# Examples of of gerbe over stacks in terms of manifolds

I am looking for some examples of gerbes over stacks (as defined in Understanding definition of gerbe over a stack) that comes from manifolds.

Let $$M$$ be a manifold then $$\underline{M}$$ is a stack associated to $$M$$ (I can give details if any one needs).

We know when to call a map of stacks $$\mathcal{D}\rightarrow \mathcal{C}$$ a gerbe over stack. Suppose $$\mathcal{D}$$ is coming from a manifold $$M$$ and $$\mathcal{C}$$ is coming from a manifold $$N$$ i.e., we have $$\underline{M}\rightarrow \underline{N}$$.

Given a map of stacks $$F:\underline{M}\rightarrow \underline{N}$$, we have unique map $$F(id):M\rightarrow N$$ associated to $$F$$.

First condition says that $$F:\underline{M}\rightarrow \underline{N}$$ is an epimorphism. Suppose $$f:M\rightarrow N$$ is a surjective submersion, then it follows that $$F:\underline{M}\rightarrow \underline{N}$$ is an epimorphism.

Second condition says that the diagonal map $$\underline{M}\rightarrow \underline{M}\times_{\underline{N}}\underline{M}$$ is an epimorphism. Suppose diagonal map $$M\rightarrow M\times_N M$$ is a surjective submersion, then $$\underline{M}\rightarrow \underline{M}\times_{\underline{N}}\underline{M}$$ is an epimorphism.

But then, $$M\rightarrow M\times_N M$$ is surjective implies that $$f$$ is injective.

An injective, surjectve submersion is a diffeomorphism. So, I landed up with a diffeomorphism conidtion on $$M\rightarrow N$$.

Do we have other non trivial examples?

Edit: Ycor has added algebraic geometry tag. I am also ok with algebraic geometry examples. I do not know much (not even the precise definition in that set up). Smooth manifolds are replaced by schemes (I guess). Given a scheme, there is an associated stack (algebraic stack??). Gerbe over stack in case of algebraic geometry of the form $$\underline{X}\rightarrow \underline{Y}$$ should correspond to some morphism of schemes $$X\rightarrow Y$$. It will be good if you can define gerbe over stack in algebraic geometry set up (loosely atleast) and the give examples.

• @YCor: Please see the edit. – Praphulla Koushik Nov 28 '18 at 15:06
• Short answer: no (if I understand the question correctly: when is $X \to Y$ a gerbe, for representable stacks $X$ and $Y$). Roughly speaking, the fibres of a gerbe are transitive (Lie) groupoids. So if the fibres are in fact manifolds, they are single points. Gerbes are a truly stacky phenomenon and do not appear until nontrivial stacks are around. – David Roberts Nov 28 '18 at 15:25
• You understood the question correctly. Yes. Question is : "When is a map of stacks $\underline{M}\rightarrow \underline{N}$ a gerbe over stack". I could only come up with trivial examples as mentioned in the question.. Can you give a reference for "fibres of a gerbe are transitive (Lie) groupoids".. Thank you. @DavidRoberts – Praphulla Koushik Nov 28 '18 at 15:33
• @PraphullaKoushik I added it as a parent tag, not to mean that you require algebraic geometry examples. The other two tags are too narrow; the parent tags I had in mind are ct.category-theory, at.algebraic-topology, ag.algebraic-geometry, and dg.differential-geometry, it's better with at least one of these (or another broad one) and don't hesitate changing. The tag not only yields an orientation, but makes the question visible. – YCor Nov 28 '18 at 15:50
• @PraphullaKoushik it's essentially the definition. A gerbe is a stack of groupoids that has non-empty stalks ($X\to Y$ is an epimorphism), which are groupoids, and (more or less) the stalks are transitive ($X\to X\times_Y X$ is an epimorphism). Another way of thinking about it, is that a gerbe is like a bundle where the fibres are equivalent to a 1-object groupoid. – David Roberts Nov 28 '18 at 22:46

There are no other such gerbes. If $$M$$ and $$N$$ are manifolds, and $$p\colon \underline{M}\to \underline{N}$$ is a gerbe, then the corresponding map of manifolds is a diffeomorphism. The same holds if one merely has representable stacks, rather than something of the form $$\underline{M}$$ etc.
• I am trying to say something like "If $\underline{M}\rightarrow \underline{N}$ is an epimorphism, then, $M\rightarrow N$ is a [---]" similarly, "If the diagonal map $\underline{M}\rightarrow \underline{M}\times_{\underline{N}}\underline{M}$ is an epimorphism, then, $M\rightarrow N$ is a [---]"... – Praphulla Koushik Nov 29 '18 at 5:13
• I understand your comment " [---] has non empty stacks ($X\rightarrow Y$ is an epimorphism) [---] (more or less) stalks are transitive ($X\rightarrow X\times_YX$ is an epimorphism)".. I have said something like this in my answer here mathoverflow.net/questions/307123/….. So, more or less stalks are transitive Lie groupoids and if they are manifolds you are saying they have to singletons.. I am more or less ok with this.. – Praphulla Koushik Nov 29 '18 at 5:20
• I am little confused... You said "The same holds if one merely if one merely has representable stacks, rather than something of the form $\underline{M}$".. As far as I know, a stack $\mathcal{D}$ is representable if there is an isomorphism $\mathcal{D}\rightarrow \underline{M}$ for a manifold $M$... Is there a more general definition that I am not aware of? – Praphulla Koushik Nov 29 '18 at 6:45
• What I mean is that the notion is invariant under equivalence of stacks. If a stack is merely equivalent to one of the form $\underline{M}$, then it may have, for instance, fibres that are contractible groupoids. Also, a representable stack $X$ doesn't necessarily come with the data of a manifold $M$ and an equivalence $\underline{M}\simeq X$. (NB Isomorphisms are not enough here, since we are dealing with objects of a 2-category) – David Roberts Nov 29 '18 at 7:02