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



Paths and cylinders

Homotopy groups

Basic facts




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 EXE \to X) equipped with a global section XEX \to E. More generally, an ex-object in any category or (infinity,1)-category 𝒞\mathcal{C} is a diagram in 𝒞\mathcal{C} of the form

E X id X. \array{ && E \\ & \nearrow & \downarrow \\ X &\underset{id}{\longrightarrow}& X } \,.

(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 (,1)(\infty,1)-category and then EXE \to X is required to be a fibration.)

One may think of this as a parameterization of pointed objects over XX. As such this is a topic in parameterized homotopy theory. For instance, passing to the fiberwise suspension spectra of an ex-space EXE \to X yields a parameterized spectrum over XX.

When the ambient category 𝒞\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

  • James, …

Discussion of their model category structures includes

  • Michele Intermont, Mark Johnson, Model structures on the category of ex-spaces, Topology and its Applications Volume 119, Issue 3, 30 April 2002, Pages 325–353 (publisher)

A comprehensive account of the parameterized homotopy theory given by ex-spaces is in

Revised on January 7, 2016 04:28:28 by Urs Schreiber (