homotopy theory, (∞,1)-category theory, homotopy type theory
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see also algebraic topology
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For $X$ a suitable space of sorts, the category of vector bundles over $X$ is the category denoted $Vect(X)$ whose
objects are vector bundles over $X$,
morphisms are vector bundle homomorphisms over $X$.
Specifically for $X$ a topological space, there is the category of topological vector bundles over $X$.
Via direct sum of vector bundles and tensor product of vector bundles this becomes a symmetric monoidal category in two compatible ways, making it a distributive monoidal category, in particular a rig category.
For $X$ a compact Hausdorff space then the Grothendieck group of $Vect(X)$ is the topological K-theory group $K(X)$.
For $X = \ast$ the point space, then this is equivalently the category Vect of plain vector spaces:
An analog in homotopy theory/higher category theory is the (infinity,1)-category of (infinity,1)-module bundles.
Last revised on June 4, 2020 at 06:29:12. See the history of this page for a list of all contributions to it.