Monoid and Topological groupoid Mohammad Qasim Mann'a Department of Mathematics College of Science, Al-Qassim University P.O. Box: 6644-Buraidah: 51452, Saudi Arabia e-mail:[email protected]
Abstract., Here we introduce some new results which is relative to the concept of topological monoid-groupoid and prove that the category of topological monoid coverings of X is equivalent to the category covering groupoids of the monoid- groupoid 1 X . Also, show that the monoid structure of monoid-groupoid lifts to a universal covering groupoid. Keywords. Fundamental groupoid, covering groupoids, topological groupoid, topological semigroup and monoid. Mathematical Subject Classification 2010: 22A05, 20L05, 22A22, 22A15, 06F05.
0.Introduction The groupoid is one of important subjects of category theory. In this work we introduced the concepts of topological monoid-groupoids and discuss some properties which relative to this concept. Also we find the fundamental groupoid of topological monoid whose underlying space has a universal covering is a topological monoid-groupoid. Moreover, we show that the category of topological monoid covering of X is equivalent to the category of covering groupoids of monoid-groupoids. Further, let G be monoid-groupoid ~ ~ and G be a groupoid and p : G G be a universal covering on the underlying ~ ~ groupoids such that G and G are transitive groupoids such that OP (e) e , then the monoid structure of G lifts.
1.Preliminaries
A groupoid [1, 2, 7]G is small category consists of two sets G andOG , called respectively the set of elements (or arrows) and the set of objects (or vertices) , of the groupoid, together with, two maps : G OG , called respectively
the source and target maps, the map :OG G, written as (x)= 1 x , where O 1x is called the identity element at x in G , and is called the object map and G G G the partial multiplication map; : , (g, h) g h, where G G (g, h)G G :(g) (h) .
These terms must satisfy the followings axioms: ( g h ) = (h), ( g h ) = (g),
1 ( g) ( g h ) k=g (h k), (1x ) (1x ) x , g 1 ( g) =g and g=g. For all g, h, k
1 G and xOG . The map : G G , g g is called the inversion map and each element g has an inverse g 1 such that (g 1 ) (g) , (g 1 ) (g) ,
1 1 g g 1 (g ) , g g 1 (g ).
y Further, for we write G(x, y) Gx as the set of all morphisms g G such that st (x) 1 (x) G (g) x and (g) y . each x, y OG . We will write G x
1 y y and costG (y) (y) G and Gx G(x, y) stG (x) costG (y). The vertex
group at x isG(x) G(x, x) stG (x) costG (x). We say G is transitive if for
y each x, y OG , Gx is non-empty. A morphism of groupoids H and G is a functor, that is, it consists of a pair of
functions f : H G and O f :OH OG preserving all structures. A subgroupoid of G is a subcategory N of G which is also a groupoid. We say that, N
y y is full (wide) subgroupoid if for all x, y ON then,Gx N x (ON OG ).Moreover, N
is called normal subgroupoid if N wide and a N(x) a 1 N(y) or a 1 N(y) a N(x) a G y for any x, y OG and x .
A topological groupoid is a groupoid G together with topologies on G and OG such that all the structure maps of G are continuous, that is: the source map , the target map , the object map , the inversion and the partial multiplication map are all continuous. A morphism of topological groupoids is a pair of maps f :G H and
O f :OG OH such that f and O f are continuous. Note that in this definition the partial multiplication map and the inversion map are continuous if and only if the map :G G G, (g h) g h 1 , called the difference map, is continuous, where G G (g, h)G G :(g) (h) has a subspace topology fromG G . Further, if the source, target and the inversion are continuous then the other one is continouos. Moreover, in topological groupoid G,
y x y for each g Gx left-translation lg : G G ,lg (h) g h and the right-translation
rg :Gy Gx ,rg (h) h g are homeomorphisms. ~ A morphism P : G G of groupoids is called a covering morphism [2, p.365-372] ~ ~ ~ and G a covering groupoid of G if each x OG then the restriction of mapping ~ ~ (G) ~ G ~ x OP (x ) of P is a bijective. Let P : G G be a morphism of groupoids . Then ~ ~ ~ ~ ~ for an object x OG the subgroup P(G(x))of G(OP (x)) ) is called the characteristic ~ ~ ~ group of P at x . So if P is the covering morphism then P maps G(x) isomorphically ~ ~ ~ to P(G(x)) . We say that a covering morphism P : G G of transitive groupoids is a ~ universal covering morphism if G is covers every cover of G, i.e., if for every ~ covering q : A G there is a unique morphism of groupoids q : G A such that
qq P (and hence q is also a covering morphism). This is equivalent to saying that
~ ~ ~ ~ ~y ~ ~ ~ for x, y OG the set G(x, y) G~x has one element at most. We recall from Howie [5] that a semigroup G is said to be monoid if there is a unique e (called the identity) such that g = eg ge for every g in G. From 6 A monoid-groupoid G is a groupoid endowed with monoid structure such that the following maps: 1. m : GG G , defined by (g, h) gh, a semigroup multiplication,
2. e : G , where is singleton. If the identityeOG , then 1e G is the of G, are morphisms of groupoids. A morphism f :G H of monoid-groupoids is a morphism of underlying groupoids preserving the monoid structure i.e. f (gh) f (g) f (h), and f (1 ) 1 ,for eG eH all g, hG. In a monoid-groupoid. The source, target and object maps are morphisms (homomorphisms) of monoid and the inversion map is an isomorphism of monoid. Also, the interchange law holds, i.e. (b a)(g h) (bg) (ah) .
Let X be a topological space then 1 (X ) is called a fundamental groupoid of X which is the set of all relative to end points homotopy classes of paths in topological space[2, p.213]. Let X be a connected and locally path connected X has a universal covering if and only if X semi-locally 1-connected [8, Corollary 10.37]. Assume [1] that a topological space X are locally path connected and semi-locally 1- connected, so that each path component of X admits a simply connected cover. Recall ~ that a covering map p : X X of connected spaces is called universal if it covers ~ every cover of X in the sense that if q :Y X is another cover of X then there exists a ~ ~ map r : X Y such that p = qr (hence r becomes a cover). A covering map ~ ~ p : X X is called simply connected if X is simply connected. So a simply connected cover is a universal cover. A subset U of the space X is called liftable (see section 2, in 1) if it is open, path connected and U lifts to each cover of X, that is, if ~ ~ p : X X is a covering map, i :U X is the inclusion map, and ~x X satisfies ~ ~ ~ p(x) x U, then there exists a map (necessarily unique) i :U X such that ~ ~ pi i and i(x) x. It is easy to see that U is liftable if and only if it is open, path connected and for each x U, the fundamental group1(U, x) , is mapped to a singleton by the morphism induced by the inclusion map i :U X . Note that if a connected and locally path connected space X has a universal covering then each point x X has a liftable neighborhood.
2. Topological Monoid-groupoid The concept of the following definition from taken from [6]. Definition2.1 A topological monoid-groupoid G is a toplogical groupoid endowed with a topological monoid structure such that the following monoid structure maps are morphism of topological groupoids: Tmg1. m :GG G , the monoid multiplications,
Tmg2. e : G , where is singleton. A morphism of topological monoid-groupoids f : G H is a morphism of the underlying topological groupoids preserving the topological monoid structures.
Example 2.2 Let G be a topological monoid . Then G G is a topological monoid- groupoid with the object set G and the morphims are the pair(a,b) . The source, target, object and the inversion maps are defined respectively by (a,b) b and (a,b) a , (a) (a,a) and (a,b) (b, a).The composition of groupoid is defined by (a,b) (b,c) (a,c) , the monoid multiplication is defined by (a,b)(d,c) (ad,bc) and the identity (e,e) of G G where e is the identity of G. Observe that all of the structure maps are continuous.
Proposition 2.3 Let G be a topological monoid-groupoid. Then the set of arrows of G and the set of objects OG are topological monoid. Proof. It’s clear from Definition 2.1.
Definition 2.4 Let G be monoid-groupoid and H G . We call H submonoid- (H,O ,,) groupoid if H form a monoid-groupoid. Further, H is wide if OH OG and
y y full if H x Gx for all x, y OH .
Proposition 2.5 Let H be submonoid-groupoid of monoid-groupoid G. Then the set of arrows of H and the set of objects OH are monoids.
Definition 2.6 Let G be a topological monoid-groupoid and H be submonoid- groupoid then H is called topological submonoid- groupoid.
Proposition 2.7Let G be a topological monoid-groupoid. Then G G andGG are topological submonoids of G G.
Proof. Let (a,b),(c,d) G G then, (ac,bd) G G and since (1e ) (1e )
Then we have, (1e ,1e ) G G. Further, since (a,b) (1 ,1 )(a.b) (a,b)(1 ,1 ). e e e e
So, G G is topological submonoid. Similar, we can prove that GG is topological submonid of G G.
Example 2.8 The set of identities A 1x : xOG is a topological wide submonoid- groupoid of a topological monoid-groupoid G.
Example 2.9 Let Hi :i I be a family of submonoid-groupoids in a topological monoid-groupoid G. Then H i is a topological submonoid-groupoid. i
In topological space G if H G then the closure of H denoted by cl(H ) .
Proposition 2.10 Let H be submonoid-groupoid of a topological monoid-groupoid G. If the set of arrows of H is an open set in the set of arrows of G and G G is dense in GG , then cl(H ) is submonoid-groupoid with Ocl(H ) cl(OH ).
Proof. Since the set of arrows of G and the set of objects OG are topological monoid then the set of arrows of cl(H ) , and the set of objects cl(OH ) are submonoids. Since the set of arrows of H is open and G G is dense in GG , then
cl(H) cl(H ) clG G (H H ) It remains to argue that cl(H ) is a subgroupoid. Since ,, and are continuous maps, then
(cl(H ) cl(H )) (clG G (H H)) cl(H) ,
(cl(H )) cl((H ) cl(OH ) ,
(cl(H)) cl( (H) cl(OH ) ,
(cl(OH ) cl( (OH )) cl(H ) . Therefore, cl(H ) is submonoid-groupoid. Example 2.11 Let f :G H be a morphism of topological monoid-groupoids. Then ker( f ) a G : f (a) 1x is a topological submomoid-groupoid.
Proposition 2.12 Let G be topological monoid then G(e) is topological submonoid- groupoid.
3. Some Properties of Topological Monoid-groupoid
From2 we have that, if X is topological space then 1 (X ) is a groupoid. In the following propositions we assume that all spaces are connected and a locally path connected.
Proposition 3.1 Let X be a semi-locally 1-connected topological monoid. Then the fundamental groupoid 1 (X ) becomes a topological monoid-groupoid. Proof. Since X is topological monoid, then the group multiplication m : X X X ,
defined by (x, y) xy, is continuous. Since 1 (X X ) 1 (X ) 1 (X ) then we have induced maps,
1 (m) : 1 (X ) 1 (X ) 1 (X ) , (a,b) ab), Also, associative law satisfied as,
a(bc) abc a(bc) (ab)c (ab)c. Further, a a1 a 1 e e , a 1 a 1 a e e .
Therefore, 1 (X ) becomes monoid-groupoid as required.
Moreovere since 1 (X ) is a topological groupoid [3] and m : X X X is a continuous map then it’s enough to show that the induced map is continuous,
1 (m) : 1 (X ) 1 (X ) 1 (X ) , (a,b) ab Let be an open cover of X consisting of all liftable neighborhoods. The elements of will be called canonical neighborhoods. For each U in and x U , define sx :U 1 (X ) by choosing for each y U a path in U from x to y and take sx (y) to be the homotopy class of this path. Since the path class of the paths in U ~ between the same points is unique, then sx is well-defined. Let sx (U ) U x
~ ~ 1 ' anda 1 X (x, y) . Then the sets Vy aU x , for all U, V in such that x U, y V form a set of basic open neighborhoods of lifted topology on 1 (X ) . This easily from description of the lifted topology in [10.5, 2].
' Now want to prove the map 1 (m) is continuous; Let ab1 (xk, yt) and U, U be canonical neighborhoods of xk, yt respectively. Then since the multiplication m : X X X is continuous, there is an canonical neighborhoods B H, B ' H ' of (x,k),(y,t) respectively such that m(B H ) U , m(B ' H ' ) U ' .Then we have
~ ' ~ 1 ' ~ ' ~ 1 ~ ' ~ ' ~ 1 ~ 1 ~ ' ~ 1 1m(By a Bx , H t b H k ) By H t ab Bx H k U yt abU xk . So, the map 1m is continuous. So, 1 (X ) is topological monoid-groupoid.
Proposition 3.2 Let f : X Y be a continuous map of a semi-locally 1-connected of topological monoid-groupoids. Then 1 ( f ) : 1 (X ) 1 (Y) is continuous.
Proof. It’s known that from2, 1 ( f ) is morphism of groupoids. We need first to show that 1 ( f ) is a morphism of monoids,
1 ( f )(ab) 1 ( f )(ab) = f (ab) f (a) f (b) = f (a) f (b)
= 1 ( f )(a) 1 ( f )(b). Also, we have
1 ( f )([1e ]) [ f (1e )] [1 f (e) ] [1e' ] Further,
1 ( f )([a]) [ f (a1e )] [ f (a)1 f (e) ] [ f (a)1e' ] [ f (a)][1e' ] = 1 ( f )(g) 1 ( f )(1e ).
Similar, we have 1 ( f )(a) 1 ( f )(1e ) 1 ( f )(a). Further, to show the continuity
' ' ' let a 1 X (x, x ) and f (a) 1Y(y, y ) . Let UandU be a canonical neighborhoods of y and y ' respectively. Then there is a canonical neighborhoods V and V ' of x and x ' respectively such that f (V ) U and f (V ' ) U ' .This implies that, ~ ~ ~ ~ f (V ' a V 1 ) U 1 f (a) U . 1 x' x y' y
So, 1 ( f ) is continuous and preserving the monoid structures.
4. Topological Coverings Let X be a topological space see section 10.6 in [2]. Then we have a category denoted ~ by TCov/X whose objects are covering maps p : X X and a morphism from ~ ~ ~ ~ p : X X to q :Y X is a map f : X Y ( f is a covering map) such that p qf . Further, we have a groupoid 1 (X ) and have a category denoted by ~ GdCov/1 X whose objects are the groupoid coverings p : G 1 X of 1 X and a ~ ~ ~ ~ morphism from p : G 1 X to q : H 1 X is a morphism f : G H of groupoids ( f is a covering morphism)such that p qf . We recall the following result from [2, 10.6.1].
Proposition 4.1 Let X be a space which has a universal covering ( connected, locally path connected and semi-locally 1-connected). Then the category TCov/X of topological covering of X is equivalent to the category of GpdCov/ 1 X of the fundamental groupoid of 1 X.
~ ~ Let X and X are topological monoids. A map p : X X is called a covering morphism of topological monoid if p is a morphism of monoid and p is a covering map on the underlying spaces. For a topological monoid X, we have the category ~ denoted by TMCov/X whose objects are topological monoid coverings p : X X ~ ~ ~ ~ and a morphism from p : X X to q : Y X is a map f : X Y such that p qf . For a topological monoid X , the fundamental groupoid 1 X is an inverse semigroup-groupoid and so we have a category denoted by MGpdCov/ 1 X whose ~ objects are monoid-groupoid coverings p : G 1 X of 1 X and a morphism from ~ ~ ~ ~ p : G 1 X to q : H 1 X is a morphism of f : G H monoid-groupoids such that p qf .
In the following proposition we assume that a space X is connected and locally path connected. Proposition 4.2 Let X be a topological monoid which has a universal covering. Then the category TMCov/X of topological monoid coverings of X is equivalent to the category MGpdCov/ 1 X of covering groupoids of the monoid-groupoid 1 X .
Proof. Define a functor 1 :TMCov / X MGpdCov / 1 X as follows: ~ Let p : X X be a covering morphism of topological monoid. Then the induced ~ morphism 1 p : 1 X 1 X is a covering morphism of groupoids2 . Moreover, the morphism 1 p preserves the monoid structures as: ~~ ~~ ~ ~ ~ ~ ~ ~ 1 p(ab ) =p(ab )) p(a) p(b ) =p(a)p(b )= 1 p(a) 1 p(b ).Also, we have ~ ~ ~ ~~ ~ ~ 1 p([1e~ ]) p(1e~ ) [1P(e~) ] [1e ]. Therefore, 1 pa 1 p(a 1e~ ) =p(a) p(1e~ ) ~ ~ ~ ~ ~ ~ ~ p(a)p(1e~ )= 1 p(a) 1 p([1e~ ]) and similar we have, 1 pa 1 p(1~e ) 1 p(a)
So 1 p becomes a covering morphism of monoid-groupoids.
We define a functor : MGpdCov/ 1 X TMCov / X as follows: ~ If q : G 1 X is a covering morphism of monoid-groupoid, then we have a covering ~ ~ ~ map p : X X , where p Oq and X OG . Further, we shall show that the monoid ~ ~ ~ multiplication m~ : X X X , (~x, ~y) ~x~y is continuous. Since X has a universal ~ covering, we can choose a cover of liftable subsets of X. Since the topology on X is lifted topology2, 10.5.5 the set consisting of all liftings of the sets in forms a ~ ~ basis for the topology on X . LetU be an open set of ~x~y and a lifting of U in . Since the multiplication, m : X X X , (x, y) xy Is continuous, there is a basic canonical neighborhood H T of (x, y) in X X such ~ ~ that m(H T ) U . Assume that H andT are liftable subsets. Let H andT be a ~ ~ ~ lifting of H and T respectively. Then pm(H T ) m(H T ) U and so we have ~ ~ ~ ~ m(H T ) U. So, m~ is continuous. Further, since p is continuous, p(e~) e and m( p p) pm~ , then p is a morphism of topological monoids. Since by Proposition (4.1) the category of topological space covering is equivalent to the category of groupoid coverings, the proof is completed by the following digram:
1 TMCov / X MGpd / 1 X 1 TCov / X Gpd / 1 X.
~ Definition 4.3 [7, Definition 2.4] Let p : G G a covering morphism of groupoids and q : H G a morphism of groupoids. If there exist a unique morphism ~ q~ : H G such that q pq~ we say q lifts to q~ by p.
We recall the following theorem from [2, 10.3.3] which is an important result to have the lifting maps on covering groupoids.
~ Theorem 4.4 Let p : G G be a covering morphism of groupoids, x OG . and ~ ~ ~ x OG . such thatO p (x) x . Let q : H G be a morphism of groupoids such
O (~y) x that H is transitive and such that q . Then the morphism q : H G ~ ~ ~ ~ uniquely lifts to a morphism q : H G such thatOq~ (y) x if and only if ~ q[H (~y)] p[G(~x)] , where H (~y) and G(~x) are the object groups.
~ Let G be monoid-groupoid, e the identity element of OG , and letG be just a ~ ~ ~ groupoid, e OG . Let p : G G be a covering morphism of groupoids such ~ ~ that O p (e ) e . We say the monoid structure of G lifts toG if there exist ~ ~ ~ ~ monoid structure on G with identity element e OG such that G is a groupoid ~ and p : G G is a morphism of monoid-groupoids.
~ ~ Theorem 4.5 Let G be monoid-groupoid and G be a groupoid. Let p : G G be a ~ universal covering on the underlying groupoids such that G and G are transitive ~ ~ ~ groupoids. Let e be the identity of OG and e OG such thatOP (e ) e . Then the monoid structure of G lifts. Proof. Since G is a monoid-groupoid then it has the following map, m : GG G , m(g,h) gh . ~ ~ SinceG is universal covering, the object groupG(e~) has one element at most. So, by Theorem(4.4) the map m lift to the map, ~ ~ ~ ~ ~ ~ ~~ m~ : GG G , m(g, h ) gh
~ by p : G G such that ~~ ~ ~ p(gh ) p(g) p(h ).
Since the multiplication m is associative then we have m(m I) m(I m) , where I denotes to the identity map. Then by Theorem (4.4) these maps m(m I) and ~ ~ m(I m) respectively lift to m~(m~ I ) and m~(I m~) which coincide on (e~,e~,e~) . So, by the uniqueness of the lifting we have, m~(m~ I) = m~(I m~) this means that m~ is ~ ~ ~ associative. In similar way we can prove that e the identity of e OG and ~ ~ p(1~ ) 1 ~ 1 .Then we conclude thatG be a monoid-groupoid. e OP (e ) e
From above we have also,
~ Proposition 4.6 Let X be a topological monoid and X be a topological space. Let ~ ~ p : X X be a simply connected covering. such that X and X are path connected . ~ Let e X the identity element. If e~ X such that p(e~) e , then the topological ~ monoid structure of X lifts to X. ~ Proof. Since p : X X is simply connected covering, the induced morphism ~ 1 p :1 X 1 X is a universal covering morphism of groupoids. Then by ~ Proposition (3.1) 1 X becomes monoid-groupoid and by Theorem (4.5) we have 1 X ~ becomes monoid-groupoid. Further, by (4.2) we have, X is topological monoid.
References
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