Let $M,N$be two smooth manifolds. Let $f,g:M\rightarrow N$ be two smooth maps. We have the notion of a homotopy (smooth homotopy) from the maps $f$ to the map $g$.

Is there a similar concept for morphisms of Lie groupoids?

Suppose $\mathcal{G}=(\mathcal{G}_1\rightrightarrows \mathcal{G}_0)$ and $\mathcal{H}=(\mathcal{H}_1\rightrightarrows \mathcal{H}_0)$ be Lie groupoids.

Let $(\phi_1,\phi_0),(\psi_1,\psi_0): \mathcal{G}\rightarrow \mathcal{H}$ be two morphisms of Lie groupoids. Is there any notion of “a homotopy” from $(\phi_1,\phi_0)$ to $(\psi_1,\psi_0)$?

Further, is there a notion of when two Lie groupoids are Homotopy equivalent?

Is there a notion for topological groupoids?

Let $M,N$be two smooth manifolds. Let $f,g:M\rightarrow N$ be two smooth maps. We have the notion of a homotopy (smooth homotopy) from the maps $f$ to the map $g$.

Is there a similar concept for morphisms of Lie groupoids?

Suppose $\mathcal{G}=(\mathcal{G}_1\rightrightarrows \mathcal{G}_0)$ and $\mathcal{H}=(\mathcal{H}_1\rightrightarrows \mathcal{H}_0)$ be Lie groupoids.

Let $(\phi_1,\phi_0),(\psi_1,\psi_0): \mathcal{G}\rightarrow \mathcal{H}$ be two morphisms of Lie groupoids. Is there any notion of a “homotopy” from $(\phi_1,\phi_0)$ to $(\psi_1,\psi_0)$?

Further, is there a notion of when two Lie groupoids are homotopy equivalent?

Is there a notion for topological groupoids?