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edited Feb 5, 2021 at 16:54

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Let me try to answer this part of your question:

"Does it imply some specific condition on the maps $\phi$ and $\varphi$?"

The quick answer: $\phi$ and $\varphi$ have to be surjective submersions.

I recommend e.g. the paper Stacky Lie groups (arXiv:0702399) by Blohmann for more details, but let me try to explain the basic idea here. The definition of Morita equivalence for Lie groupoids involves the concept of a biprincipal bibundle. We can break this down into the following notions:

Groupoid actions. Let $G\rightrightarrows G_0$ be a Lie groupoid, and let $l_X:X\to G_0$ be a smooth map. A groupoid action of $G$ on $X$ along $l_X$ is a smooth map $$G\times_{G_0}^{\mathrm{src},l_X}X\longrightarrow X;\qquad (g,x)\longmapsto gx$$ satisfying some further properties that straightforwardly generalise the notion of a group action. Sometimes you'll see these written as $G\curvearrowright^{l_X} X$. Actions from the right are defined analogously.
Groupoid bundles. This is a notion that will eventually allow us to generalise Lie group principal bundles. A $G$-bundle is a smooth map $\pi:X\to B$ that is invariant under some action $G\curvearrowright^{l_X} X$. You might write this as $G\curvearrowright^{l_X} X\xrightarrow{\pi}B$.
Principality. A Lie groupoid bundle $G\curvearrowright^{l_X} X\xrightarrow{\pi}B$ is called principal if $\pi$ is a surjective submersion, and the $G$-action is free and transitive on the $\pi$-fibres.
Bibundles. A bibundle between two Lie groupoids $G\rightrightarrows G_0$ and $H\rightrightarrows H_0$ is a pair of actions $G\curvearrowright^{l_X}X$ and $X{~}^{r_X}\curvearrowleft H$ that interact well together; in particular we must have two groupoid bundles $G\curvearrowright^{l_X} X\xrightarrow{r_X}H_0$ and $G_0\xleftarrow{l_X}X{~}^{r_X}\curvearrowleft H$. Here the smooth maps $l_X$ and $r_X$ are the arrows $\phi$ and $\varphi$ from your question, respectively.
Biprincipality. A bibundle is called biprincipal if both the groupoid bundles described in the previous bullet point are themselves principal groupoid bundles. From the definition of principality for groupoid bundles it follows that $l_X$ and $r_X$ (i.e. $\phi$ and $\varphi$) have to be surjective submersions.

Also note that there is a relation between the formalism of (biprincipal) bibundles and Morita morphisms (also known as weak equivalences). It is known in the literature that (one-sided) principality of bibundles corresponds to essential surjectivity and full faithfulness of Morita morphisms.

Let me try to answer this part of your question:

"Does it imply some specific condition on the maps $\phi$ and $\varphi$?"

The quick answer: $\phi$ and $\varphi$ have to be surjective submersions.

I recommend e.g. the paper arXiv:0702399 by Blohmann for more details, but let me try to explain the basic idea here. The definition of Morita equivalence for Lie groupoids involves the concept of a biprincipal bibundle. We can break this down into the following notions:

Groupoid actions. Let $G\rightrightarrows G_0$ be a Lie groupoid, and let $l_X:X\to G_0$ be a smooth map. A groupoid action of $G$ on $X$ along $l_X$ is a smooth map $$G\times_{G_0}^{\mathrm{src},l_X}X\longrightarrow X;\qquad (g,x)\longmapsto gx$$ satisfying some further properties that straightforwardly generalise the notion of a group action. Sometimes you'll see these written as $G\curvearrowright^{l_X} X$. Actions from the right are defined analogously.
Groupoid bundles. This is a notion that will eventually allow us to generalise Lie group principal bundles. A $G$-bundle is a smooth map $\pi:X\to B$ that is invariant under some action $G\curvearrowright^{l_X} X$. You might write this as $G\curvearrowright^{l_X} X\xrightarrow{\pi}B$.
Principality. A Lie groupoid bundle $G\curvearrowright^{l_X} X\xrightarrow{\pi}B$ is called principal if $\pi$ is a surjective submersion, and the $G$-action is free and transitive on the $\pi$-fibres.
Bibundles. A bibundle between two Lie groupoids $G\rightrightarrows G_0$ and $H\rightrightarrows H_0$ is a pair of actions $G\curvearrowright^{l_X}X$ and $X{~}^{r_X}\curvearrowleft H$ that interact well together; in particular we must have two groupoid bundles $G\curvearrowright^{l_X} X\xrightarrow{r_X}H_0$ and $G_0\xleftarrow{l_X}X{~}^{r_X}\curvearrowleft H$. Here the smooth maps $l_X$ and $r_X$ are the arrows $\phi$ and $\varphi$ from your question, respectively.
Biprincipality. A bibundle is called biprincipal if both the groupoid bundles described in the previous bullet point are themselves principal groupoid bundles. From the definition of principality for groupoid bundles it follows that $l_X$ and $r_X$ (i.e. $\phi$ and $\varphi$) have to be surjective submersions.

Let me try to answer this part of your question:

"Does it imply some specific condition on the maps $\phi$ and $\varphi$?"

The quick answer: $\phi$ and $\varphi$ have to be surjective submersions.

Groupoid actions. Let $G\rightrightarrows G_0$ be a Lie groupoid, and let $l_X:X\to G_0$ be a smooth map. A groupoid action of $G$ on $X$ along $l_X$ is a smooth map $$G\times_{G_0}^{\mathrm{src},l_X}X\longrightarrow X;\qquad (g,x)\longmapsto gx$$ satisfying some further properties that straightforwardly generalise the notion of a group action. Sometimes you'll see these written as $G\curvearrowright^{l_X} X$. Actions from the right are defined analogously.
Groupoid bundles. This is a notion that will eventually allow us to generalise Lie group principal bundles. A $G$-bundle is a smooth map $\pi:X\to B$ that is invariant under some action $G\curvearrowright^{l_X} X$. You might write this as $G\curvearrowright^{l_X} X\xrightarrow{\pi}B$.
Principality. A Lie groupoid bundle $G\curvearrowright^{l_X} X\xrightarrow{\pi}B$ is called principal if $\pi$ is a surjective submersion, and the $G$-action is free and transitive on the $\pi$-fibres.
Bibundles. A bibundle between two Lie groupoids $G\rightrightarrows G_0$ and $H\rightrightarrows H_0$ is a pair of actions $G\curvearrowright^{l_X}X$ and $X{~}^{r_X}\curvearrowleft H$ that interact well together; in particular we must have two groupoid bundles $G\curvearrowright^{l_X} X\xrightarrow{r_X}H_0$ and $G_0\xleftarrow{l_X}X{~}^{r_X}\curvearrowleft H$. Here the smooth maps $l_X$ and $r_X$ are the arrows $\phi$ and $\varphi$ from your question, respectively.
Biprincipality. A bibundle is called biprincipal if both the groupoid bundles described in the previous bullet point are themselves principal groupoid bundles. From the definition of principality for groupoid bundles it follows that $l_X$ and $r_X$ (i.e. $\phi$ and $\varphi$) have to be surjective submersions.

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created Feb 5, 2021 at 11:40

Nesta

Let me try to answer this part of your question:

"Does it imply some specific condition on the maps $\phi$ and $\varphi$?"

The quick answer: $\phi$ and $\varphi$ have to be surjective submersions.

Groupoid actions. Let $G\rightrightarrows G_0$ be a Lie groupoid, and let $l_X:X\to G_0$ be a smooth map. A groupoid action of $G$ on $X$ along $l_X$ is a smooth map $$G\times_{G_0}^{\mathrm{src},l_X}X\longrightarrow X;\qquad (g,x)\longmapsto gx$$ satisfying some further properties that straightforwardly generalise the notion of a group action. Sometimes you'll see these written as $G\curvearrowright^{l_X} X$. Actions from the right are defined analogously.
Groupoid bundles. This is a notion that will eventually allow us to generalise Lie group principal bundles. A $G$-bundle is a smooth map $\pi:X\to B$ that is invariant under some action $G\curvearrowright^{l_X} X$. You might write this as $G\curvearrowright^{l_X} X\xrightarrow{\pi}B$.
Principality. A Lie groupoid bundle $G\curvearrowright^{l_X} X\xrightarrow{\pi}B$ is called principal if $\pi$ is a surjective submersion, and the $G$-action is free and transitive on the $\pi$-fibres.
Bibundles. A bibundle between two Lie groupoids $G\rightrightarrows G_0$ and $H\rightrightarrows H_0$ is a pair of actions $G\curvearrowright^{l_X}X$ and $X{~}^{r_X}\curvearrowleft H$ that interact well together; in particular we must have two groupoid bundles $G\curvearrowright^{l_X} X\xrightarrow{r_X}H_0$ and $G_0\xleftarrow{l_X}X{~}^{r_X}\curvearrowleft H$. Here the smooth maps $l_X$ and $r_X$ are the arrows $\phi$ and $\varphi$ from your question, respectively.
Biprincipality. A bibundle is called biprincipal if both the groupoid bundles described in the previous bullet point are themselves principal groupoid bundles. From the definition of principality for groupoid bundles it follows that $l_X$ and $r_X$ (i.e. $\phi$ and $\varphi$) have to be surjective submersions.