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I have heard that one can think of stacks on a site as taking sheaves but instead of the restrictions being equal, we just loosen it to isomorphic, and treat the sheaf conditions with the "obvious" coherence relations.

How seriously can one take this analogy? Note that my background is stacks is feeble at best.

I hope this question isn't too vague. One may choose to respond to this question with "That analogy is stupid because of __ gotcha".

Thanks in advance!

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Maybe you will find useful (and links there). – Grigory M Jul 25 '10 at 10:02
Yea, I have taken a look at that, in fact part of this intuition came from reading that, but I find it difficult to extract the answer to my question from that discussion because it is so rough. – BBischof Jul 25 '10 at 19:16
I think the title of this question is incorrect/misleading, and should be changed/fixed. Maybe to "Stacks are sheaves of groupoids"? – Kevin H. Lin Jul 29 '10 at 9:52
up vote 8 down vote accepted

What you are referring to is the "stacks as sheaves of groupoids" point of view.

To illustrate where it comes from, imagine for example that we are talking about the moduli stack of elliptic curves (on the category of schemes). To give an elliptic curve over a scheme, it is not just enough to specify the elliptic curve over the members of the open cover; we have to explain how we glue the restrictions of the curves on the various opens on their overlaps, and this gluing has to be coherent over triple overlaps.

The reason for this is that elliptic curves can have non-trivial automorphisms, so that there is no a priori determined way to make the identifications on the overlaps (because having non-trivial automorphisms is the same as saying that when two curves are isomorphic, they can be isomorphic in more than one way), so it is your job to choose these identifications, and to make sure that you do it in a coherent way.

(Here elliptic curves can be replaced by any other moduli problem you can think of, of course.)

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