(see also Chern-Weil theory, parameterized homotopy theory)
vector bundle, (∞,1)-vector bundle
topological vector bundle, differentiable vector bundle, algebraic vector bundle
direct sum of vector bundles, tensor product of vector bundles, inner product of vector bundles?, dual vector bundle
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
An “ex-space” (terminology due to James, adopted in May-Sigurdsson 04) is a general bundle of topological spaces/homotopy types (hence any map of spaces $E \to X$) equipped with a global section $X \to E$. More generally, an ex-object in any category or (infinity,1)-category $\mathcal{C}$ is a diagram in $\mathcal{C}$ of the form
(In an (infinity,1)-category this diagram is filled with a 2-morphism, a homotopy. Often this is considered in a category of fibrant objects, or similar, presenting an $(\infty,1)$-category and then $E \to X$ is required to be a fibration.)
One may think of this as a parameterization of pointed objects over $X$. As such this is a topic in parameterized homotopy theory. For instance, passing to the fiberwise suspension spectra of an ex-space $E \to X$ yields a parameterized spectrum over $X$.
When the ambient category $\mathcal{C}$ is suitably monoidal, then the category of ex-objects with its fiberwise smash product forms a monoidal fibration. See at dependent linear type theory for more on this.
The terminology “ex-spaces” is due to
Discussion of their model category structures includes
A comprehensive account of the parameterized homotopy theory given by ex-spaces is in
Last revised on January 7, 2016 at 04:28:28. See the history of this page for a list of all contributions to it.