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Let's say we have a $k$ group scheme $G$ acting on a $k$ scheme $X$, we can consider its quotient in the category of stacks, the usual definition of it would be the quotient stack $[X/G]$ defined by

$[X/G](S)$ is the groupoid of $G$ bundles over $S$ together an equivariant maps to $X$.

My Question 1 is why don't we take the categorical quotient?

I think the categorical quotient exists because we can take the categorical quotient in the categories of 2-functors from Schemes to Groupoids, and then sheafify (stackify).

I understand that the stack quotients enjoy some very good properties, e.g. it makes every action look like a free action and it the theory of sheaves downstair is equivalent to the theory of sheaves.

but why people (or do they) don't study categorical quotient as well?

My Question 2 is how do stack quotients differentiate from categorical quotients?

Stack quotient is equal to the categorical quotient in the case free action, and I have the feeling that the two notions agree only when the action is free, is it true?

Thank you very much!

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  • $\begingroup$ I am not sure if I understand your question correctly.. You say "Schemes to Groupoids, and then sheafify (stackify)"... I thought this is what happening.. Isn't it so? See page 29 for definition of $\widetilde{y}(\mathcal{G})$ for a Lie groupoid $\mathcal{G}$ and page number 69 how sheafification/stackification comes into picture when defining the stack $B\mathcal{G}$... This is about stacks coming from Lie groupoids.. I am sure the same can be done for stacks coming through schemes... $\endgroup$ Apr 27, 2020 at 3:51
  • $\begingroup$ See mathoverflow.net/a/339953/118688 It looks like it holds for schemes as well $\endgroup$ Apr 27, 2020 at 3:52
  • $\begingroup$ answer in mathoverflow.net/questions/319038/… may be of useful.. $\endgroup$ Apr 27, 2020 at 4:09
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    $\begingroup$ It sounds like you are asking for a map from $X$ to a stack that is $G$-invariant and universal with respect to $G$-invariant maps to stacks. The quotient map $X \to [X/G]$ satisfies this property, so stack quotients and categorical quotients in stacks are the same thing. $\endgroup$
    – S. Carnahan
    Apr 27, 2020 at 4:12
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    $\begingroup$ You seem to be working with some sort of 1-category of stacks. I'm unclear on how exactly it is defined, but regardless that is not the right thing to consider. What you want to consider is the (2,1)-category of stacks, and there $BG$ is the categorical quotient. In general any notion that does not identify two equivalent stacks is unlikely to have geometric meaning. $\endgroup$ Apr 27, 2020 at 8:35

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