closer to Lie’s original concepts of Lie group, in the case where the transformations involved depend on a finite number of parameters, than the contemporary definition via manifolds. Examples of infinite-dimensional pseudogroups abound, beginning with the pseudogroup of all diffeomorphism of E. The interest in mainly in sub-pseudogroup of the diffeomorphisms, and therefore with objects that have a Lie algebra analogue of vector fields. The methods proposed by Lie and Cartan for studying these objects have become more practical given the progress of computer algebra. A generalization of the

∗Corresponding author Email addresses: [email protected] (A. Dehghan Nezhad ), [email protected] (S. Shahriyari) Received: 12 January 2015 Accepted: 26 January 2015 http://dx.doi.org/10.20454/jast.2015.901 2090-8288 c 2015 Modern Science Publishers. All rights reserved. A. D. Nezhad, S. Shahriyari, Journal of Advanced Studies in Topology 6:2 (2015), 43–55 44 notion of transformation group is a pseudogroup. We summarize and develop some basics of pseudomonoids in chapter two. Here, we introduce the notion of pseudogroup, following chapter 3 of W. Thurston [12] and [8, 9]. Deﬁnition 1.1. (Pseudogroup) Let X be a topological space. Let Γ be a local homeomorphisms set f : Uf −→ Vf where Uf ,Vf are open subsets of X. The set Γ is called a pseudogroup on X if satisfying the following conditions: (1) The identity belong to Γ.

(2) If f ∈ Γ, then f |U ∈ Γ for any open U ⊆ Uf . −1 (3) If f ∈ Γ, then f : Vf −→ Uf is in Γ.

(4) If f, g ∈ Γ and Vf ⊆ Ug, then g ◦ f ∈ Γ. (5) If f is a local homeomorphism of M such that each point in the domain of f has a neighborhood U such that f |U ∈ Γ, then f ∈ Γ. Example 1.2. [5, 6] The natural examples of pseudogroups are: (1) Let G be a Lie group acting on a manifold X. Then we define the pseudogroup as the set of all pairs (G|U, U) where U is the set of all open subsets of X, (2) Pseudogroup of automorphisms of a tensor field, (3) Pseudogroup of analytical (or symplectic) diffeomorphisms,

n (4) The pseudogroup PL of piecewise linear homeomorphisms between open subsets of R . Next we are going to deﬁne the notion of a pseudomonoid is a collection of transformations which is closed under certain properties.

2. Pseudomonoid We now change our focus from pseudogroup to pseudomonoid. We want to introduce a generalization of pseudogroups. In the new concept, we change the homeomorphism maps to continuous maps. In this case, the elements of Γ aren’t necessarily invertible. So part (3) of the Definition 1.1 can’t be defined in the new concept. Thus we define the pseudomonoid following.

Deﬁnition 2.1. Let X be a topological space and let Γ be a set of locally continuous functions f : Uf −→ X where Uf is an open subset of X. The set Γ is called a pseudomonoid on X if satisfying the following conditions: (1) The identity belong to Γ.

(2) If f ∈ Γ, then f|U ∈ Γ for any open U ⊆ Uf .

(3) If f, g ∈ Γ and im(f) ⊆ Ug, then g ◦ f ∈ Γ. The pair (X, Γ) is called a pseudomonoid. Remark 2.2. Any pseudogroup on X is a pseudomonoid on X also. Example 2.3. We can construct many examples: (1) For any topology space X all locally continuous functions on X form a pseudomonoid on its. This pseudomonoid is largest pseudomonoid on X. In the other words, any pseudomonoid on X is a subset of its.

(2) All linear maps on a Banach space E and their restriction to open subsets. A. D. Nezhad, S. Shahriyari, Journal of Advanced Studies in Topology 6:2 (2015), 43–55 45

2.1. Basic definitions and notations Before we define some concepts of pseudomonoid for a topological space, we need to introduce the following definitions. See below for more details. A Cr-pseudomonoid, is a pseudomonoid Γ on a Cr-manifold that Γ contains Cr-maps (with 0 ≤ r ≤ ∞). We say S ⊂ Γ generates Γ and write Γ =< S >, if composition and restriction to open subsets in S obtain Γ (expect identity). The orbit of a point x ∈ X is the set

Γ(x) := {f(x): f ∈ Γ and x ∈ Uf }.

We will denote by X˜ the set of all orbits of (X, Γ). The stability or isotropy of x, show by Γx, contain all f ∈ Γ leaving x ﬁxed: Γx := {f ∈ Γ: x ∈ Uf , f(x) = x} .

A path of x ∈ X to y ∈ X is a sequence x = y1, ··· , yn = y of points in X such that Γ(yi) ∩ Γ(yi+1) 6= ∅ for all i = 1, ..., n − 1. We will deﬁne on X the following equivalence relation: ”xRy if and only if there exists a path of x to y”. The equivalence classes of R are said leaves of (X, Γ). We will denote by Lx the leaf contain of x ∈ X. The notation X/Γ means the quotient space of R with quotient topology.

The union of leaves that meet X0 ⊆ X is called the saturation of X0 and denoted by Γ(X0). We say the set X0 invariant or saturated, if X0 = Γ(X0). Let π : X → X/Γ is the quotient projection. Then −1 Γ(X0) = π (π(X0)) = ∪x∈X0 Lx, where Lx shows the leaf of Γ that x ∈ Lx. Γ is called transitive if for any x, y ∈ X, there exists f ∈ Γ that f(x) = y. A sub-pseudomonoid of Γ is a subset Σ ⊆ Γ which is a pseudomonoid on X also.

Deﬁnition 2.4. Let Γ1 and Γ2 are two pseudomonoids on X1 and X2 (resp). The structure (X2, Γ2) is called a strong sub-pseudomonoid of (X1, Γ1) if X2 is a subspace of X1, and {i ◦ g : g ∈ Γ2} ⊆ {f ◦ i : f ∈ Γ1} where i : X2 → X1 is inclusion map. Example 2.5. (1) Let Γ be the set of all locally continuous maps of X. Then any pseudomonoid on X is a sub-pseudomonoid of Γ. (2) Let Γ be contain of identity and all identity restrictions to open subsets of X. Then Γ is a sub- pseudomonoid of any pseudomonoid on X. (3) Let M be a smooth manifold and let Cr(M) be all locally Cr-maps on M. Then Cr1 (M) is a sub- r pseudomonoid of C 2 (M) where 0 ≤ r2 ≤ r1 ≤ ∞. m ∞ m n ∞ n (4)( R ,C (R )) is a strong sub-pseudomonoid of (R ,C (R )) for all m ≤ n.

Lemma 2.6. Let Γ be a pseudomonoid on X. If Γx is isotropy of x, then the pseudomonoid is generated by Γx is the set {g|U : U is an open subset of Ug, x ∈ Ug, g(x) = x}.

Proof. Λ stands for the set. We have divided the proof into two parts: First, we show that Λ is a pseudomonoid, next we will prove that Λ =< Γx >. Clearly, identity belongs to Λ and (2) of pseudomonoid definition is valid by the definition of Λ. Let f1, f2 belong to Λ and f2 ◦ f1 is definition. Then there are g1, g2 ∈ Γ such that fi = gi|Ui ,Ui ⊂ Ugi , x ∈ Ugi and gi(x) = x ,i = 1, 2. Define g = g2 ◦ g1, there is a U ⊂ Ug that f2 ◦ f1 = g|U . Thus (3) of pseudomonoid definition is valid. Finally, we conclude that Λ is a pseudomonoid. Obviously, Γx is a subset of Λ. Since Λ is a pseudomonoid and Γx is a subset of Λ, we have < Γx >⊆ Λ. By definition, Λ ⊆< Γx >, So Λ =< Γx > and this finishes the proof.

Lemma 2.7. Let S be a subset of Γ and S generates Γ. Then

Γ(x) = {f1 ◦ · · · ◦ fn(x)| fi ∈ S, n ∈ N, if the composition is deﬁned on Ux} ∪ {x} for any x ∈ X. A. D. Nezhad, S. Shahriyari, Journal of Advanced Studies in Topology 6:2 (2015), 43–55 46

Proposition 2.8. If X be T1-space and if π is a closed map, then any leaf of Γ is a closed subset of X.

Proof. The singleton {x} is a closed set, because X is T1-space. It is evident that π is a closed map. −1 Thus π(x) is a closed set. From continuously of π, we conclude π (π(x)) = Lx is a closed set in X which completes the proof.

c Proposition 2.9. If X0 ⊆ X is an invariant subset under Γ, then X0 = X − X0 is invariant under Γ too.

c Proof. Since X0 is invariant. Consequently, X0 is the union of some leaves of Γ. Thus X0 is union some c leaves in Γ that do not intersect with X0, therefore X0 is invariant.

Example 2.10. In following, we give several examples of pseudomonoids. n (1) Consider a pseudomonoid (X, Γ) where X = R with Γ =< f >, which f is deﬁned by

n n f : R −→ R n X (x1, x2, ··· , xn) 7−→ ( xi, 0, ··· , 0). i=1

n n Then the isotropy a = (a1, ··· , an) ∈ R is the set Γa = {id|U : a ∈ U ⊆ R } when aj 6= 0 for some n j ≥ 2 and Γa = {g ∈ Γ: a ∈ Ug} when aj = 0 for all j ≥ 2. The orbit of a = (a1, ··· , an) ∈ R Pn n Pn is Γ(a) = a, ( i=1 ai, 0, ··· , 0) and the leaf contains a ∈ R is La = {(x1, ··· , xn)| i=1 xi = Pn ∼ i=1 ai}. So X/Γ = R. n (2) Let X = S and Γ is generated by

n n f : S −→ S x 7−→ −x.

n n ∼ n Then the isotropy of a ∈ S is Γa = {id|U : a ∈ U ⊆ S } , and we have X/Γ = RP , because La = Γ(a) = {a, −a}. n (3) Suppose that X = R \{0} and consider the pseudomonoid < f > on X, where

n n f : R \{0} −→ R \{0} x 7−→ (kxk, 0, ··· , 0).

n For any a = (a1, ··· , an) ∈ R \{0} the isotropy of its Γa = {g ∈ Γ: a ∈ Ug} if a1 > 0 and aj = 0 n for all j ≥ 2, other wise Γa = {id|U : a ∈ U} . For any a ∈ R \{0}, we have the orbit of its Γ(a) = {a, (kak, 0, ··· , 0)}. The leaf contain of a is

n La = x ∈ R \{0} : kxk = kak . ∼ Therefore, we conclude X/Γ = R. n (4) Assume (X, Γ) is a pseudomonoid where, X = R \{0}. The set Γ is generated by

n n f : R \{0} −→ R \{0} x x 7−→ . kxk

Then the isotropy set Γa = {g ∈ Γ: a ∈ Ug} if kak = 1, and other wise Γa = {id|U : a ∈ U} . We have + ∼ n−1 Γ(a) = {a, a/kak}, La = {λa : λ ∈ R }. Finally, we earn that X/Γ = S . (5) Choose X = R and Γ =< f >, which f(x) = x + 1. It is a simple matter to check that Γa = ∼ 1 {id|U : a ∈ U} , Γ(a) = {a + n|n ∈ N ∪ {0}} and La = {a + n|n ∈ Z}. Therefore X/Γ = S . A. D. Nezhad, S. Shahriyari, Journal of Advanced Studies in Topology 6:2 (2015), 43–55 47

n (6) Let X = R and Γ =< fi, gi : i = 1, ··· , n >, where

n n fi, gi : R −→ R fi(x1, ··· , xn) = (x1, ··· , xi + 1, ··· , xn),

gi(x1, ··· , xn) = (x1, ··· , xi + αi, ··· , xn).

Then by using the Lemma 2.7 we have:

Γx = {id|U : x ∈ U} Γ(x) = {(x1 + m1 + k1α1, ··· , xn + mn + knαn): mi, ki ∈ N ∪ {0}} Lx = {(x1 + m1 + k1α1, ··· , xn + mn + knαn): mi, ki ∈ Z}. ∼ n Now, if αi ∈ Q for all i = 1, ··· , n, then X/Γ = T (n-torus). If αi ∈/ Q for some i = 1, ··· , n, then ∼ n X/Γ = Tα = Tα1 × · · · × Tαn . Example 2.11. (a) The nth symmetric product of topological space Y [13]. Let (Y, ∗) be a pointed n topological space. Suppose that X = Πi=1Y and Γ is a pseudomonoid such that it is generated by the maps

fi,j : X −→ X

(x1, ··· , xi, ··· , xj, ··· , xn) 7−→ (x1, ··· , xj, ··· , xi, ··· , xn)

for any 1 ≤ i, j ≤ n and i 6= j. Then Γ (x1, ··· , xn) = (xσ(1), ··· , xσ(n)): σ ∈ Sn ,

n where Sn is the symmetric group. Consequently, we have X/Γ = SP Y. (b) The inﬁnite symmetric product of topological space Y [13]. Let Y is a topological space. Consider

the pseudomonoid (X, Γ) such that X = Πm∈NY . Here the set Γ is a pseudomonoid such that it is generated by the following maps on X.

fi,j : X −→ X

(x1, ··· , xi, ··· , xj, ··· ) 7−→ (x1, ··· , xj, ··· , xi, ··· )

n for all i, j ∈ N and i 6= j. Therefore, we have X/Γ = SPY = colim (SP Y )

Lemma 2.12. Let {Γi}i∈I is a collection of pseudomonoids on X. Then ∩i∈I Γi is a pseudomonoid on X too.

Proof. Because any Γi contain the identity, therefore the ∩i∈I Γi contain the identity. Let f, g ∈ ∩i∈I Γi that im(f) ⊆ Ug. Then f, g ∈ Γi for any i ∈ I. thus g ◦ f ∈ Γi for all i ∈ I. Hence g ◦ f ∈ ∩i∈I Γi. Let f ∈ ∩i∈I Γi and U ⊆ Uf . Then f ∈ Γi for any i ∈ I, thus f|U ∈ Γi for any i ∈ I. Thus f|U ∈ ∩i∈I Γi and this ﬁnishes the proof.

Proposition 2.13. Let S is a family of locally continuous maps on X. Then the intersection all pseudomonoids on X that contain S is the pseudomonoid such that it is generated by S.

Proof. Suppose Γ1 is the pseudomonoid is generated by S and Γ2 is the intersection all pseudomonoids on X and contain S. By the Lemma 2.12, the set Γ2 is a pseudomonoid. Because Γ1 contain S, thus by to deﬁne Γ2 ⊆ Γ1. From any pseudomonoid contain S is a superset of Γ1. We have Γ1 ⊆ Γ2. Thus, we showed that Γ1 = Γ2. A. D. Nezhad, S. Shahriyari, Journal of Advanced Studies in Topology 6:2 (2015), 43–55 48

Proposition 2.14. If Γ is a pseudomonoid on X and

Γ|U = {f ∈ Γ: Uf ⊆ U, im(f) ⊆ U} where U an open subset of X, then Γ|U is a pseudomonoid on U.

The restriction of Γ to a subspace X0 ⊆ X,Γ|X0 , contains continuous maps between open subsets of X0 that can be locally extended to maps in Γ. If X0 is open in X, then Γ|X0 consists of maps in Γ whose domains and images are subsets of X0 (the Proposition 2.14).

Lemma 2.15. If X0 ⊆ X is a subspace of X, then

Γ|X0 (x) ⊆ Γ(x) ∩ X0 for any x ∈ X0.

Proof. Let y ∈ Γ|X0 (x). Thus there is a f ∈ Γ|X0 (x) that y = f(x). It follows that f can be locally 0 0 extended to f ∈ Γ, therefore y = f(x) = f (x) ∈ Γ(x). Finally, we get Γ|X0 (x) ⊆ Γ(x) ∩ X0.

Lemma 2.16. If U is an open subset of X, then

Γ|U (x) = Γ(x) ∩ U for any x ∈ U.

Proof. We can prove Γ|U (x) ⊆ Γ(x) ∩ U as the Lemma 2.15. Now, assume y ∈ Γ(x) ∩ U, there is a continuous map f ∈ Γ that y = f(x). Because y and x are in U. We can restrict f to an open subset U 0 ⊆ U 0 0 0 and contain x, where f(U ) ⊆ U. So f = f|U 0 is a member of Γ|U . Thus, we have y = f (x) ∈ Γ|U (x).

Note 2.17. In the Lemma 2.15 may not be equal (see the following example).

n Example 2.18. Assume X = R and Γ =< f > where,

n n f : R −→ R (x1, ··· , xn) 7→ (1, x1 + x2, x1 + x2 + x3, ··· , x1 + ··· + xn)

n−1 and choose X0 = R × {0} . Simplify, we get Γ|X0 =< idX0 >=⇒ Γ|X0 (0, ··· , 0) = {(0, ··· , 0)}. But (1, 0, ··· , 0) ∈ Γ (0, ··· , 0) . Therefore Γ (0, ··· , 0) ∩ X0 = {(0, ··· , 0), (1, 0, ··· , 0)}, which is not equal Γ|X0 (0, ··· , 0) .

Let Γ1, ··· , Γn are pseudomonoid on X1, ··· ,Xn, resp. Then the pseudomonoid is generated by S = {f1 × · · · × fn : fi ∈ Γi with 1 ≤ i ≤ n} and it is called product of Γ1 × · · · × Γn on X1 × · · · × Xn.

n n Proposition 2.19. If Γ = Πi=1Γi is product pseudomonoid on Πi=1Xi, then

n Γ(x1, ··· , xn) = Πi=1Γi(xi)

n for any (x1, ··· , xn) ∈ Πi=1Xi. A. D. Nezhad, S. Shahriyari, Journal of Advanced Studies in Topology 6:2 (2015), 43–55 49

Proof. From the Lemma 2.7, the proposition is proved.

Given X = ti∈I Xi that any Γi is a pseudomonoid on Xi, for i ∈ I. The pseudomonoid is generated by ∪i∈I Γi is called composition of pseudomonoids Γi and show by ti∈I Γi.

Lemma 2.20. Under the above assumptions, for x ∈ Xj where, j ∈ I we have

(ti∈I Γi)(x) = Γj(x).

Proof. We can prove from the Lemma 2.7 and the composition deﬁnition.

Proposition 2.21. If (X, Γ) is a pseudomonoid and if Γ0 is a sub-pseudomonoid of Γ on X, then the following properties are being held: 0 0 (1) Isotropy: For any x ∈ X,Γx ∩ Γ = Γx. (2) Orbit: Γ0(x) ⊆ Γ(x), for every x ∈ X. 0 0 0 (3) Leaves: Suppose that Lx and Ly are the leaves of Γ contain of x and y in X, resp. Then, the following statements are being held: 0 0 • if Lx = Ly, then Lx = Ly. 0 • Lx ⊆ Lx. 0 0 (4) Subspace: If X0 is a subspace of X, then Γ |X0 ⊆ Γ|X0 . Thus, for any open subset U of X,Γ |U ⊆ Γ|U . 0 (5) Product: Let Γi is a sub-pseudomonoid of Γi on topological space Xi, for any i = 1, ··· , n. Then 0 0 Γ1 × · · · × Γn ⊆ Γ1 × · · · × Γn. 0 0 6) Composition: If for any i ∈ I,Γi to be sub-pseudomonoid of Γi, then ti∈I Γi ⊆ ti∈I Γi.

0 0 Proof. Simplify, all statements is proved expect (3). For the ﬁrst part of (3) proof, suppose that Lx = Ly, 0 0 then there is a sequence x = x1, ··· , xn = y of X that Γ (xj) ∩ Γ (xj+1) 6= ∅ with (j = 1, ··· , n − 1). From the (2) we have Γ(xj) ∩ Γ(xj+1) 6= ∅ with (j = 1, ··· , n − 1). We thus get Lx = Ly. For proved the second 0 0 0 part of (3) we assume y is a member of Lx. We have Lx = Ly. By the ﬁrst part, we have y ∈ Lx. Therefore, 0 here Lx ⊆ Lx.

Deﬁnition 2.22. Let Γ is a pseudomonoid on X. Then, a subset µ ⊆ X is called minimal, if it satisﬁes the following properties:

(1) µ is nonempty, closed and invariant,

0 0 0 (2) if µ ⊆ µ is a closed, invariant subset, then either µ = ∅ or µ = µ. Proposition 2.23. If Γ is a pseudomonoid on Hausdorﬀ compact space X, then there is a minimal subset of X.

Proof. Show Z the family of nonempty, compact and invariant subsets of X. Clearly Z is nonempty. ∞ Given a sequence µ1 ⊃ µ2 ⊃ · · · of members of Z, thus µ = ∩i=1µi is nonempty, invariant, compact and µ ⊆ µi for every i ∈ N. By Zorn’s lemma, Z have a minimal member, what is a minimal subset of X.

We want to define étalémorphisms by involving arbitrary local maps, precisely, we introduce the following concept.

Deﬁnition 2.24. Let Γ1 and Γ2 are pseudomonoid on X1 and X2, respectively. A morphism Φ : (X1, Γ1) → (X2, Γ2) is a maximal collection of continuous maps of open subsets of X1 to X2 satisfying the following properties:

(1) If φ ∈ Φ, h ∈ Γ1 and h2 ∈ Γ2, then h2 ◦ φ ◦ h1 ∈ Φ. A. D. Nezhad, S. Shahriyari, Journal of Advanced Studies in Topology 6:2 (2015), 43–55 50

(2) The domains of members of Φ cover X1.

(3) If φ, ψ ∈ Φ and x ∈ domφ ∩ domψ, then there is h2 ∈ Γ2 with φ(x) ∈ dom h2 that h2 ◦ φ = ψ on some neighborhood of x.

Proposition 2.25. If Φ : (X1, Γ1) → (X2, Γ2) is a morphism, then [ Γ2 ψ(x) = φ Γ1(x) ∩ dom φ φ∈Φ for all x ∈ X and for every ψ ∈ Φ that x ∈ dom ψ. Proof. Fix ψ ∈ Φ. Let y ∈ Γ2 ψ(x) is an arbitrary element. Then for some h2 ∈ Γ2 we have y = h2(ψ(x)). S S Since h2 ◦ ψ is a member of Φ, y ∈ φ∈Φ φ Γ1(x) ∩ dom φ . Therefore Γ2 ψ(x) ⊆ φ∈Φ φ Γ1(x) ∩ dom φ . S If y ∈ φ∈Φ φ Γ1(x) ∩ dom φ , then we have y ∈ φ Γ1(x) ∩ dom φ for some φ ∈ Φ. There is a h1 ∈ Γ1 that y = φ(h1(x)). By the property (3) of morphism deﬁnition, there is h2 ∈ Γ2 that h2 ◦ ψ = φ ◦ h1 on some S neighborhood of x, therefore y ∈ Γ2 ψ(x) . Thus, φ∈Φ φ Γ1(x) ∩ dom φ ⊆ Γ2 ψ(x) . The proof is now complete.

Corollary 2.26. The morphism Φ induces a map Φorb : X˜1 −→ X˜2 is deﬁned by Φorb Γ1(x) = Γ2 φ(x) . for any φ ∈ Φ that x ∈ dom φ. S Proof. Deﬁne Φorb Γ1(x) = φ∈Φ φ Γ1(x) ∩ dom φ .

Proposition 2.27. Let Φ : (X1, Γ1) → (X2, Γ2) is a morphism. Then Φ induces a continuous map from X1/Γ1 into X2/Γ2 that be deﬁned by:

Φ: X1/Γ1 −→ X2/Γ2 0 Lx 7−→ Lφ(x) such that φ ∈ Φ where, x ∈ dom φ, and L and L0 are leaves in Γ and Γ0, resp.

0 0 Proof. We first show that, if Γ1(x) ∩ Γ2(y) 6= ∅, then Γ2(ψ(x)) ∩ Γ2(ψ (y)) 6= ∅ for any ψ, ψ ∈ Φ where, 0 0 0 x ∈ domψ and y ∈ domψ . Let t ∈ Γ1(x)∩Γ2(y), then there are h1 and h1 in Γ1 that t = h1(x) = h1(y). By the property (2) morphism definition, there is φ that t ∈ domφ. From the property (3) of morphism, there are two 0 0 0 0 0 0 maps h2 and h2 in Γ2 where, ψ(x) ∈ domh2, ψ (y) ∈ domh2 and h2 ◦ψ = φ◦h1, h2 ◦ψ = φ◦h1. Hence φ(t) ∈ 0 Γ2(ψ(x))∩Γ2(ψ (y)). Now, if Lx = Ly, then there are x = x1, ··· , xn = y in X1 that Γ1(xj)∩Γ1(xj+1) 6= ∅ for j = 1, ··· , n−1. By the first part of the proof we have Γ2 φj(xj) ∩Γ2 φj+1(xj+1) 6= ∅ for j = 1, ··· , n−1 where, x ∈ dom φ . Then L0 = L0 for any φ , φ in Φ that x ∈ dom φ and y ∈ dom φ . j j φ1(x) φn(y) 1 n 1 n

The set of morphisms (X1, Γ1) −→ (X2, Γ2) will be denoted by C(Γ1, Γ2). A morphism Φ : (X1, Γ1) −→ ∞ ∞ ∞ (X2, Γ2) is said to be of class C if Γ1 and Γ2 are C pseudomonoids and Φ consists of C maps; the set ∞ ∞ of C morphisms (X1, Γ1) −→ (X2, Γ2) will be denoted by C (Γ1, Γ2).

Proposition 2.28. Let Φ be a continuous map collection of open subsets of X1 to X2 satisfying the properties (1) - (3) of Deﬁnition 2.24. Then Φ is a morphism (X1, Γ1) −→ (X2, Γ2) if and only if Φ is closed under combinations of maps. A. D. Nezhad, S. Shahriyari, Journal of Advanced Studies in Topology 6:2 (2015), 43–55 51

Proof. Suppose that Φ : (X1, Γ1) → (X2, Γ2) is a morphism. Let Ψ contain all possible combinations of maps in Φ. Then Ψ satisfies (1)-(3) of Definition 2.24 and Ψ contains Φ, by the maximality of Φ, Φ = Ψ. Now, Suppose that Φ is closed under map combinations of and let us show the maximality of Definition 2.24. Assume Φ is contained in another collection Ψ of continuous maps of open subsets of X1 to X2 satisfying the (1)-(3) of Definition 2.24. Take any ψ ∈ Ψ. By property (2) for Φ, for every x ∈ dom ψ there is some φ ∈ Φ with x ∈ dom φ. From property (3) of Ψ, there is some h2 ∈ Γ2 with φ(x) ∈ dom h2 that h2 ◦ φ = ψ on some neighborhood of x. But h2 ◦ φ ∈ Φ because Φ satisfies property (1). Therefore, every germ of Ψ is some member germ of Φ, yielding that ψ is a combination of members of Φ. Thus ψ ∈ Ψ as desired because Φ is closed under combinations of maps.

0 Proposition 2.29. Let Φ is a morphism from (X1, Γ1) into (X2, Γ2). If Γ1 is sub-pseudomonoid of Γ1 on 0 X1, then Φ : (X1, Γ1) → (X2, Γ2) is a morphism too.

Proof. Because Φ is a morphism, satisfies (1)-(3) of Definition 2.24 and it is closed under combination, 0 by the Proposition 2.28. Φ : (X1, Γ1) → (X2, Γ2) is closed under combination and satisfying (1)-(3) of 0 Definition 2.24. Therefore Φ : (X1, Γ1) → (X2, Γ2) is a morphism.

Proposition 2.30. If Φ : (X1, Γ1) −→ (X2, Γ2) is a morphism, then Γ = Γ1 ∪ Γ2 ∪ Φ ∪ {idX } is a pseudomonoid on the topological sum X = X1 t X2. So Γ|X1 = Γ1 and Γ|X2 = Γ2.

Lemma 2.31. Let Φ0 be a collection of continuous maps of open subsets of X1 to X2 satisfying the following properties:

(2)’Γ 1-saturation of the domains of maps in Φ0 cover X.

(3)’ There is a subset S of generators of Γ1 that satisﬁes the following property: If φ, ψ ∈ Φ0, h ∈ S and −1 0 0 0 x ∈ dom φ ∩ h (dom ψ) then there is some h ∈ Γ2 with φ(x) ∈ dom h and so h ◦ φ = ψ ◦ h on some neighborhood of x.

Then there is a unique morphism Φ : (X1, Γ1) → (X2, Γ2) containing Φ0. In this case, we say that Φ0 generates Φ.

Proof. Let Φ be the family of continuous maps from open subset X to X0 that satisfying the following conditions:

0 0 • All composites h ◦ φ ◦ h with φ ∈ Φ0, h ∈ Γ1 and h ∈ Γ2, wherever defined. • All possible combinations of composites of the above type are in Φ. Clearly, Φ satisfies properties (1)-(3) of Definition 2.24. By the Lemma 2.28, Φ is closed under combinations of maps, it is a morphism. Finally, the uniqueness of Φ follows because, if Ψ is another morphism (X1, Γ1) → (X2, Γ2) containing Φ0, then is also contains Φ by property (i) for Ψ and its maxinoimality, and thus Φ = Ψ by the maximality of Φ.

The composition of two morphisms,

Φ Ψ (X1, Γ1) / (X2, Γ2) / (X3, Γ3) is the morphism Ψ ◦ Φ:(X1, Γ1) → (X3, Γ3) be generated by all compositions of maps in Φ with maps in Ψ. With this operation, the pseudomonoids morphisms form a category PsMo. The identity morphism id(X,Γ) of PsMo at X is the morphism be generated by idX, note that Γ ⊆ id(X,Γ). 0 0 The restriction of a morphism Φ : (X, Γ) → (X , Γ ) to a subspace X0 ⊂ X is the morphism Φ|X0 : 0 0 (X0, Γ|X0 ) → (X , Γ ) consisting of all maps of open subsets of X0 to X that can be locally extended to maps in Φ. A. D. Nezhad, S. Shahriyari, Journal of Advanced Studies in Topology 6:2 (2015), 43–55 52

If X0 is open in X, then Γ|X0 consists of all maps in Φ whose domain is contained in X0. The inclusion map

X0 ,→ X generates a morphism (X0, Γ|X0 ) → (X, Γ), whose composition with Φ is Φ|X0 . 0 0 Given X = ∪iXi and suppose that Φi :(Xi, Γ|Xi ) → (X , Γ ) for each i. Let Φ be the family of continuous 0 maps φ : U → X , where U is an open subset of X, such that φ|U∩Xi ∈ Φi for all i. If Φ is a morphism and the maps in each Φi can be locally extended to maps in Φ, then Φ|Xi = Φi for all i, and Φ is called the combination of the morphisms Φi.

When every Xi is open in X, then the combination of morphisms Φi is defined just when Φi|Xi∩Xj = Φj|Xi∩Xj 0 0 for all i and j. We define the image of a morphism Φ : (X, Γ) → (X , Γ ) be the set im Φ = ∪φ∈Φim φ. If Γ is a pseudomonoid on X and X0 ⊂ X, define direct image [ Φ(X0) = im Φ|X0 = φ(X0 ∩ dom φ) φ∈Φ

0 0 0 0 We say Φ is constant if im Φ is one orbit (Φorb is constant). If Γ is a pseudomonoid on X and X0 ⊂ X , deﬁne the inverse image

−1 0 [ −1 0 Φ (X0) = φ (X0 ∩ imφ) φ∈Φ

0 0 0 0 Proposition 2.32. Let X0 is a subspace of X and im Φ ⊂ X0. Then the restrictions φ : dom φ → X0, 0 0 for any φ ∈ Φ, form a morphism that we denote by Φ : (X, Γ) → (X , Γ | 0 ). The morphism is called the 0 X0 restriction of Φ too.

0 0 The product of two morphism, Φi :(Xi, Γi) → (Xi, Γi), i = 1, 2, is the morphism

0 0 0 0 Φ1 × Φ2 :(X1 × X2, Γ1 × Γ2) −→ (X1 × X2, Γ1 × Γ2)

0 0 be generated by product maps in Φ1 and Φ2. The pair of two morphism Φi :(X, Γ) → (Xi, Γi), for i = 1, 2, 0 0 0 0 is the morphism (Φ1, Φ2):(X, Γ) → (X1 × X2, Γ1 × Γ2) be generated by the pairs (φ1, φ2), where φ1 ∈ Φ1 and φ2 ∈ Φ2 have the same domain.

3. Top, PsGr and PsMo Categories

Here we focus on Top, PsGr and PsMo Categories and want to prove the following results. Let Top denote the category of topological spaces and continuous maps between them. There is a canonical injective covariant functor Top → PsMo which the pseudomonoid be generated by idX to each space X, and assigns the morphism be generated by f to each map. Conclude, we can consider Top as a subcategory of PsMo. If PsGr be the pseudogroup category [1], then

Top ⊂ PsGr ⊂ PsMo

. Let Y be a topological space and Γ is a pseudomonoid on X. Then any continuous map Y → X generates a morphism (Y, < idY >) → (X, Γ).

Lemma 3.1. A continuous map f : X → Y generates a morphism (X, Γ) → (Y, < idY >) if and only if f is constant on the Γ-orbits. A. D. Nezhad, S. Shahriyari, Journal of Advanced Studies in Topology 6:2 (2015), 43–55 53

A pseudomonoid Γ is said to be path connected if, for any pair of leaf L and L0 of Γ, there is a morphism Φ: I → Γ with Φ(0) = L and Φ(1) = L0, where I = [0, 1] is a pseudomonoid in the above sense. Clearly, if Γ is path connected, then X/Γ is path connected too. 0 0 0 0 A homotopy between two morphism Φ0, Φ1 :(X, Γ) → (X , Γ ) is a morphism Ψ : (X, Γ)×(I, idI) → (X , Γ ) such that Φ0 and Φ1 can be identiﬁed to the restrictions of Ψ to X × {0} and X × {1}. Since the restriction of X × I to each slice X × {t} can be identiﬁed with X, a homotopy Ψ can be considered as a family of its 0 0 restrictions Ψt = Ψ|T ×{t} :(X, Γ) → (X , Γ ). A morphism Ψ : (X, Γ) → (X0, Γ0) is said a homotopy equivalence if there is a morphism Φ0 :(X0, Γ0) → 0 0 (X, Γ) that Φ ◦ Φ and Φ ◦ Φ are homotopic to the identity morphisms id(X,Γ) and id(X0,Γ0), respectively.

If id(X,Γ) is homotopic to a constant morphism, then (X, Γ) is called to be contractible.

4. (X, Γ)-structure

Let M is a non-empty set and X is a topological space. Then a parametrization of X into M is a map φ : U → M where, U ⊂ X is open subset. A cover for M is a collection A = {(φα,Uα)}α∈I of parametrizations of M where, M = ∪α∈I φα(Uα).

Deﬁnition 4.1. Let Γ be a pseudomonoid on X. A(X, Γ)-atlas on M, is a cover for M, A = {(Uα, φα)}α∈I , that for any φα and for any f ∈ Γ, φα ◦ f ∈ A where φα ◦f is deﬁned. A set endowed with a (X, Γ)-atlas is called a (X, Γ)-structure and denote by (M, A ) where, A is the (X, Γ)-atlas on M. Proposition 4.2. Any pseudomonoid (X, Γ) is a (X, Γ)-structure too.

Remark 4.3. Let (X, Γ) be a pseudomonoid. Suppose that C = {ψ : U ⊆ X → M} is a cover for M. Then {ψ ◦ f : ψ ∈ C , f ∈ Γ} is called the (X, Γ)-atlas generated by C and denote by < C > .

n Example 4.4. (n-manifolds) Let Γ be all local diﬀeomorphisms on R . If M is a n-manifold with maximal n n atlas A . Then A is a (Γ, R )-atlas for M, from here M has a (R , Γ)-structure. n Remark 4.5. According to the above example, the (R , Γ)-structures category is greater than the n-manifolds category.

n n Lemma 4.6. Let Γ is the pseudomonoid of previous example on R and suppose that A is a (R , Γ)-atlas for M. Then (M, A ) is a smooth manifold if and only if A satisﬁes the following properties:

(1) All elements of A are one-to-one and onto.

(2) For any φ, ψ ∈ A and p ∈ im ψ ∩ im φ there is f ∈ Γ, where φ = ψ ◦ f on an open subset contain φ−1(p).

Example 4.7. (Diﬀeological spaces [7]) A diﬀeology on a non-empty set M is a parametrizations collection n D of R ’s(n ∈ N) to M such that: (1) Any constant parametrization is an element of D (Covering).

(2) If P : ∪i∈I Ui → M is a parametrization and if P |Ui ∈ D for all i ∈ I. Then P ∈ D (Locality).

m (3) For every parametrization P : U → M and for any open subset V of R , if F : V → U is a smooth map, P ◦ F ∈ D (Smooth compatibility). A. D. Nezhad, S. Shahriyari, Journal of Advanced Studies in Topology 6:2 (2015), 43–55 54

∞ n A set M equipped with a diffeology is called a diffeological space. Now, consider X = tn=1R and Γ is all n m smooth maps f : U ⊆ R → V ⊆ R that U and V are open subsets. If M is a diffeological space be equipped ∞ n by the diffeology D. Then D is a (X, Γ)-atlas for M. Consequently (M, D) is a (tn=1R , Γ)-structure. ∞ n n m Lemma 4.8. Suppose that X = tn=1R and Γ is all smooth maps f : U ⊆ R → V ⊆ R that U and V are open subsets. Then a (X, Γ)-structure, on (M, A ), is a diffeology if and only if A satisfies the following conditions:

(1) A contain all constant parametrizations,

(2) A is closed under combination maps.

Deﬁnition 4.9. Let A be a (X, Γ)-atlas on M. Then the weakest topology on M such that all parametrizations in A are continuous is called (X, Γ)-topology on M induced by A .

Lemma 4.10. Suppose that (M, A ) is a (X, Γ)-structure. If V ⊆ M is open subset, then

−1 A |V = (φ (V ), φ|φ−1(V )): φ ∈ A is a (X, Γ)-atlas on V. Therefore, (V, A |V ) is a (X, Γ)-structure.

Deﬁnition 4.11. Let (M1, A1) and (M2, A2) are (X, Γ)-structure. Then α : M1 → M2 is called (X, Γ)-map if for any φ ∈ A1, α ◦ φ ∈ A2.

Deﬁnition 4.12. Let (M1, A1) and (M2, A2) are (X, Γ)-structure. Then α : M1 → M2 is called (X, Γ)- equivalent if α satisﬁes the following properties:

(1) α is one-to-one and onto,

(2) α and α−1 are (X, Γ)-maps.

Remark 4.13. We say (M1, A1) is local (X, Γ)-equivalent to (M2, A2) if for p ∈ M1 there are open subsets U ⊆ M1,V ⊆ M2 such that p ∈ U and there is a (X, Γ)-equivalent β : U → V.

n n Lemma 4.14. Assume that Γ is all local diffeomorphism maps on R . Suppose that A is a (R , Γ)-atlas 0 0 n on M. Then, there is A ⊆ A that (M, A ) is a n-manifold if and only if (M, A ) is local (R , Γ)-equivalent n to (R , Γ). Example 4.15. (Frölicher spaces[4]) A Frölicher structure on a set S is a couple (C,F ) where C ⊆ SR and S ∗ F ⊆ R such that C = D∗F and F = D C holds, with

∞ D∗F = {c : R → S : f ◦ c ∈ C (R, R) for all f ∈ F } ∗ ∞ D C = {f : S → R : f ◦ c ∈ C (R, R) for all c ∈ C} .

A Frölicher space is a triple (S, C, F ) where S is a set and the couple(C,F ) is a Frölicher structure on it. ∞ Let X = R and Γ be pseudomonoid generated by C (R, R). Therefore C is a cover for S. So by Lemma 4.3 the (S, < C >) is a (X, Γ)-structure. Hence, any Frölicher spaces equipped a (X, Γ)-atlas.

∞ Lemma 4.16. Let (S, A ) is a (R, < C (R, R) >)-structure. Then S admits a Fr¨olicher structure (C,F ) ∗ such that C = D∗F,F = D C0 and C0 = {φ ∈ A : domain(φ) = R} . Example 4.17. (Sikorski spaces[11]) Suppose that S is a nonempty set. A Sikorski structure on S is a non-empty collection F = {f : S → R}, with weakest topology on S that all elements of F are continuous, satisfying the following conditions:

∞ m (1) For any m ∈ N, if f1, ··· , fm ∈ F and G ∈ C (R ), then G(f1, ··· , fm) ∈ F . A. D. Nezhad, S. Shahriyari, Journal of Advanced Studies in Topology 6:2 (2015), 43–55 55

(2) Let g : S → R is a function on S. Suppose that for any p ∈ S there are fp ∈ F and an open subset

Up ⊂ S such that g|Up = fp|Up . Then g ∈ F .

A Sikorski space is a couple (S, F ) where S is a non-empty set and F is a Sikorski structure on S. Now, ∗ get C = D∗F and F = D C. Thus, (C,F ) is a Fr¨olicher structure on S such that F ⊆ F. By the Example ∞ 4.15, (S, < C >) is a (R, < C (R, R) >)-structure. Lemma 4.18. Any Sikorski structure on a non-empty set S generate a Fr¨olicher structure S. Therefore S ∞ admit a (R, < C (R, R) >)-atlas. Lemma 4.19. Consider M is a n-manifold and C∞(M) contain all smooth functions on M. Then C∞(M) is a Sikorski structure on M. Therefore, any manifold admits a Sikroski structure.

Remark 4.20. Now we continue considering the above lemma and in particular, we can conclude that, M admits other Sikroski structure. For example, suppose F ⊆ C∞(M) contain all functions such that df = 0.

Lemma 4.21. Let (X0, Γ0) and (X, Γ) are two pseudomonoids where X0 is an open subset of X and Γ0 ⊆ Γ. If (M, A 0) is a (X0, Γ0)-structure. Then there exists a (X, Γ)-atlas, denoted by A , on M such that A 0 ⊆ A . Proposition 4.22. Any Fr¨olicher space admits a diﬀeology.

Corollary 4.23. We conclude from this paper that, any n-manifold admits a Sikroski structure. Hence, any Sikroski space admits a Frölicher structure. Finally, any Frölicher space admits a diffeology.

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