(see also Chern-Weil theory, parameterized homotopy theory)
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
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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 ) equipped with a global section . More generally, an ex-object in any category or (infinity,1)-category is a diagram in 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 -category and then is required to be a fibration.)
One may think of this as a parameterization of pointed objects over . As such this is a topic in parameterized homotopy theory. For instance, passing to the fiberwise suspension spectra of an ex-space yields a parameterized spectrum over .
When the ambient category 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 Ioan James. See for instance:
Discussion of their model category structures includes
A comprehensive account of the parameterized homotopy theory given by ex-spaces is in
Last revised on February 2, 2020 at 09:48:48. See the history of this page for a list of all contributions to it.