Who gives a sheaf? Part 7: Sheafification

In this video, we begin to discuss the sheafification of a presheaf.

Stack exchange thread:

A video on vector bundles:

Overview (0:00)
(re)Constructing sheaves (7:38)
We love Abelian (19:25)
Recap and looking ahead (25:30)

[lexinwonderland]

The video made sense enough to me although i had to rewatch the middle a bit, I would especially like to see some explicit examples of the objects being defined (an example of a particular pre-sheaf being sheafified) but anyway awesome stuff! Thanks for letting us know we're bullying you juuuust enough ;)

[hubertmasson]

The reason why the empty set condition is in the definition because the sheaf axioms can be put in terms of a certain diagram being an equalizer diagram for any open cover. If you take the empty set and cover it with only the empty set, the equalizer diagram will ask that the sheaf evaluated at the empty set must be a product indexed by the empty set, which is the same as a terminal object, such as zero in Ab or a one-point set in Set.

[Schieber]

In your definition of the sheafification, shouldn't ∀t rather be an ∃t? ∀t seems way to restrictive, it forces all the elements of F(V) to have identical germs I think.
Otherwise great video, thanks :D I usually think of that condition in the sheafification as making sure the germs picked out by s fit together nicely. It basically says that locally on V we need s to "look like an element t of F(V)", in the sense that it has the same germs. Considering the presheaf of bounded continuous functions usually helps with the intuition here, at least for me.