On Bitopological Lie Groupoids

1. Introduction

Groupoids provide a natural generalization of groups in which multiplication is defined only for composable pairs of arrows. They occur throughout differential geometry, foliation theory, Poisson geometry, and the study of geometric structures with symmetries. A systematic account of Lie groupoids and their associated Lie algebroids can be found in [4,5].

A bitopological space is a set equipped with two topologies. This concept was introduced and developed as a generalization of ordinary topological spaces; see Kelly [1] and Lane [2]. The presence of two topologies makes it possible to distinguish two different notions of continuity and to study structures that are not adequately described by a single topology. The author introduced and and investigated basic properties of bitipological groupoids [7].

The purpose of this paper is to combine these two ideas. We consider Lie groupoids whose object and arrow spaces carry two smooth manifold structures. Each smooth structure determines a topology, and the groupoid operations are required to be smooth with respect to each of the two structures.

This framework contains ordinary Lie groupoids as a special case. It also provides a setting in which one can investigate the compatibility of two differential structures on the same underlying groupoid.

The main objectives of this paper are as follows.

(i)

To define bitopological manifolds and bitopological Lie groupoids.

(ii)

To establish basic properties of the groupoid structure maps under the two smooth structures.

(iii)

To describe the associated Lie algebroids and formulate compatibility conditions between them.

(iv)

To provide examples and discuss restrictions, pullbacks, and morphisms.

The definitions below are intended as a framework for further study. In particular, the existence of two smooth structures does not automatically imply that the resulting Lie algebroids are isomorphic. Additional compatibility assumptions will therefore be stated whenever they are needed.

2. Preliminaries

We first recall the notions of bitopological spaces and Lie groupoids.

Definition 1. 

Abitopological spaceis a triple

$$(X,{\tau }_1,{\tau }_2),$$

where X is a set and ${\tau }_1,{\tau }_2$ are topologies on X. 

The two topologies need not be comparable. In particular, neither

$${\tau }_1\subseteq {\tau }_2\mathrm{nor}{\tau }_2\subseteq {\tau }_1$$

is required.

Definition 2. 

Let $(X,{\tau }_1,{\tau }_2)$ and $(Y,{\sigma }_1,{\sigma }_2)$ be bitopological spaces. A map $f:X\to Y$ is calledpairwise continuousif

$$f:(X,{\tau }_i)⟶(Y,{\sigma }_i)$$

is continuous for $i=1,2$. 

We recall the standard groupoid notation.

Definition 3. 

A groupoid consists of a set $G_1$ of arrows, a set $G_0$ of objects, and structure maps

$$s,t:G_1⟶G_0,$$

$$u:G_0⟶G_1,\iota :G_1⟶G_1,$$

together with a multiplication

$$m:G_2⟶G_1,$$

where

$$G_2=\{(g,h)\in G_1\times G_1:s\left(g\right)=t\left(h\right)\}.$$

These maps satisfy the usual groupoid axioms. 

Here s and t are the source and target maps, u is the unit map, ι is inversion, and $m(g,h)=gh$ denotes composition. 

A Lie groupoid is a groupoid for which the object and arrow sets are smooth manifolds, the source and target maps are surjective submersions, and all structure maps are smooth. We use the standard convention that an arrow g goes from $s\left(g\right)$ to $t\left(g\right)$; see [4].

3. Bitopological Manifolds

We now introduce the differential structure used throughout the paper.

Definition 4. 

Abitopological smooth manifold[8] is a triple

$$(M,{\mathcal{A}}_1,{\mathcal{A}}_2),$$

where ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ are smooth manifold structures on the same underlying set M. 

We write

$${\tau }_i=\tau \left({\mathcal{A}}_i\right),i=1,2,$$

for the topologies induced by these structures. The pair $({\tau }_1,{\tau }_2)$ is called the associated bitopology. 

We assume throughout that the smooth manifolds are Hausdorff and second countable, unless explicitly stated otherwise.

Remark 1. 

The two smooth structures may induce the same topology without having identical smooth atlases. Conversely, the topologies induced by the two structures may be different. Thus the bitopological structure contains information about both topological and differentiable compatibility. 

Definition 5. 

Let M and N be bitopological smooth manifolds. A map $f:M\to N$ is calledbitopologically smoothif

$$f:(M,{\mathcal{A}}_i)⟶(N,{\mathcal{B}}_i)$$

is smooth for both $i=1,2$. 

Proposition 1. 

Every bitopologically smooth map is pairwise continuous with respect to the associated bitopologies. 

Proof. 

For each $i\in \{1,2\}$, the map

$$f:(M,{\mathcal{A}}_i)\to (N,{\mathcal{B}}_i)$$

is smooth. Every smooth map between smooth manifolds is continuous. Hence f is continuous from $(M,{\tau }_i)$ to $(N,{\sigma }_i)$ for both values of i. □

Definition 6. 

A bitopological smooth manifold M is calledcompatibleif the identity map

$${\mathrm{id}}_M:(M,{\mathcal{A}}_1)⟶(M,{\mathcal{A}}_2)$$

is a diffeomorphism. 

Remark 2. 

Compatibility is a strong condition: it means that the two smooth structures coincide in the sense of smooth manifold structures. In this case, the bitopological manifold reduces to an ordinary smooth manifold with a duplicated topology. 

4. Bitopological Lie Groupoids

We are ready to define the central object of this paper.

Definition 7. 

Abitopological Lie groupoidis a groupoid

$$\mathcal{G}=(G_1\rightrightarrows G_0)$$

such that $G_1$ and $G_0$ are bitopological smooth manifolds and the following conditions hold for $i=1,2$: 

(i)

The source and target maps

$$s,t:G_1\to G_0$$

are smooth surjective submersions with respect to the i-th smooth structures. 

(ii)

The unit map

$$u:G_0\to G_1$$

is smooth with respect to the i-th smooth structures. 

(iii)

The inversion map

$$\iota :G_1\to G_1$$

is smooth with respect to the i-th smooth structure. 

(iv)

The composable-pair space $G_2$ carries the induced smooth manifold structure for each i, and the multiplication

$$m:G_2\to G_1$$

is smooth with respect to that structure. 

The requirement that $G_2$ be a smooth manifold for both structures is meaningful because the source and target maps are submersions. In each smooth structure, $G_2$ is the fiber product of s and t.

Remark 3. 

The definition requires the same underlying groupoid operations to be smooth in both differential structures. It does not require the two smooth structures to be compatible, nor does it assert that their tangent bundles are canonically identical. 

Proposition 2. 

Let $\mathcal{G}$ be a bitopological Lie groupoid. Then each of the two smooth structures makes

$$G_1\rightrightarrows G_0$$

an ordinary Lie groupoid. 

Proof. 

Fix $i\in \{1,2\}$. By definition, $G_1$ and $G_0$ are smooth manifolds with respect to ${\mathcal{A}}_i$. The source and target maps are surjective submersions, and the unit, inverse, and multiplication maps are smooth. The groupoid identities hold independently of the smooth structures because they are identities of the underlying groupoid. These are precisely the axioms of a Lie groupoid. □

Corollary 1. 

Every bitopological Lie groupoid determines two ordinary Lie groupoids on the same underlying groupoid. 

Remark 4. 

The converse construction is not automatic. Two Lie groupoid structures on the same underlying groupoid define a bitopological Lie groupoid only when the structure maps satisfy the stated conditions for both smooth structures. 

5. Examples

5.1. Ordinary Lie Groupoids

Example 1. 

Let

$$\mathcal{G}=(G_1\rightrightarrows G_0)$$

be an ordinary Lie groupoid. Give $G_1$ and $G_0$ the same smooth structures in both positions:

$${\mathcal{A}}_1={\mathcal{A}}_2.$$

Then $\mathcal{G}$ is a bitopological Lie groupoid. We call this thediagonal bitopological structure.

5.2. The Pair Groupoid

Example 2. 

Let M be a smooth manifold. Its pair groupoid is

$$M\times M\rightrightarrows M,$$

with

$$s(x,y)=y,t(x,y)=x,$$

$$u\left(x\right)=(x,x),\iota (x,y)=(y,x),$$

and

$$(x,y)(y,z)=(x,z).$$

Equip M with two smooth structures ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$. Equip $M\times M$ with the corresponding product structures ${\mathcal{A}}_1\times {\mathcal{A}}_1$ and ${\mathcal{A}}_2\times {\mathcal{A}}_2$. 

The pair groupoid is then a bitopological Lie groupoid. Indeed, the source and target maps are projections, the unit is the diagonal map, inversion exchanges the two factors, and multiplication is smooth in each product structure. 

5.3. Lie Groups as Groupoids

Example 3. 

Every Lie group H determines a Lie groupoid

$$H\rightrightarrows \{*\}$$

with a single object. If H is equipped with two smooth Lie group structures on the same underlying group, and multiplication and inversion are smooth for both structures, then it defines a bitopological Lie groupoid. 

In particular, taking the same Lie group structure twice gives the diagonal example. 

5.4. Action Groupoids

Action groupoids provide an important class of examples of Lie groupoids. They encode the action of a Lie group on a smooth manifold and are particularly useful in the study of transformation groups, orbit spaces, foliations, and differentiable stacks. The ordinary construction is standard in the theory of Lie groupoids; see, for example, [4,5].

Let H be a Lie group and let M be a smooth manifold. Suppose that H acts smoothly on M. Thus, there is a smooth map

$$\alpha :H\times M⟶M,\alpha (h,x)=h·x,$$

satisfying

$$e·x=x$$

for every $x\in M$, and

$$\left(hk\right)·x=h·(k·x)$$

for all $h,k\in H$ and $x\in M$.

The action determines a Lie groupoid, called the action groupoid or transformation groupoid, denoted by

$$H⋉M\rightrightarrows M.$$

Its arrow space is

$${(H⋉M)}_1=H\times M,$$

while its object space is

$${(H⋉M)}_0=M.$$

We use the convention that $(h,x)$ is an arrow from x to $h·x$. The source and target maps are therefore defined by

$$s(h,x)=x,t(h,x)=h·x.$$

The unit map is

$$u:M⟶H\times M,u\left(x\right)=(e,x),$$

where e denotes the identity element of H. The inverse map is

$$\iota :H\times M⟶H\times M,\iota (h,x)=(h^{-1},h·x).$$

Two arrows $(h,x)$ and $(k,y)$ are composable precisely when

$$s(h,x)=t(k,y).$$

Since

$$s(h,x)=x$$

and

$$t(k,y)=k·y,$$

the composability condition is

$$x=k·y.$$

In this case the multiplication is given by

$$(h,k·y)(k,y)=(hk,y).$$

Thus, the space of composable pairs is

$${(H\times M)}^{\left(2\right)}=\left\{\left((h,x),(k,y)\right)\in (H\times M)\times (H\times M):x=k·y\right\}.$$

The groupoid identities follow directly from the group action. For example,

$$u\left(t\right(h,x\left)\right)(h,x)=(e,h·x)(h,x)=(h,x),$$

and

$$(h,x)u\left(s\right(h,x\left)\right)=(h,x)(e,x)=(h,x).$$

Furthermore,

$$\iota (h,x)(h,x)=(h^{-1},h·x)(h,x)=(e,x)=u\left(s(h,x)\right),$$

and

$$(h,x)\iota (h,x)=(h,x)(h^{-1},h·x)=(e,h·x)=u\left(t(h,x)\right).$$

We now adapt this construction to the bitopological setting.

Suppose that the Lie group H is equipped with two smooth structures

$${\mathcal{H}}_1,{\mathcal{H}}_2,$$

and that the manifold M is equipped with two smooth structures

$${\mathcal{M}}_1,{\mathcal{M}}_2.$$

We write

$$H_i=(H,{\mathcal{H}}_i),M_i=(M,{\mathcal{M}}_i),i=1,2.$$

The corresponding product manifolds are

$$H_i\times M_i.$$

Assume that the same underlying action

$$\alpha :H\times M⟶M,\alpha (h,x)=h·x,$$

is smooth with respect to both product smooth structures. In other words, assume that

$${\alpha }_i:H_i\times M_i⟶M_i$$

is smooth for $i=1,2$.

Under these assumptions, we obtain two ordinary action groupoids

$$H_1⋉M_1\rightrightarrows M_1$$

and

$$H_2⋉M_2\rightrightarrows M_2$$

on the same underlying groupoid.

This leads to the following definition.

Definition 8. 

Let H and M be equipped with two smooth structures, and suppose that the action

$$\alpha :H\times M⟶M$$

is smooth with respect to both corresponding product structures. If the source and target maps

$$s,t:H\times M⟶M$$

are surjective submersions for both smooth structures, then

$$H⋉M\rightrightarrows M$$

is called abitopological action Lie groupoid.

The source map is particularly simple:

$$s:H\times M⟶M,s(h,x)=x.$$

For either product smooth structure, this is simply the projection onto the second factor. Hence s is a smooth surjective submersion.

The target map is

$$t:H\times M⟶M,t(h,x)=h·x.$$

The smoothness of t is precisely the smoothness of the corresponding action map.

Moreover, t is automatically a submersion. To see this, fix $(h,x)\in H\times M$. The restriction of t to $\left\{h\right\}\times M$ is the diffeomorphism

$$M⟶M,x⟼h·x,$$

whose inverse is

$$y⟼h^{-1}·y.$$

Consequently, the differential

$$d{t}_{(h,x)}:T_{h}H\oplus T_{x}M⟶T_{h·x}M$$

is surjective. Indeed, its restriction to the second factor is

$$d_x(h·-):T_{x}M⟶T_{h·x}M,$$

which is an isomorphism.

Thus, once the action is smooth, the source and target maps satisfy the submersion requirement for each of the two smooth structures.

Proposition 3. 

Let H be a Lie group and M a manifold, both equipped with two smooth structures. Suppose that the action

$$\alpha :H\times M⟶M$$

is smooth with respect to both product smooth structures. Then

$$H⋉M\rightrightarrows M$$

is a bitopological Lie groupoid. 

Proof. 

For each $i=1,2$, consider the smooth structures ${\mathcal{H}}_i$ on H and ${\mathcal{M}}_i$ on M.

The arrow space is

$$H_i\times M_i,$$

and the object space is $M_i$. The source map

$$s(h,x)=x$$

is the projection onto the second factor and is therefore a smooth surjective submersion.

By assumption, the action map

$$t(h,x)=h·x$$

is smooth. As explained above, for every fixed $h\in H$, the map

$$x⟼h·x$$

is a diffeomorphism of $M_i$, with inverse

$$x⟼h^{-1}·x.$$

Hence $dt$ is surjective, and t is a submersion.

The unit map

$$u\left(x\right)=(e,x)$$

is smooth because it is the product of the constant map $x\mapsto e$ and the identity map of $M_i$.

The inverse map

$$\iota (h,x)=(h^{-1},h·x)$$

is smooth because inversion in $H_i$ and the action map are smooth.

Finally, multiplication is

$$(h,k·x)(k,x)=(hk,x).$$

It is smooth because multiplication in $H_i$ and the action map are smooth.

Therefore, for each $i=1,2$, the underlying groupoid is an ordinary Lie groupoid. Hence it is a bitopological Lie groupoid. □

Example 4. 

Let H be a Lie group acting on itself by left multiplication:

$$h·x=hx.$$

The corresponding action groupoid is

$$H⋉H\rightrightarrows H,$$

with

$$s(h,x)=x,t(h,x)=hx.$$

If H carries two smooth Lie group structures for which multiplication and inversion are smooth, and the action above is smooth for both structures, then this is a bitopological action Lie groupoid. 

Example 5. 

Suppose that a Lie group H acts trivially on a manifold M:

$$h·x=x$$

for all $h\in H$ and $x\in M$. Then

$$s(h,x)=x,t(h,x)=x.$$

The resulting action groupoid is

$$H\times M\rightrightarrows M.$$

The isotropy group at every point $x\in M$ is naturally isomorphic to H. 

If both H and M are equipped with two compatible smooth structures, this provides a simple example of a bitopological Lie groupoid with constant isotropy. 

5.5. Orbits and Isotropy in the Bitopological Setting

The orbit of $x\in M$ under the action of H is

$$H·x=\{h·x:h\in H\}.$$

The orbit relation is therefore

$$x{\sim }_{H}y⟺\mathrm{there}\mathrm{exists}h\in H\mathrm{such}\mathrm{that}y=h·x.$$

The orbit space is the set

$$M/H=\{H·x:x\in M\}.$$

In the bitopological situation, the orbit set is the same set-theoretic object for both smooth structures, but the topological behavior of the quotient may differ.

Indeed, if

$${\tau }_1,{\tau }_2$$

are the two topologies induced on M, the quotient topologies on $M/H$ obtained from the two actions may be different. Thus one obtains, in general, a pair of quotient topologies

$${\tau }_{M/H}^{\left(1\right)},{\tau }_{M/H}^{\left(2\right)}.$$

The isotropy group at $x\in M$ is

$$H_x=\{h\in H:h·x=x\}.$$

Equivalently, in the action groupoid,

$$H_x=\{(h,x)\in H\times M:s(h,x)=t(h,x)=x\}.$$

When the action is smooth for both structures, the isotropy group can be studied relative to each of the two resulting Lie group structures.

The orbit and isotropy constructions show that a bitopological action groupoid simultaneously records the algebraic action of H and two differential descriptions of the same transformation geometry.

6. Morphisms

We next define morphisms between bitopological Lie groupoids.

Definition 9. 

Let

$$\mathcal{G}=(G_1\rightrightarrows G_0),\mathcal{H}=(H_1\rightrightarrows H_0).$$

be bitopological Lie groupoids. Abitopological Lie groupoid morphism$F:G⟶H$ 

is a pair of bitopologically smooth maps

$$F_1:G_1\to H_1,F_0:G_0\to H_0$$

that commute with all the groupoid structure maps. In particular,

$$s_H\circ F_1=F_0\circ s_G,$$

$$t_H\circ F_1=F_0\circ t_G,$$

$$F_1\circ u_G=u_H\circ F_0,$$

and

$$F_1\circ {\iota }_G={\iota }_H\circ F_1.$$

Moreover, for every composable pair $(g,h)$,

$$F_1\left(gh\right)=F_1\left(g\right)F_1\left(h\right).$$

Proposition 4. 

The identity morphism of a bitopological Lie groupoid is bitopologically smooth. The composition of two bitopological Lie groupoid morphisms is again a bitopological Lie groupoid morphism. 

Proof. 

The identity maps on the object and arrow manifolds are smooth for both smooth structures and preserve all groupoid operations.

Now let $F:G\to H,K:H\to \mathcal{K}$ be bitopological Lie groupoid morphisms. The component maps of their composition are

$${(K\circ F)}_1=K_1\circ F_1,{(K\circ F)}_0=K_0\circ F_0.$$

For each $i\in \{1,2\}$, these maps are compositions of smooth maps and hence are smooth. Since both F and K preserve the groupoid operations, their composition does as well. □

6.1. The Category of Bitopological Lie Groupoids

The preceding definitions naturally lead to a category whose objects are bitopological Lie groupoids and whose morphisms are bitopological Lie groupoid morphisms. We denote this category by

$$\mathbf{BiLieGpd}.$$

More precisely, an object of $\mathbf{BiLieGpd}$ is a groupoid

$$\mathcal{G}=(G_1\rightrightarrows G_0)$$

equipped with two smooth structures on $G_1$ and two smooth structures on $G_0$, such that all the groupoid structure maps are smooth with respect to both smooth structures. Thus, for $i=1,2$, we have Lie groupoids

$${\mathcal{G}}_i=(G_1^{\left(i\right)}\rightrightarrows G_0^{\left(i\right)}),$$

where the superscript $\left(i\right)$ indicates the i-th smooth structure.

Let

$$\mathcal{G}=(G_1\rightrightarrows G_0)\mathrm{and}\mathcal{H}=(H_1\rightrightarrows H_0)$$

be objects of $\mathbf{BiLieGpd}$. A morphism

$$F:\mathcal{G}⟶\mathcal{H}$$

consists of a pair of maps

$$F_1:G_1⟶H_1,F_0:G_0⟶H_0,$$

such that $F_1$ and $F_0$ are smooth with respect to both smooth structures and preserve all the groupoid operations.

In particular, the source and target maps satisfy

$$s_{\mathcal{H}}\circ F_1=F_0\circ s_{\mathcal{G}},$$

and

$$t_{\mathcal{H}}\circ F_1=F_0\circ t_{\mathcal{G}}.$$

The unit maps satisfy

$$F_1\circ u_{\mathcal{G}}=u_{\mathcal{H}}\circ F_0,$$

while inversion satisfies

$$F_1\circ {\iota }_{\mathcal{G}}={\iota }_{\mathcal{H}}\circ F_1.$$

Finally, whenever $g,h\in G_1$ are composable, we require

$$F_1\left(gh\right)=F_1\left(g\right)F_1\left(h\right).$$

The compatibility with the source and target maps implies that composable arrows are sent to composable arrows. Indeed, if

$$s_{\mathcal{G}}\left(g\right)=t_{\mathcal{G}}\left(h\right),$$

then

$$\begin{array}{cc}\hfill s_{\mathcal{H}}\left(F_1\left(g\right)\right)& =F_0\left(s_{\mathcal{G}}\left(g\right)\right)\hfill \\ & =F_0\left(t_{\mathcal{G}}\left(h\right)\right)\hfill \\ & =t_{\mathcal{H}}\left(F_1\left(h\right)\right).\hfill \end{array}$$

Hence the product

$$F_1\left(g\right)F_1\left(h\right)$$

is defined in $\mathcal{H}$.

For each $i=1,2$, the pair

$$(F_1,F_0)$$

is therefore an ordinary smooth Lie groupoid morphism

$$F_i:{\mathcal{G}}_i⟶{\mathcal{H}}_i.$$

Thus, a morphism in $\mathbf{BiLieGpd}$ can equivalently be viewed as a pair of compatible Lie groupoid morphisms

$$F_1^{\mathrm{Lie}}:{\mathcal{G}}_1⟶{\mathcal{H}}_1,F_2^{\mathrm{Lie}}:{\mathcal{G}}_2⟶{\mathcal{H}}_2$$

defined on the same underlying groupoid maps.

Proposition 5. 

The identity maps

$${id}_{G_1}:G_1⟶G_1,{id}_{G_0}:G_0⟶G_0$$

define an identity morphism

$${id}_{\mathcal{G}}:\mathcal{G}⟶\mathcal{G}$$

in $\mathbf{BiLieGpd}$. 

Proof. 

The identity map on a smooth manifold is smooth with respect to every smooth structure under consideration. Consequently,

$${id}_{G_1}$$

and

$${id}_{G_0}$$

are smooth for both smooth structures.

Moreover, the identity maps preserve all groupoid structure maps. For example,

$$s_{\mathcal{G}}\circ {id}_{G_1}=s_{\mathcal{G}},$$

and

$${id}_{G_0}\circ s_{\mathcal{G}}=s_{\mathcal{G}}.$$

The same argument applies to the target, unit, inverse, and multiplication maps. Hence

$${id}_{\mathcal{G}}$$

is a morphism in $\mathbf{BiLieGpd}$. □

Composition of morphisms is defined componentwise. Suppose that

$$F:\mathcal{G}⟶\mathcal{H}$$

and

$$K:\mathcal{H}⟶\mathcal{K}$$

are morphisms in $\mathbf{BiLieGpd}$. Their composition is the morphism

$$K\circ F:\mathcal{G}⟶\mathcal{K}$$

whose components are

$${(K\circ F)}_1=K_1\circ F_1$$

and

$${(K\circ F)}_0=K_0\circ F_0.$$

Since $F_1,F_0,K_1,$ and $K_0$ are smooth with respect to both smooth structures, their compositions are also smooth with respect to both structures.

Proposition 6. 

The composition of two morphisms in $\mathbf{BiLieGpd}$ is again a morphism in $\mathbf{BiLieGpd}$. 

Proof. 

Let

$$F:\mathcal{G}⟶\mathcal{H}$$

and

$$K:\mathcal{H}⟶\mathcal{K}$$

be morphisms.

For the source map, we have

$$\begin{array}{cc}\hfill s_{\mathcal{K}}\circ {(K\circ F)}_1& =s_{\mathcal{K}}\circ K_1\circ F_1\hfill \\ & =K_0\circ s_{\mathcal{H}}\circ F_1\hfill \\ & =K_0\circ F_0\circ s_{\mathcal{G}}\hfill \\ & ={(K\circ F)}_0\circ s_{\mathcal{G}}.\hfill \end{array}$$

Similarly,

$$t_{\mathcal{K}}\circ {(K\circ F)}_1={(K\circ F)}_0\circ t_{\mathcal{G}}.$$

For the multiplication, let $g,h\in G_1$ be composable. Then

$$\begin{array}{cc}\hfill {(K\circ F)}_1\left(gh\right)& =K_1\left(F_1\left(gh\right)\right)\hfill \\ & =K_1\left(F_1\left(g\right)F_1\left(h\right)\right)\hfill \\ & =K_1\left(F_1\left(g\right)\right)K_1\left(F_1\left(h\right)\right)\hfill \\ & ={(K\circ F)}_1\left(g\right){(K\circ F)}_1\left(h\right).\hfill \end{array}$$

The compatibility with the unit and inverse maps follows in exactly the same way.

Therefore $K\circ F$ is a bitopological Lie groupoid morphism. □

Theorem 1. 

The collection of bitopological Lie groupoids together with bitopological Lie groupoid morphisms forms a category, denoted by

$$\mathbf{BiLieGpd}.$$

Proof. 

We have shown that every bitopological Lie groupoid $\mathcal{G}$ possesses an identity morphism

$${id}_{\mathcal{G}}:\mathcal{G}⟶\mathcal{G},$$

and that the composition of two morphisms is again a bitopological Lie groupoid morphism.

It remains to verify the identity and associativity axioms. For every morphism

$$F:\mathcal{G}⟶\mathcal{H},$$

we have

$$F\circ {id}_{\mathcal{G}}=F$$

and

$${id}_{\mathcal{H}}\circ F=F,$$

because these identities hold componentwise on both the arrow spaces and the object spaces.

Now let

$$F:\mathcal{G}⟶\mathcal{H},K:\mathcal{H}⟶\mathcal{K},L:\mathcal{K}⟶\mathcal{P}$$

be morphisms. Then

$${\left((L\circ K)\circ F\right)}_1=(L_1\circ K_1)\circ F_1=L_1\circ (K_1\circ F_1)={\left(L\circ (K\circ F)\right)}_1,$$

and similarly,

$${\left((L\circ K)\circ F\right)}_0={\left(L\circ (K\circ F)\right)}_0.$$

Thus

$$(L\circ K)\circ F=L\circ (K\circ F).$$

Hence composition is associative, and all the category axioms are satisfied. Therefore

$$\mathbf{BiLieGpd}$$

is a category. □

Remark 5. 

There is a natural forgetful construction

$$\mathbf{BiLieGpd}⟶\mathbf{LieGpd}\times \mathbf{LieGpd},$$

which sends a bitopological Lie groupoid $\mathcal{G}$ to the pair of Lie groupoids

$$({\mathcal{G}}_1,{\mathcal{G}}_2)$$

determined by its two smooth structures. A morphism

$$F:\mathcal{G}⟶\mathcal{H}$$

is sent to the pair

$$(F_1^{\mathrm{Lie}},F_2^{\mathrm{Lie}}).$$

The two components are not independent, since they arise from the same underlying set maps

$$F_1:G_1\to H_1,F_0:G_0\to H_0.$$

This compatibility distinguishes $\mathbf{BiLieGpd}$ from an arbitrary product of two categories of Lie groupoids. 

Remark 6. 

The category $\mathbf{BiLieGpd}$ contains the ordinary category of Lie groupoids as a natural subcategory. Indeed, given an ordinary Lie groupoid

$$\mathcal{G}=(G_1\rightrightarrows G_0),$$

we may equip both $G_1$ and $G_0$ with the same smooth structure in the two positions:

$${\mathcal{A}}_1={\mathcal{A}}_2.$$

This gives the diagonal bitopological Lie groupoid

$${\mathcal{G}}^{\mathsf{\Delta }}.$$

A morphism between two such diagonal objects is precisely an ordinary Lie groupoid morphism. 

Remark 7. 

The categorical formulation is useful because it allows one to study constructions such as products, subgroupoids, pullbacks, and equivalences in a unified setting. In particular, whenever such a construction is compatible with both smooth structures, it can be regarded as a construction internal to $\mathbf{BiLieGpd}$. 

7. The Associated Lie Algebroids

Every Lie groupoid has an associated Lie algebroid. For the standard construction and its properties, we refer to [4].

Let

$$\mathcal{G}=(G_1\rightrightarrows G_0)$$

be a bitopological Lie groupoid. For each $i\in \{1,2\}$, define

$$A_i\left(\mathcal{G}\right)=ker\left(d{s}_i\right){|}_{u\left(G_0\right)},$$

where $d{s}_i$ denotes the differential of the source map with respect to the i-th smooth structure.

The anchor is

$${\rho }_i=d{t}_i{|}_{A_i\left(\mathcal{G}\right)}:A_i\left(\mathcal{G}\right)⟶T{G}_0^{\left(i\right)}.$$

The Lie bracket on sections is obtained by identifying sections of $A_i\left(\mathcal{G}\right)$ with right-invariant vector fields on the corresponding Lie groupoid.

Definition 10. 

The pair

$$\left(A_1\left(\mathcal{G}\right),A_2\left(\mathcal{G}\right)\right)$$

is called theassociated pair of Lie algebroidsof the bitopological Lie groupoid $\mathcal{G}$. 

Remark 8. 

The two algebroids need not be equal. Their underlying vector bundles may have different smooth structures, and their anchor maps take values in tangent bundles defined by different smooth structures on $G_0$. 

Thus, in general, it is more accurate to speak of a pair of Lie algebroids than of a single Lie algebroid. 

Definition 11. 

Acompatible Lie algebroid identificationis a vector bundle isomorphism

$$\mathsf{\Phi }:A_1\left(\mathcal{G}\right)⟶A_2\left(\mathcal{G}\right)$$

over the identity on $G_0$ that preserves the Lie brackets on sections and intertwines the anchors through a specified tangent bundle identification

$$\mathsf{\Psi }:T{G}_0^{\left(1\right)}⟶T{G}_0^{\left(2\right)}.$$

Thus,

$${\rho }_2\circ \mathsf{\Phi }=\mathsf{\Psi }\circ {\rho }_1.$$

The tangent bundle identification $\mathsf{\Psi }$ is additional data; it is not supplied merely by the existence of two smooth structures on the same set.

Proposition 7. 

Suppose that the two smooth structures on $G_1$ and $G_0$ coincide. Then

$$A_1\left(\mathcal{G}\right)=A_2\left(\mathcal{G}\right)$$

as Lie algebroids. 

Proof. 

If the smooth structures coincide, their tangent bundles and the differentials of the source and target maps coincide. Therefore,

$$ker\left(d{s}_1\right){{|}_{u\left(G_0\right)}=ker\left(d{s}_2\right)|}_{u\left(G_0\right)}.$$

The anchors are equal, and the right-invariant vector fields used to define the brackets are the same. Consequently, the two Lie algebroid structures agree. □

8. Restrictions and Subgroupoids

Definition 12. 

Let $\mathcal{G}$ be a bitopological Lie groupoid. A subgroupoid

$$\mathcal{K}=(K_1\rightrightarrows K_0)$$

is called abitopological Lie subgroupoidif $K_1$ and $K_0$ are embedded submanifolds of $G_1$ and $G_0$, respectively, for both smooth structures, and the restricted structure maps make $\mathcal{K}$ a bitopological Lie groupoid. 

The embedded-submanifold condition is imposed separately for each smooth structure.

Proposition 8. 

Let $\mathcal{K}$ be a bitopological Lie subgroupoid of $\mathcal{G}$. Then the inclusion

$$\mathcal{K}\hookrightarrow \mathcal{G}$$

is a bitopological Lie groupoid morphism. 

Proof. 

The inclusions

$$K_1\hookrightarrow G_1,K_0\hookrightarrow G_0$$

are smooth for both smooth structures because the subgroupoid is embedded in both. Since the structure maps of $\mathcal{K}$ are restrictions of those of $\mathcal{G}$, the inclusions commute with source, target, units, inversion, and multiplication. Hence they define a bitopological Lie groupoid morphism. □

Remark 9. 

An arbitrary set-theoretic subgroupoid need not be a bitopological Lie subgroupoid. The required embedded submanifold and submersion conditions must be checked. 

9. Pullbacks

Pullback constructions are fundamental in Lie groupoid theory. We formulate a sufficient condition for preserving the bitopological structure.

Let

$$f:M⟶G_0$$

be a map from a bitopological smooth manifold to the object space of a bitopological Lie groupoid

$$\mathcal{G}=(G_1\rightrightarrows G_0).$$

The pullback arrow space is defined by

$$f^{!}\mathcal{G}=\left\{(x,g,y)\in M\times G_1\times M|f\left(x\right)=t\left(g\right),f\left(y\right)=s\left(g\right)\right\}.$$

Here, $f^{!}\mathcal{G}$ denotes the pullback Lie groupoid of $\mathcal{G}$ along the smooth map

$$f:M⟶G_0.$$

The exclamation mark in $f^{!}$ is part of the notation and does not denote a factorial.

An element $(x,g,y)\in f^{!}\mathcal{G}$ consists of $x,y\in M$ and an arrow $g\in G_1$ such that

$$t\left(g\right)=f\left(x\right),s\left(g\right)=f\left(y\right).$$

Thus, g determines an arrow from y to x in the pullback groupoid.

The source and target maps are given by

$$s_{f^{!}\mathcal{G}}(x,g,y)=y,t_{f^{!}\mathcal{G}}(x,g,y)=x.$$

The unit map is

$$u_{f^{!}\mathcal{G}}\left(x\right)=\left(x,u_{\mathcal{G}}\left(f\left(x\right)\right),x\right),$$

and the inverse of an arrow is

$${(x,g,y)}^{-1}=(y,g^{-1},x).$$

The multiplication is defined by

$$(x,g,y)(y,h,z)=(x,gh,z),$$

whenever the arrows are composable.

Consequently, $f^{!}\mathcal{G}$ is a groupoid over M. Under the usual smoothness and transversality assumptions, it inherits a Lie groupoid structure.

Proposition 9. 

Suppose that f is smooth for both smooth structures and is transverse to the source and target maps in each structure. Assume that the resulting fiber products carry the corresponding smooth manifold structures. Then the pullback groupoid $f^{!}\mathcal{G}$ inherits a bitopological Lie groupoid structure. 

Proof. 

For each $i\in \{1,2\}$, the transversality assumptions ensure that the relevant fiber products are smooth manifolds in the i-th structure.

The source and target maps of the pullback groupoid are the projections

$$s(x,g,y)=y,t(x,g,y)=x.$$

The unit map is

$$u\left(x\right)=(x,u(f\left(x\right)),x),$$

and inversion is

$${(x,g,y)}^{-1}=(y,g^{-1},x).$$

Multiplication is induced by the multiplication in $\mathcal{G}$:

$$(x,g,y)(y,h,z)=(x,gh,z).$$

All these maps are smooth in each structure because the corresponding maps on M and $\mathcal{G}$ are smooth and the fiber-product structures are induced from the product manifolds. The groupoid identities follow from those of $\mathcal{G}$. Thus the pullback is a bitopological Lie groupoid. □

10. Further Properties and Questions

The preceding constructions suggest several directions for further investigation.

10.1. Compatibility of the Two Smooth Structures

One natural question is to determine conditions under which the identity map

$${\mathrm{id}}_M:(M,{\mathcal{A}}_1)\to (M,{\mathcal{A}}_2)$$

is smooth, or is a diffeomorphism. Such conditions would relate the two associated tangent bundles and could lead to a canonical comparison of the associated Lie algebroids.

11. Conclusion

We have introduced bitopological Lie groupoids as groupoids whose object and arrow spaces carry two smooth manifold structures, with all groupoid structure maps satisfying the corresponding smoothness and submersion conditions.

The definition produces two ordinary Lie groupoids on the same underlying groupoid. Consequently, it also gives rise to a pair of Lie algebroids. These algebroids need not coincide, and comparing them requires additional data relating the two tangent bundle structures.

We have described basic morphisms, subgroupoids, pullbacks, and several examples. The diagonal construction shows that the ordinary theory is contained in this framework, while the general setting allows two potentially distinct differential structures.

Further work should address bitopological Morita equivalence, cohomology, integration, and the conditions under which the two associated Lie algebroids are equivalent.