(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**

**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

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

(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 $(\infty,1)$-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 Ioan James. See for instance:

- Ioan Mackenzie James,
*Introduction to fibrewise homotopy theory*in Ioan Mackenzie James (ed.),*Handbook of Algebraic Topology*(1995)

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

- Peter May, J. Sigurdsson, section 1.3 and 8.5 of
*Parametrized Homotopy Theory*(arXiv:math/0411656)

Last revised on February 2, 2020 at 09:48:48. See the history of this page for a list of all contributions to it.