nLab ex-space

Idea

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.

References

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)