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Differentiable stack
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<p>A differentiable stack is the analogue in <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a> of an <a href="/facts/Algebraic_stack/flrtwvwU">algebraic stack</a> in <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>. It can be described either as a <a href="/facts/Stack_(mathematics)/MkASN1rA">stack</a> over <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifolds</a> which admits an atlas, or as a <a href="/facts/Lie_groupoid/11ARLE9M">Lie groupoid</a> up to <a href="/facts/Morita_equivalence/vEVRJSJw">Morita equivalence</a>. </p><p>Differentiable stacks are particularly useful to handle spaces with <a href="/facts/Singularity_(mathematics)/Y5sMQY4I">singularities</a> (i.e. orbifolds, leaf spaces, quotients), which appear naturally in differential geometry but are not differentiable manifolds. For instance, differentiable stacks have applications in <a href="/facts/Foliation_theory/cxml9vyJ">foliation theory</a>, <a href="/facts/Poisson_geometry/LvAmEzul">Poisson geometry</a> and <a href="/facts/Twisted_K-theory/s9sYhLNa">twisted K-theory</a>. </p>