Today's lecture is another algebraic interlude (this will be the last such interlude of the semester). We introduce the notion of a sheaf, and use this to unify many of the concepts we've looked at during so far.

Just as with the category theory from Lecture 14, we will never actually use any genuine theorems in sheaf theory—for us it will merely be a convenient way to concisely formulate other concepts.

Roughly speaking, a presheaf is a way to assign data locally to open subsets of a topological space in such a way that it is compatible with restrictions. A sheaf is a presheaf for which it is possible to go backwards and reassemble global data from local data.

After giving the relevant definitions, we give a “sheaf-theoretic” definition of a manifold and a vector bundle, and outline the proof that there is an equivalence of categories between the category of vector bundles over $M$ and the category of finite rank locally free $C^{ \infty}(M)$-modules.

##### Remark

This is a course on geometry, not algebra! Large swathes of this lecture are completely non-examinable. As usual, I urge you to ignore everything marked with a $( \clubsuit)$.