This post is based on (wanted to write after reading) Lie Groupoids and Differentiable stacks by Matias L. del Hoyo. I would suggest this for any one who wants to know about Lie groupoids and Differentiable stacks. This is well written.
By a manifold, we always mean a smooth manifold. A Groupoid is a category where every arrow is invertible. A Lie groupoid is a groupoid with additional smooth structures on object set/morphism set and maps between them.
Definition : A Lie groupoid consists of a manifold \(\mathcal{G}_0\) of objects, a manifold \(\mathcal{G}_1\) of arrows and following maps :Proposition : Given a Lie groupoid \(\mathcal{G}_1\rightrightarrows \mathcal{G}_0\) and \(x,y\in \mathcal{G}_0\),with some compatibility conditions. We denote this Lie groupoid by \(\mathcal{G}_1\rightrightarrows \mathcal{G}_0\). Definition : Let \(\mathcal{G}_1\rightrightarrows \mathcal{G}_0\) be a Lie groupoid and \(x\in \mathcal{G}_0\).
- \(s:\mathcal{G}_1\rightarrow \mathcal{G}_0\), a submersion, called the source map.
- \(t:\mathcal{G}_1\rightarrow \mathcal{G}_0\), a submersion, called the target map.
- \(m:\mathcal{G}_1\times_{s,\mathcal{G}_0,t}\mathcal{G}_1\rightarrow \mathcal{G}_1\), a smooth map, called the multiplication map.
- \(u:\mathcal{G}_0\rightarrow \mathcal{G}_1\), a smooth map, called the unit map.
- \(i:\mathcal{G}_1\rightarrow \mathcal{G}_1\), a smooth map, called the inverse map.
- The set \(s^{-1}(x)=:G(x,-)\) is called the \(s\)-fibre of \(x\) .
- The set \(t^{-1}(x)=:G(x,-)\) is called the \(s\)-fibre of \(x\)
- The set \(s^{-1}(x)\cap t^{-1}(x)=:G_x\) is called the Isotropy group of \(x\).
- The set \(t(s^{-1}(x))=\{y:x\rightarrow y\in \mathcal{G}_1\}=:O_x\) is called the orbit of \(x\).
- the subset \(G(y,x)\subseteq G\) is an embedded submanifold. In particular, \(G_x\) is a Lie group.
- the subset \(O_x\) is a (may not be embedded) submanifold in a canonical way.