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Stream: learning: questions

Topic: Morphisms of differentiable stacks

ADITTYA CHAUDHURI (Aug 21 2021 at 00:24):

Let $\mathcal{X}, \mathcal{X}'$ be two differentiable stacks. Is a morphism $p: \mathcal{X} \rightarrow \mathcal{X}'$ just a usual morphism of stacks?
Or,
do we need to add any extra condition on it to make it compatible with the atlases?

David Michael Roberts (Aug 21 2021 at 06:18):

Nope. No condition. The assumption on the existence of an atlas is a pure existence statement, not part of the data. Another way to put it is that the 2-category of differentiable stacks is a full sub-2-category of the category of stacks.

ADITTYA CHAUDHURI (Aug 21 2021 at 09:30):

@David Michael Roberts Thank you for the answer. I also found this statement "2-category of differentiable stacks is a full sub-2-category of the category of stacks" in the paper "the Differentiable Stacks and Gerbes" by Behrend and Xu https://arxiv.org/pdf/math/0605694.pdf but still was feeling little uncomfortable to accept it, as in the classical set up, generally atlas is taken as a part of data in the definition of smooth manifold.

But yes, I realise now, that since the Lie groupoids representing the differentiable stacks (corresponding to two different atlas on the same underlying stack) are Morita equivalent, hence only the existential criterion is sufficient. But such previledge does not exist in the classicial level. I think in the Remark 4.17 of the paper Orbifold as stacks by Eugene Lerman https://arxiv.org/pdf/0806.4160.pdf the author remarked along that line of thoughts:
Note a loss: if we think of smooth manifolds as stacks, we lose the way to talk about maps between manifolds that are not smooth.

Fawzi Hreiki (Aug 21 2021 at 11:28):

Its easier to see this one level down with (pre)sheaves: a scheme is just a Zariski sheaf with an atlas. Since this condition is internal to the topos, maps of schemes are just natural transformations.

Fawzi Hreiki (Aug 21 2021 at 11:29):

Same story for smooth manifolds regarded as sheaves on the site of open balls.