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 $\mathbf{TX}$ is the stack associated to the tangent groupoid $\mathcal{TG}$.

A more formal way is the following:

Consider the composite

$\text{Mfd} \stackrel{T}{\rightarrow} \text{VectorBundles} \stackrel{\text{forget}}{\rightarrow} \text{Mfd} \stackrel{\text{yoneda}}{\rightarrow} \text{St}(\text{Mfd}).$

by restricting its weak left Kan extension to stacks we get a 2-functor (by abuse of notation)

$\text{St}(\text{Mfd}) \stackrel{T}{\rightarrow} \text{St}(\text{Mfd}).$

Now, suppose $\mathbf{X}$ is a stack coming from a Lie groupoid $\mathcal{G}$. Then $\mathbf{X}$ is the weak colimit of the composition

$\Delta^{\rm op} \stackrel{N(\mathcal{G})}{\rightarrow}Mfd\stackrel{\text{yoneda}}{\rightarrow}\text{St}(\text{Mfd})$

(in fact even of its 2-truncation.)

Since $T$ is weak-colimit preserving, it follows that $T \mathbf{X}$ is the weak colimit of

$\Delta^{\rm op} \stackrel{N(\mathcal{TG})}{\rightarrow}\text{Mfd}\stackrel{\text{yoneda}}{\rightarrow}\text{St}(\text{Mfd})$

which is in turn just the stack associated to the tangent groupoid $\mathcal{TG}$. So both definitions agree.

Now suppose you wanted to define the cotangent stack. The first line of attack, done naively, seems to fail. The cotangent functor $T^*$ is contravariant so it doesn't send groupoid objects to groupoid objects. However, less naively, in the literature (for instance here: http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.4318v2.pdf), one can define the cotangent groupoid of $\mathcal{G}$ to be a certain symplectic groupoid $\mathcal{T^{*}G}_1 \rightrightarrows \text{Lie}(\mathcal{G})^{*}$, where $\text{Lie}(\mathcal{G})^{*}$ is the dual Lie algebroid of $\mathcal{G}$. (Is this invariant under Morita equivalence?)

The second way would seem to be:

Consider the opposite of the composite

$\text{Mfd}^{\rm op} \stackrel{T^*}{\rightarrow} \text{VectorBundles} \stackrel{\text{forget}}{\rightarrow} \text{Mfd} \stackrel{\text{yoneda}}{\rightarrow} \text{St}(\text{Mfd}),$

$\text{Mfd} \to \text{St}(\text{Mfd})^{\rm op}$

and take its weak left Kan extension. (So $\mathbf{T^{*}X}$ is the weak limit of $T^{*} M$ over all $M \to \mathbf{X}$.)

The opposite of this 2-functor goes $\text{St}(\text{Mfd})^{\rm op} \to \text{St}(\text{Mfd})$.

How do these two notions of cotangent stack relate to each other? Is either reasonable? Is there a better notion than either of these?

I would like, for instance, if you have a Riemannian metric on an orbifold, to get an equivalence between its tangent stack and its cotangent stack.

Along these lines, the underlying space of the cotangent bundle of an orbifold is an orbifold (or is it a manifold, like its frame bundle?), so it should be represented by a orbifold groupoid. Is this the same as the cotangent groupoid?

Keep in mind, the correct definition of tangent stack should produce the correct notion of differentiable forms on a stack, and the cotangent stack should somehow have a symplectic structure.

`$Mfd$`

and`$\text{Mfd}$`

. But feel free to roll back whatever I messed up. (In particular, I forget that jsMath gets it wrong when asked to do text-mode in scriptstyle, so the words "yoneda" are too large right now, sorry.) – Theo Johnson-Freyd May 13 '10 at 19:14