For my research I would like to read all the known proofs of the very classical result "Coherent topoi have enough points", by Deligne.

Ref 1: D3.3.13 in Sketches of an Elephant

provides a very logic-rooted proof of the statement, I would like to see a more geometric or a more category theoretic proof.

Suggested by Christian Espindola

Ref 2: 7.44 in Topos Theory by Johnstone.

Ref 3: 9.11.3 in Sheaves in Geometry and Logic.

  • 3
    $\begingroup$ The original proof by Deligne appeared in SGA4, but it might be a little bit hard to found: it is in the appendix of exposé VI, (which is part II) $\endgroup$ – Simon Henry Aug 21 at 15:49

The proof you cite is certainly based in the completeness theorem for coherent logic, but that book contains a fully categorical proof of such a theorem, based on ideas of Joyal, so it can qualify as category-theoretic. It makes use of previous categorical constructions from other parts of the book.

For a more geometric proof you can refer to Deligne's original proof, which is actually exposed in detail in Johnstone's other book "Topos theory", section 7.4. It is actually essentially the same argument, though you can distinguish the geometric flavour here.

And there is of course the proof given in Maclane-Moerdijk "Sheaves in geometry and logic" which is based on Barr's theorem and is also quite geometric.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.