# Tagged Questions

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### Homotopy theory of topological stacks/orbifolds

Motivation $\newcommand{\T}{\mathscr{T}}$ I have many times found myself saying some variant of the following. Let $\T_g$ be the Teichmüller space of a surface of genus $g$, and $\Gamma_g$ its ...
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### how to construct a $C^\infty$ stack from a holomorphic stack

Given a complex manifold, you can `weaken' its structure to give a smooth manifold. Is there an analogous construction that constructs a stack over the category of smooth manifolds from a stack over ...
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### Ramification formula for orbifolds

It's well known for smooth curves that if $\pi:X\to Y$ is a finite map, $K_X=\pi^*K_Y+Ram(\pi)$, this is just the Riemann-Hurwitz formula at the level of line bundles. I've been told that this ...
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### When do maps of ineffective orbifolds descend to their effective part?

If $$f:\mathscr{X} \to \mathscr{Y}$$ is a map between (possibly ineffective) orbifolds (in the sense of differentiable stacks, or orbifold groupoids), does it follow that $f$ induces a map between ...
Recall that an orbifold is an etale and proper differentiable stack $X$. Etale means that it admits an etale atlas $M \to X$ from a manifold $M$ (which is to say it is represented by an etale Lie ...
If you ever wanted to construct the tangent bundle of a differentiable stack, it's relatively simple: First, if $\mathbf{X}$ is a stack coming from a Lie groupoid $\mathcal{G}$, you could just say ...