Morita equivalent Lie groupoids

Question

Suppose $[X_1\rightrightarrows X_0]$ and $[Y_1\rightrightarrows Y_0]$ are Morita equivalent Lie groupoids. This means, there exists another Lie groupoid $[Z_1\rightrightarrows Z_0]$ and Morita morphisms $[Z_1\rightrightarrows Z_0]\rightarrow [X_1\rightrightarrows X_0]$ and $[Z_1\rightrightarrows Z_0]\rightarrow [Y_1\rightrightarrows Y_0]$.

Another alternative description is, there exists a $[X_1\rightrightarrows X_0]-[Y_1\rightrightarrows Y_0]$ ``biprincipal bibundle" $X_0\xleftarrow{\phi} P\xrightarrow{\varphi} Y_0$.

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

For example, can the differentials $\phi_{*,a}:T_aP\rightarrow T_{\phi(a)}X_0$ and $\varphi_{*,a}:T_aP\rightarrow T_{\varphi(a)}Y_0$ have some common properties for $a\in P?$ In particular, does $\ker(\varphi_{*,a})=\ker (\phi_{*,a})$ for all $a\in P$?

Answer 1

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.

Answer 2

I will answer the new version of the question:

Does $\ker(\varphi_{*,a})=\ker (\phi_{*,a})$ for all $a\in P$, where $\phi_{*,a}:T_aP\to T_{\phi(a)}X_0$ and $\varphi_{*,a}:T_aP\to T_{\varphi(a)}Y_0$ are the differentials?

For Morita equivalences, the left and right moment maps $\phi$ and $\varphi$ are surjective submersions. Accordingly, $\phi_{*,a}$ and $\varphi_{*,a}$ are surjective linear maps. Hence, if $X_0$ and $Y_0$ have different dimensions, we cannot have $\ker(\varphi_{*,a})=\ker (\phi_{*,a})$.