# Stack associated to Groupoid object in category $\text{Sch}/S$

Consider the category of manifolds $$\text{Man}$$.

A groupoid object in the category of manifolds is called a Lie groupoid, denoted by $$\mathcal{G}$$. There is a way to associate a stack (over the category $$\text{Man}$$) for this $$\mathcal{G}$$, the stack of principal $$\mathcal{G}$$ bundles, denoted by $$B\mathcal{G}$$. Any stack which is isomorphic to $$B\mathcal{G}$$ is called a geometric stack. So, geometric stacks are same as Lie groupoids; the ”geometric” stacks of interest over the category of manifolds are precisely the stacks associated to groupoid objects in the category of manifolds.

Now, fix a scheme $$S$$ and consider the category, $$\text{Sch}/S$$, of schemes over $$S$$. Fix some Grothendieck topology (I do not know many, so, you can fix what ever is comfortable). In this category also I can talk about groupoid objects, take one such and denote by $$\mathcal{G}$$. Is there a way to associate a stack over the category $$\text{Sch}/S$$.

Is it the case, similar to the case of manifolds, that the ”geometric” stacks of interest over the category $$\text{Sch}/S$$ are precisely the stacks associated to groupoid objects in the category of $$\text{Sch}/S$$? I mean to ask are Algebraic stack precisely the stacks coming from groupoid object in category $$\text{Sch}/S$$? I guess the answer is No (please feel free to add some comments regarding this).

Are stacks over $$\text{Sch}/S$$ associated to groupoids over $$\text{Sch}/S$$ of any interest? Do they cover “most” of Algebraic stacks over the stack $$S$$ with what ever topology on $$\text{Sch}/S$$? Are there (basic) examples of stacks associated to groupoids over $$\text{Sch}/S$$ that are not algebraic?

• Just after reading the title I knew it was a question by you! :-) If you allow your "geometric" groupoids to be internal to algebraic spaces instead of just schemes, and consider the lisse (smooth) topology, then I think all the answers become "yes" by Laumon--Moret-Bailly Champs algébriques, (4.3) and Proposition (4.3.1). – Qfwfq Sep 4 '19 at 18:09
• @Qfwfq I am taking “Just after reading the title I knew it was a question by you!” as a nice comment, not sure if this was supposed to be nice :D.. Can you please give some English reference... Does it follow as straight forward as in the case of Lie groupoids or it is more serious (I do not think there is no definite answer for this, just asking)... – Praphulla Koushik Sep 4 '19 at 18:09
• "I am taking [...]" No offence intended! - You could have a look at the Stacks Project; looks like there's a whole chapter on groupoids in algebraic spaces (stacks.math.columbia.edu/tag/0437). The association groupoid in alg sp $\mapsto$ corresponding stack is (I think) stacks.math.columbia.edu/tag/044O. For the reverse association, I think you just take the groupoid $U\times_{\mathscr{X}}U\to U$ of an atlas $U\to\mathscr{X}$. – Qfwfq Sep 4 '19 at 18:26
• @Qfwfq I knew no offence was intended :) :) I saw that chapter of groupoids on Algebriac spaces just now.. it looks good, I will see what I can understand from that... I will also read about quotient stacks as well. – Praphulla Koushik Sep 4 '19 at 18:33
• As a general reference, these notes seem to me to be good, especially Appendix C: arxiv.org/abs/1708.08124 – David Roberts Sep 6 '19 at 2:46

For any algebraic stack $$X$$, there is a groupoid in schemes whose fppf stackification is equivalent to $$X$$. You can construct such a groupoid following the stacks project, by choosing a smooth presentation in algebraic spaces (Lemma 04T5), then choosing an étale presentation of the component algebraic spaces (Lemma 0262).
• Ok, I am not sure if I am understanding correctly or not.. For an algebriac stack $X$, we some how get a groupoid in scheme... For this groupoid, we associate a stack (in the usual sense we associate stack for Lie groupoid) over some Grothendieck topology... But, this is not a stack with respect to the fppf topology... so, we stackify (tag/06WP) to get a stack/fppf... Then, this stack is isomorphic to $X$.. Is this correct? – Praphulla Koushik Sep 5 '19 at 15:49