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A continuously varying family of vector spaces of the same dimension over a topological space. If the vector spaces are one-dimensional, the term line bundle is used and has the associated tag line-bundles.
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Is the theory of vector bundles just linear algebra done in a suitable topos?
I'll gather all the snippets from the various comments here (and mark this post as community wiki).
The answer to the question "Is the theory of vector bundles just linear algebra done in a suitable …