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In chapter 13 (Quasicoherent and coherent sheaves) of Ravi Vakil's wonderful notes, the author starts by discussing vector bundles, supposedly for motivation. Having understood that each locally free sheaf gives rise to a vector bundle and vice versa, I don't understand how vector bundles "help" tell the story at all. The definition of a locally free sheaf seems to me far more natural, while vector bundles feel (to me) opaque and forced.

In what sense do vector bundles motivate locally free sheaves? Is this simply because most people encounter vector bundles in their mathematical life earlier than sheaves?

As I mentioned in this question, quasi-coherent sheaves are an enlargement of locally-free sheaves into an abelian category. Hence there is a homological algebra of quasi-coherent sheaves and in particular of locally-free ones. So perhaps there are some categories of vector bundles which tempt one to do homological algebra, thereby motivating the sheaf POV..

What are some simple examples of categories of vector bundles that tempt you to do homological algebra?

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Arrow
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    To your second question, the whole category of vector bundles tempts one to do homological algebra! But perhaps I've misunderstood. As to the first, yeah, I think people like vector bundle because they're easier to visualize than sheaves and because most people learn topology before algebraic geometry. – Kevin Carlson Aug 17 '15 at 17:24
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    @KevinCarlson my route through math has been pretty unorthodox so far.. I'm very ignorant of vector bundles. Could you point me to some books which give clear geometric intuition for them? Why/in what sense does the category of vector bundles tempt one to do homological algebra? – Arrow Aug 17 '15 at 17:33
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    Basically the same reason as locally free sheaves do, or as any additive category does. You want to have kernels and exact sequences and cohomology. Most treatments of vector bundles will be in topology, not algebraic geometry. Hatcher has a book on vector bundles and K-theory, but it's hard to know what to recommend if you haven't had much topology. Usually one picks it up on the way to knowing either algebraic or geometric topology. – Kevin Carlson Aug 17 '15 at 18:33

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