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.

  • $\begingroup$ 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. $\endgroup$ – Andreas Blass Jul 10 '11 at 23:54
  • $\begingroup$ 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. $\endgroup$ – Tom Leinster Jul 11 '11 at 4:56

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

  • $\begingroup$ Exposé IV, of course. :) $\endgroup$ – JBorger Jul 11 '11 at 1:45
  • $\begingroup$ Mais oui! The 4 was for Exposée IV. $\endgroup$ – Benjamin Steinberg Jul 11 '11 at 2:06
  • 2
    $\begingroup$ Then you forgot one of the other 4s! It's SGA4, exposé IV, proposition 4.9.4. I do love my SGA4. :) $\endgroup$ – JBorger Jul 11 '11 at 3:18
  • $\begingroup$ Thanks. But do they give it a name? $\endgroup$ – Mike Shulman Jul 12 '11 at 14:52

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