nLab parameterized homotopy theory



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






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

Parameterized point-set topology

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


For B𝒞B \,\in\, \mathcal{C} an object in any category, write 𝒞 /B\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)BkwHauskTop. B \;\in\; kwHaus \xhookrightarrow{\;} kTop \,.

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

(X,p X)×(Y,p Y)(X× BY,p Xpr X=p Ypr Y)kTop /B (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:B 1B 2f \,\colon\, B_1 \xrightarrow{\;\;} B_2 a continuous map between such base spaces, notice the usal base change adjoint triple:



  1. f *f^\ast is pullback in kTopkTop along ff;

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

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

(3)f !((U,p U)×f *(X,p X))(f !(U,p U))×(X,p X). 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 ff is the terminal map to the point space, which we denote

p B:B*. p_B \,\colon\, B \xrightarrow{\;} \ast \,.

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


In this case

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

    (5)(p B *)X 0=B×X 0pr BB, (p_B^\ast) X_0 \;=\; B \times X_0 \xrightarrow{\; pr_B \;} B \,,
  2. (p B) !(p_B)_! gives the total space of a fibration:

    (p B) !(XB)=X (p_B)_! \big( X \to B\big) \;=\; X
  3. (p B) *(p_B)_* gives the space of sections of a fibration.

Eventually we consider pointed objects

(X,p X,σ X)(kTop /B) B/=kTop /B B/ (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 “bundlesXp XBX \xrightarrow{p_X} B (in the most general sense, without any condition on the bundle projection, except continuity) equipped with a fixed section σ X\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]


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

Xι XX˜ X \xhookrightarrow{\;\; \iota_X \;\;} \widetilde{X}

together with X˜\widetilde{X} itself.

(MaSi06, Def. 1.3.6)


(fiberwise mapping space)

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

a pair of k-spaces over BB (1) their fiberwise mapping space is the pullback (in kTopkTop):

(6)Map((X,p X),(A,p A)) Map(X,A˜) (pb) Map(X,p A˜) B b(x{b ifxX b otherwise) Map(X,B˜), \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 /BkTop_{/B}.

Here Map(X,A˜)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).


(on notation)
Contrary to most references, Def. is intentionally not using a subsript “ B{}_B” in the notation for the fiberwise mapping space: This is because “Map(X,A) BMap(X,A)_B” is also standard notation for Map(X,A)×Map(X,B){p B}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(,)Map(-,-) is always of the same type as its arguments, as befits an internal hom.


(fiberwise mapping space satisfies the exponential law)
With BB 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((X,p X),Map((Y,p Y),(A,p A)))kTop /B((X,p X)×(Y,p Y),(A,p A)) 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× BYX \times_B Y in kTopkTop).

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


(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(p B *X 0,p B *A 0)p B *Map(X 0,A 0). Map \big( p_B^\ast X_0 ,\, p_B^\ast A_0 \big) \;\simeq\; p_B^\ast Map\big(X_0,\, A_0\big) \,.


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

kTop /B((U,p U),Map(p B *X 0,p B *A 0)) kTop /B((U,p U)×(B×X 0,pr B),p B *A 0) kTop /B((U×X 0,p Upr U),p B *A 0) kTop(U×X 0,A 0) kTop(U,Map(X 0,A 0)) kTop((p B) !(U,p U),Map(X 0,A 0)) kTop /B((U,p U),p B *Map(X 0,A 0)) \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)(U, p_U), the claim follows by the Yoneda lemma (over the large category (kTop /B) op\big(kTop_{/B}\big)^{op}).



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

f *Map((X,p X),(A,p A))Map(f *(X,p X),f *(A,p A)). 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 *f^\ast is a closed functor with respect to fiberwise mapping spaces.


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

kTop /B((U,p U),f *Map((X,p X),(A,p A))) kTop /B(f !(U,p U),Map((X,p X),(A,p A))) kTop /B((f !(U,p U))×(X,p X),(A,p A)) kTop /B(f !((U,p U)×f *(X,p X)),(A,p A)) kTop /B((U,p U)×f *(X,p X),f *(A,p A)) kTop /B((U,p U),Map(f *(X,p X),f *(A,p A))) \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)(U,p_U), the claim follows by the Yoneda embedding (for the large category (kTop /B) op\big( kTop_{/B'}\big)^{op}).



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

Map((X,p x),(A,p A)) bMap(X b,A b). 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)

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}B = \{b\}.

Homotopy theory of the fiberwise mapping space


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

p X,p AhFibp Map((X,p X),(A,p A))hFib. 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.)


(fiberwise mapping space does not preserve weak Hausdorffness)
Even if XX and AA are weak Hausdorff spaces over the weak Hausdorff space BB (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).


Exponential law for parameterized topological spaces

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

And with an eye towards parameterized homotopy theory:

Parameterized (“fiberwise”) homotopy theory

On the homotopy theory of such parameterized topological spaces:

On model structures for parameterized spectra:

Discussion in (∞,1)-category theory:

Discussion as a linear homotopy type theory:

Last revised on May 25, 2023 at 09:40:08. See the history of this page for a list of all contributions to it.