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Idea

In parameterized homotopy theory, by a retractive space (older terminology: “ex-space” [[James (1995)](#James95)], cf. footnote 1, p. 19 in May & Sigurdsson (2006)) one means a retraction in a given category $\mathcal{S}$ of models for homotopy types (usually in TopologicalSpaces or SimplicialSets), to be thought of as a bundle $p \colon E \to B$ of pointed topological spaces over a base space $B$, where the section $i \colon B \to E$ 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.

Definition

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

\begin{tikzcd}[column sep=10pt] & E \ar[d, { p }] \ B \ar[r, equals] \ar[ur, i] & B \mathrlap{\,.} \end{tikzcd}

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

\begin{tikzcd}[sep=20pt] B \ar[d, {i}] \ar[rr, {f}] && B’ \ar[d, { i }]
\ E \ar[d, {p}] \ar[rr, {\widehat f}] && E’ \ar[d, {p}] \ B \ar[rr, {f}] && B’ \end{tikzcd}

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

which sends

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

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

\begin{tikzcd}[row sep=30pt] B \ar[d, {i}{description}] \ar[rr, {f}{description}] \ar[drr, phantom, { \scalebox{ (po) } }{pos=.7}] && B’ \ar[d, { f_!(i) }{description}]
\ E \ar[d, {p}{description}] \ar[rr, {\widehat f}{description}] \ar[drr, phantom, { \scalebox{ (po) } }{pos=.7}] && f_! E \ar[d, {f_!(p)}{description}] \ B \ar[rr, {f}{description}] && B’ \end{tikzcd}

i.e.

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

Related concepts

• retraction

• parameterized homotopy theory

• parameterized spectrum

• model structure for parameterized spectra

References

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

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

Early discussion of their model category structures includes

• Michele Intermont, Mark Johnson, Model structures on the category of ex-spaces, Topology and its Applications 119 3 (2002) 325-353 []

Discussion in the context of model structures for parameterized spectra:

• Peter May, Johann Sigurdsson, Sections 1.3 and 8.5 of: Parametrized Homotopy Theory, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (ISBN:978-0-8218-3922-5, arXiv:math/0411656, pdf)

• Vincent Braunack-Mayer, Combinatorial parametrised spectra, Algebr. Geom. Topol. 21 (2021) 801-891 [[arXiv:1907.08496](https://arxiv.org/abs/1907.08496), doi:10.2140/agt.2021.21.801]

  (based on the PhD thesis, 2018)

• Fabian Hebestreit, Steffen Sagave, Christian Schlichtkrull, Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces, Forum of Mathematics, Sigma 8 (2020) e16 [[arXiv:1904.01824](https://arxiv.org/abs/1904.01824), doi:10.1017/fms.2020.11]

• Cary Malkiewich, Section 2.1 in: Parametrized spectra, a low-tech approach [[arXiv:1906.04773](https://arxiv.org/abs/1906.04773), user guide: pdf, pdf]