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Is there an accepted name or attribution by which to refer to the following well-known theorem?

If C is a small site, then the topos of sheaves on C is the classifying topos for flat cover-preserving functors out of C.

In the case when C has a trivial topology, the corresponding assertion for its presheaf topos is usually called Diaconescu's theorem. But I don't think I have ever seen a name given to the more general version.

flag	asked Jul 10 2011 at 23:27 Mike Shulman 17.2k●20●70

	I thought the more general version was also due to Diaconescu, but that might just be because the two versions seem to occupy the same neurons in my head. – Andreas Blass Jul 10 2011 at 23:54
	I thought so too, Andreas, but section B3.2 of Sketches of an Elephant and section 4.5 of Borceux's Handbook of Categorical Algebra 3 both agree with Mike. – Tom Leinster Jul 11 2011 at 4:56

Benjamin Steinberg · Answer 1 · 2011-07-11 00:48:55Z UTC

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If the site has finite limits, then this can be found in SGA4 Proposition 4.9.4.

edited Jul 11 2011 at 0:48

answered Jul 11 2011 at 0:43

Benjamin Steinberg
10.7k●1●14●49

Exposé IV, of course. :) – James Borger Jul 11 2011 at 1:45

Mais oui! The 4 was for Exposée IV. – Benjamin Steinberg Jul 11 2011 at 2:06

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Then you forgot one of the other 4s! It's SGA4, exposé IV, proposition 4.9.4. I do love my SGA4. :) – James Borger Jul 11 2011 at 3:18

Thanks. But do they give it a name? – Mike Shulman Jul 12 2011 at 14:52

What sheaf topoi classify: attribution request

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