What itinerary would a gentle introduction to sheaves have to take? I would suggest the following:

A basic tour of the category Set, introducing objects, arrows, arrow composition, unique arrows and limits. (OK, that’s actually quite a lot to start with).

Introduction to bundles and sections, with a nicely-motivated example.

Enough topology to know what open sets are and what a continuous map is, topologically speaking.

Now we can talk about product spaces and fiber bundles.

Now we can talk about the sheaf of sections on a fiber bundle.

Now we back up and talk about order structures – posets, lattices, Heyting algebras, and their relationship to the lattice of open sets in a topological space. We note in passing that a poset can be seen as a kind of category.

Functors, covariant and contravariant.

That’s probably enough to get us to a categorial description of first presheaves (contravariant functor from a poset category to e.g. Set) and sheaves (presheaves plus gluing axiom, etc). Show how this captures the fundamental characteristics of the sheaf of sections we met earlier.

Then to applications outside of the sheaf of sections; sheaf homomorphisms and sheaf categories; applications in logic and so on. This is actually where my understanding of the topic falls of the edge of the cliff, but I think that rehearsing all of the material up to this point might help to make some of it more accessible.

Anything really essential that I’ve missed? Anything I’ve included that’s actually not that important?