nLab retractive space





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




In parameterized homotopy theory, by a retractive space (older terminology: “ex-space” [James (1995)], cf. footnote 1, p. 19 in May & Sigurdsson (2006)) one means a retraction in a given category 𝒮\mathcal{S} of models for homotopy types (usually in TopologicalSpaces or SimplicialSets), to be thought of as a bundle p:EBp \colon E \to B of pointed topological spaces over a base space BB, where the section i:BEi \colon B \to E exhibits the fiber-wise base points.

Just as plain pointed topological spaces serve as the basis on which to construct spectra, so retractive spaces serve as a basis on which to construct parameterized spectra.


A retractive space is a commuting diagram in a category 𝒮\mathcal{S} “of spaces”, of this form:

Taking the category 𝒮 \mathcal{S}_{\mathcal{R}} of retractive spaces to have as morphisms the evident commuting diagrams

reflecting bundle-homomorphisms respecting the base points, this is equivalent to the Grothendieck construction on the pseudo-functor

(𝒮 /()) */:𝒮Cat \big(\mathcal{S}_{/(-)}\big)^{\ast/} \;\colon\; \mathcal{S} \longrightarrow Cat

which sends

  • spaces BB \in \mathcal{B} to the pointed category (𝒮 /B) */\big(\mathcal{S}_{/B}\big)^{\ast/} of pointed objects in the slice category of 𝒮\mathcal{S} over BB (i.e. in the category of “bundles” over the fixed base space BB),

  • morphisms f:BBf \colon B \to B' of base spaces to the functor f !f_! forming pushouts of bundles along f !f_!:


𝒮 B𝒮(𝒮 /B) */. \mathcal{S}_{\mathcal{R}} \;\simeq\; \int_{B \in \mathcal{S}} \big(\mathcal{S}_{/B}\big)^{\ast/} \,.

This follows readily from the definitions, but see also Braunack-Mayer (2021), Rem. 1.15; Hebestreit, Sagave & Schlichtkrull (2020), Lem. 2.14.


The terminology “ex-spaces” is due to Ioan James, used for instance in:

Early discussion of their model category structures includes

Discussion in the context of model structures for parameterized spectra:

Last revised on April 17, 2023 at 11:25:11. See the history of this page for a list of all contributions to it.