Posted by Adi Ostrov

https://mathoverflow.net/questions/507771/counterexamples-of-nice-properties-of-the-composition-of-sheafification-with-yon

https://mathoverflow.net/q/507771

I'm reading "Sheaves in Geometry and Logic", and am stuck on the following exercise:

Give examples that illustrate that $ay : C \to Sh(C, J)$ can be full, but not faithful, faithful but not full, or neither full nor faithful.

I am really stuck on this exercise. I managed to find an example which is faithful but not full: take $C = (\mathbb Z, \le)$ with every arrow covering. Then the sheafification of all the Yoneda embeddings should all be equal to the same constant functor, thus there would be morphisms between two such sheafifications even between elements which have an empty hom-set in $C$.

However, I am not sure how to find examples for the other cases. Additionally, I would be happy to get some intuition about what does it "mean" about the site if $ay$ does not satisfy this or that property. Is a failure of faithfulness "due to" the topology being "too large"? In what way? How about failure to be full?

Thanks!

Edit:

I realised that I accidentaly made the exercise harder than it should be, by assuming that empty covers are not allowed. If they are allowed, I can take $C = B\mathbb N$, the category of a single object with an endomorphism, and take the topology with the empty cover on the object. In this case, the only sheaf is the one with $F(*) = \{*\}$, with no non-trivial endomorphisms. Sheafification takes all the maps to the identity, thus $ay$ is not faithful.

Of course, the union of both sites is an example where $ay$ is neither full nor faithful.

Having found these answers, I will still be happy to hear some intuition on the reason for such failures.

https://mathoverflow.net/questions/507771/counterexamples-of-nice-properties-of-the-composition-of-sheafification-with-yon

https://mathoverflow.net/q/507771