In the same way as groupoids $(\mathrm{src},\mathrm{trg}):X_1\rightrightarrows X_0$ are a simultaneous generalization of group actions

$$(\mathrm{pr}_1,\mathrm{act}):G\times X\rightrightarrows X$$

and equivalence relations

$$(\mathrm{pr}_1|_R,\mathrm{pr}_2|_R):R\rightrightarrows X\;,\qquad R\subseteq X\times X$$

on sets $X$, Lie groupoids are a simultaneous generalization of (smooth) Lie group actions and "smooth" equivalence relations (where "smooth" means the maps induced on $R\subseteq M\times M$ by the projections to the first and second factor are submersions) on manifolds $M$.

Group actions and equivalence relations, hence also groupoids, should have quotients, but a quotient manifold does not always literally exist; a Morita morphism between two groupoids is intuitively a smooth morphism between the putative quotient manifolds. The geometric objects that replace the non existing quotient manifolds are called (differentiable) stacks.

In the same way as groupoids $(\mathrm{src},\mathrm{trg}):X_1\rightrightarrows X_0$ are a symultaneous generalization of group actions

$$(\mathrm{pr}_1,\mathrm{act}):G\times X\rightrightarrows X$$

and equivalence relations

$$(\mathrm{pr}_1|_R,\mathrm{pr}_2|_R):R\rightrightarrows X\;,\qquad R\subseteq X\times X$$

on sets $X$, Lie groupoids are a simultaneous generalization of (smooth) Lie group actions and "smooth" equivalence relations (where "smooth" means the maps induced on $R\subseteq M\times M$ by the projections to the first and second factor are submersions) on manifolds $M$.

Group actions and equivalence relations, hence also groupoids, should have quotients, but a quotient manifold does not always literally exist; a Morita morphism between two groupoids is intuitively a smooth morphism between the putative quotient manifolds. The geometric objects that replace the non existing quotient manifolds are called (differentiable) stacks.

In the same way as groupoids are a simultaneous generalization of group actions and equivalence relations on sets, Lie groupoids are a simultaneous generalization of (smooth) Lie group actions on manifolds and "smooth" equivalence relations (where "smooth" means the maps induced on $R\subseteq M\times M$ by the projections to the first and second factor are submersions).

Group actions and equivalence relations, hence also groupoids, should have quotients, but a quotient manifold does not always literally exist; a Morita morphism between two groupoids is intuitively a smooth morphism between the putative quotient manifolds. The geometric objects that replace the non existing quotient manifolds are called (differentiable) stacks.

In the same way as groupoids $(\mathrm{src},\mathrm{trg}):X_1\rightrightarrows X_0$ are a simultaneous generalization of group actions

$$(\mathrm{pr}_1,\mathrm{act}):G\times X\rightrightarrows X$$

and equivalence relations

$$(\mathrm{pr}_1|_R,\mathrm{pr}_2|_R):R\rightrightarrows X\;,\qquad R\subseteq X\times X$$

on sets $X$, Lie groupoids are a simultaneous generalization of (smooth) Lie group actions and "smooth" equivalence relations (where "smooth" means the maps induced on $R\subseteq M\times M$ by the projections to the first and second factor are submersions) on manifolds $M$.

Group actions and equivalence relations, hence also groupoids, should have quotients, but a quotient manifold does not always literally exist; a Morita morphism between two groupoids is intuitively a smooth morphism between the putative quotient manifolds. The geometric objects that replace the non existing quotient manifolds are called (differentiable) stacks.