nLab parameterized homotopy theory

Idea

Parameterized (stable) homotopy theory is (stable) homotopy theory of bundles of homotopy types/stable homotopy types over a given base space.

For formalizations see also at

• ex-space

• parameterized spectrum, excisive functor

• (infinity,1)-module bundle

• tangent (infinity,1)-topos

• twisted cohomology

Parameterized point-set topology

The point-set topology of parametrized spaces is surprisingly subtle. [May & Sigurdsson 2006, p. 15]

Write:

• $kTop$ for the convenient category of compactly generated topological spaces (k-spaces)

• $kwHaus \subset Top_k$ for that of compactly generated weak Hausdorff spaces.

For $B \,\in\, \mathcal{C}$ an object in any category, write $\mathcal{C}_{/B}$ for the slice category over it.

In the following all bases spaces are assumed (as in MaSi06, p. 19) to be compactly generated weak Hausdorff spaces regarded among k-spaces:

(1)$$B \;\in\; kwHaus \xhookrightarrow{\;} kTop \,.$$

Notice that for $(X,p_X), (Y,p_Y) \,\in\, kTop_{/B}$, their Cartesian product in the slice category is given by the fiber product in $kTop$:

$$(X,p_X) \times (Y, p_Y) \;\simeq\; \big( X \times_B Y ,\, p_X \circ pr_X = p_Y \circ pr_Y \big) \;\;\; \in \; kTop_{/B}$$

For $f \,\colon\, B_1 \xrightarrow{\;\;} B_2$ a continuous map between such base spaces, notice the usal base change adjoint triple:

(2)

where

1. $f^\ast$ is pullback in $kTop$ along $f$;

2. $f_!$ is post-composition with $f$;

Notice the “Frobenius reciprocity law” (in its cartesian version here) which follows immediately by the pasting law in $kTop$, namely the following natural isomorphism:

(3)$$f_! \big( (U,p_U) \times f^\ast (X,p_X) \big) \;\simeq\; \big( f_!(U,p_U) \big) \times (X,p_X) \,.$$

In the special case where $f$ is the terminal map to the point space, which we denote

$$p_B \,\colon\, B \xrightarrow{\;} \ast \,.$$

we have $kTop_{/\ast} \,\simeq\, kTop$ and the above base change adjoint triple becomes

(4)$$\;$$

In this case

1. the functor $(p_B)^\ast$ is the Cartesian product with $B$, regarded as the trivial fibration:

   (5)$$(p_B^\ast) X_0 \;=\; B \times X_0 \xrightarrow{\; pr_B \;} B \,,$$

2. $(p_B)_!$ gives the total space of a fibration:

   $$(p_B)_! \big( X \to B\big) \;=\; X$$

3. $(p_B)_*$ gives the space of sections of a fibration.

Eventually we consider pointed objects

$$(X, p_X, \sigma_X) \;\in\; \big( kTop_{/B}\big)^{B/} \;=\; kTop_{/B}^{B/}$$

in the slice category of such a base space – hence topological “bundles” $X \xrightarrow{p_X} B$ (in the most general sense, without any condition on the bundle projection, except continuity) equipped with a fixed section $\sigma_X$ (sometimes called “ex-spaces”, see [May & Sigurdsson 2006, p. 19, footnote 1]).

Fiberwise mapping spaces

Parametrized mapping spaces are especially delicate [May & Sigurdsson 2006, p. 15, see Rem. below]

Definition

(partial map classifier space)
For $X \,\in\, kTop$, write $\widetilde X \,\in\, kTop$ for its continuous partial map classifier: The result of forming the disjoint union of the underlying set of $X$ with a singleton set $\{\bot\}$ and declaring the closed subsets on the result to be those of $X$ under the defining injection

$$X \xhookrightarrow{\;\; \iota_X \;\;} \widetilde{X}$$

together with $\widetilde{X}$ itself.

(MaSi06, Def. 1.3.6)

Definition

(fiberwise mapping space)
For

$$(X, p_X), \, (A,p_A) \;\in\; kTop_{B}$$

a pair of k-spaces over $B$ (1) their fiberwise mapping space is the pullback (in $kTop$):

(6)$$\array{ \phantom{---} \mathclap{ Map \big( (X,p_X) ,\, (A,p_A) \big) } \phantom{---} &\xrightarrow{\phantom{-------}}& Map \big( X, \widetilde{A} \big) \\ \big\downarrow &{}_{{}^{(pb)}}& \big\downarrow{}^{\mathrlap{ Map \big( X, \widetilde{p_A} \big) }} \\ B & \underset{ b \mapsto \left( x \mapsto \left\{ \begin{array}{ll} b & if\, x \in X_b \\ \bot & otherwise \end{array} \right. \right) }{\longrightarrow} & Map \big( X, \widetilde{B} \big) \,, }$$

regarded as an object of $kTop_{/B}$.

Here $Map(X,\widetilde{A})$ denotes the ordinary mapping space into the continuous partial map classifier from Def. .

(This is May & Sigurdsson 2006, Def. 1.3.7, following Booth & Brown 1978a).

Remark

(on notation)
Contrary to most references, Def. is intentionally not using a subsript “${}_B$” in the notation for the fiberwise mapping space: This is because “$Map(X,A)_B$” is also standard notation for $Map(X,A) \underset{Map(X,B)}{\times} \{p_B\}$ (see e.g. at space of sections), which is crucially different. Instead, with the above notation, $Map(-,-)$ is always of the same type as its arguments, as befits an internal hom.

Proposition

(fiberwise mapping space satisfies the exponential law)
With $B$ as above (1), the fiberwise mapping space (Def. ) is an exponential object (satisfies the exponential law) in that there is a natural isomorphism of hom-sets

(7)$$kTop_{/B} \Big( (X,p_X) ,\, Map \big( (Y,p_Y) ,\, (A,p_A) \big) \Big) \;\; \simeq \;\; kTop_{/B} \big( (X,p_X) \times (Y,p_Y) ,\, (A,p_A) \big)$$

(where on the right we have the Cartesian product in the slice, given by the fiber product $X \times_B Y$ in $kTop$).

(Booth & Brown 1978a, Thm. 3.5, see May & Sigurdsson 2006, (1.3.9))

Example

(fiberwise mapping space between trivial fibrations)
The fiberwise mapping space (Def. ) between trivial fibrations (5) is the trivial fibration with fiber the ordinary mapping space between the fibers:

$$Map \big( p_B^\ast X_0 ,\, p_B^\ast A_0 \big) \;\simeq\; p_B^\ast Map\big(X_0,\, A_0\big) \,.$$

Proof

This may be gleaned concretely from point-set-analysis of the defining pullback diagram (6), but it also follows abstractly by adjointness from the exponential law (Prop. ):

For any $(U, p_U) \,\in\, kTop_{/B}$ we have the following sequence of natural isomorphisms:

$$\begin{aligned} & kTop_{/B} \Big( (U, p_U) ,\, \Map \big( p_B^\ast X_0 ,\, p_B^\ast A_0 \big) \Big) \\ & \;\simeq\; kTop_{/B} \big( (U, p_U) \times (B \times X_0, pr_{B}) ,\, p_B^\ast A_0 \big) \\ & \;\simeq\; kTop_{/B} \big( (U \times X_0, p_U \circ pr_U) ,\, p_B^\ast A_0 \big) \\ & \;\simeq\; kTop \Big( U \times X_0 ,\, A_0 \big) \\ & \;\simeq\; kTop \Big( U ,\, Map \big( X_0 ,\, A_0 \big) \Big) \\ & \;\simeq\; kTop \Big( (p_B)_! (U,p_U) ,\, Map \big( X_0 ,\, A_0 \big) \Big) \\ & \;\simeq\; kTop_{/B} \Big( (U,p_U) ,\, p_B^\ast Map \big( X_0 ,\, A_0 \big) \Big) \end{aligned}$$

Here most steps are Hom-isomorphisms of the various adjoint functors: (4) and (7). Since this holds naturally for all $(U, p_U)$, the claim follows by the Yoneda lemma (over the large category $\big(kTop_{/B}\big)^{op}$).

Similarly:

Proposition

(pullback of fiberwise mapping space)
For $f \,\colon\, B' \longrightarrow B$ a map of base spaces (1), the pullback (2) along $f$ of the fiberwise mapping space (Def. ) is the fiberwise mapping space of the pullback of the arguments:

$$f^\ast Map \big( (X,p_X) ,\, (A, p_A) \big) \;\; \simeq \;\; Map \big( f^\ast (X,p_X) ,\, f^\ast (A, p_A) \big) \,.$$

In other words: Pullback $f^\ast$ is a closed functor with respect to fiberwise mapping spaces.

Proof

For any $(U, p_U) \,\in\, kTop_{/B'}$ we have the following sequence of natural isomorphisms:

$$\begin{aligned} & kTop_{/B'} \Big( (U,p_U) ,\, f^\ast Map \big( (X,p_X) ,\, (A,p_A) \big) \Big) \\ & \;\simeq\; kTop_{/B} \Big( f_! (U,p_U) ,\, Map \big( (X,p_X) ,\, (A,p_A) \big) \Big) \\ & \;\simeq\; kTop_{/B} \Big( \big( f_!(U, p_U) \big) \times (X, p_X) ,\, (A,p_A) \Big) \\ & \;\simeq\; kTop_{/B} \Big( f_! \big( (U, p_U) \times f^\ast (X, p_X) \big) ,\, (A,p_A) \Big) \\ & \;\simeq\; kTop_{/B'} \Big( (U, p_U) \times f^\ast (X, p_X) ,\, f^\ast (A,p_A) \Big) \\ & \;\simeq\; kTop_{/B'} \Big( (U, p_U) ,\, Map \big( f^\ast (X, p_X) ,\, f^\ast (A,p_A) \big) \Big) \end{aligned}$$

Here the crucial step, besides various Hom-isomorphisms, is the use of Cartesian “Frobenius reciprocity” (3).

Since these isomorphism are natural in $(U,p_U)$, the claim follows by the Yoneda embedding (for the large category $\big( kTop_{/B'}\big)^{op}$).

,

Example

(fiber of fiberwise mapping space is mapping space of fibers)
For $b \in B$, the fiber of the fiberwise mapping space fibration (Def. ) is homemorphic to the ordinary mapping space betwee the fibers:

$$Map \big( (X,p_x) ,\, (A,p_A) \big)_b \;\; \simeq \;\; Map \big( X_b ,\, A_b \big) \,.$$

(e.g. May & Sigurdsson 2006, p. 21)

Proof

This is immediate from concrete analysis of the defining pullback-diagram (6) in Def. , but it is also the special case of Prop. for $B = \{b\}$.

Homotopy theory of the fiberwise mapping space

Proposition

(fiberwise mapping space preserves h-fibrations)
If $p_X \colon X \to B$ and $p_A \colon A \to B$ are Hurewicz fibrations, then so is the map (6) out of their fiberwise mapping space (Def. ):

$$p_X, p_A \,\in\, hFib \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; p_{ Map\big( (X,p_X) , (A,p_A)\big) } \;\in\; hFib \,.$$

(This is due to Booth 1970, §6.1, see MaSi06, Prop. 1.3.11.)

Remark

(fiberwise mapping space does not preserve weak Hausdorffness)
Even if $X$ and $A$ are weak Hausdorff spaces over the weak Hausdorff space $B$ (1), their fiberwise mapping space (Def. ) need not be weak Hausdorff (Booth & Brown 1974a). Sufficient conditions for this to be the case are given in Lewis 1985, Prop. 1.5

On the other hand, the suitable cofibrant resolution of the fiberwise mapping space will again be weak Hausdorff (see MaSi06, p. 19).

Related concepts

• retractive space

• parameterized spectrum

• model structure for parameterized spectra

• rational parametrized homotopy theory

  • rational fibration lemma

• rational parameterized stable homotopy theory

References

Exponential law for parameterized topological spaces

On exponential objects (internal homs) in slice categories of (compactly generated) topological spaces – see at parameterized homotopy theory):

• Peter I. Booth, The Exponential Law of Maps I, Proceedings of the London Mathematical Society s3-20 1 (1970) 179-192 $[$doi:10.1112/plms/s3-20.1.179$]$

• Peter I. Booth, The exponential law of maps. II, Mathematische Zeitschrift 121 (1971) 311–319 $[$doi:10.1007/BF01109977$]$

• Peter I. Booth, Ronnie Brown, Spaces of partial maps, fibred mapping spaces and the compact-open topology, General Topology and its Applications 8 2 (1978) 181-195 $[$doi:10.1016/0016-660X(78)90049-1$]$

• Peter I. Booth, Ronnie Brown, On the application of fibred mapping spaces to exponential laws for bundles, ex-spaces and other categories of maps, General Topology and its Applications 8 2 (1978) 165-179 $[$doi:10.1016/0016-660X(78)90048-X$]$

• L. Gaunce Lewis, Jr., §1 of: Open maps, colimits, and a convenient category of fibre spaces, Topology and its Applications 19 1 (1985) 75-89 $[$doi.org/10.1016/0166-8641(85)90087-2$]$

And with an eye towards parameterized homotopy theory:

• Ioan Mackenzie James: §II.9 in: Fibrewise topology, Cambridge Tracts in Mathematics, Cambridge University Press (1989) $[$ISBN:9780521360906$]$

• Peter May, Johann Sigurdsson, §1.3.7-§1.3.9 in: Parametrized Homotopy Theory, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (ISBN:978-0-8218-3922-5, arXiv:math/0411656, pdf)

Parameterized (“fiberwise”) homotopy theory

On the homotopy theory of such parameterized topological spaces:

• Ioan Mackenzie James, §IV of: Fibrewise topology, Cambridge Tracts in Mathematics, Cambridge University Press (1989) [ISBN:9780521360906]

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

• Michael C. Crabb, Ioan Mackenzie James: Fiberwise homotopy theory, Springer Monographs in Mathematics, Springer (1998) [doi:10.1007/978-1-4471-1265-5, pdf ,pdf]

On model structures for parameterized spectra:

• Peter May, Johann Sigurdsson, 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, doi:10.2140/agt.2021.21.801]

  (based on the PhD thesis)

• 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, doi:10.1017/fms.2020.11]

• Cary Malkiewich, Parametrized spectra, a low-tech approach [arXiv:1906.04773, user guide: pdf, pdf]

• Cary Malkiewich, A convenient category of parametrized spectra [arXiv:2305.15327]

Discussion in (∞,1)-category theory:

• Matthew Ando, Andrew Blumberg, David Gepner, Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map, Geom. Topol. 22 (2018) 3761-3825 (arXiv:1112.2203)

Discussion as a linear homotopy type theory:

• Urs Schreiber, Quantization via Linear homotopy types [arXiv:1402.7041]

  (intended categorical semantics)

• Mitchell Riley, Eric Finster, Daniel R. Licata, Synthetic Spectra via a Monadic and Comonadic Modality [arXiv:2102.04099]

  (inference rules for the classical modality $\natural$)

• Mitchell Riley, A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory, PhD Thesis (2022) [doi:10.14418/wes01.3.139, ir:3269, pdf]

  (including inference rules for the multiplicative conjunction by bringing in the required bunched logic)