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classical model structure on topological spaces

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

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topological homotopy theory

Model category theory

model category

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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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

The classical model structure on topological spaces or Quillen-Serre model structure Top QuillenTop_{Quillen} (Quillen 67, II.3) is a model category structure on the category Top of topological spaces (also on many convenient categories of topological spaces) which represents the standard homotopy theory of CW-complexes (topological homotopy theory), in that its homotopy category of a model category is the classical homotopy category on cell complexes/CW-complexes.

Its weak equivalences are the weak homotopy equivalences, its fibrations are the Serre fibrations and its cofibrations are the retracts of relative cell complexes.

The singular simplicial complex/geometric realization adjunction constitutes a Quillen equivalence between Top QuillenTop_{Quillen} and sSet QuillensSet_{Quillen}, the classical model structure on simplicial sets. This is sometimes called part of the statement of the homotopy hypothesis for Kan complexes. In the language of (∞,1)-category theory this means that Top QuillenTop_{Quillen} and sSet QuillensSet_{Quillen} both are presentations of the (∞,1)-category ∞Grpd of ∞-groupoids.

There are also other model structures on Top itself, see at model structure on topological spaces for more. This entry here focuses on just the classical model structure on topological spaces.

Background from point-set topology

This section recalls basic relevant concepts from topology (“point-set topology”) and highlights some basic facts that may serve to motivate the Quillen model structure below.

Homotopy

The fundamental concept of homotopy theory is that of homotopy. In the context of topological spaces this is about contiunous deformations of continuous functions parameterized by the standard topological interval:

Definition

Write

I[0,1] I \coloneqq [0,1] \hookrightarrow \mathbb{R}

for the standard topological interval, a compact connected topological subspace of the real line.

Equipped with the canonical inclusion of its two endpoints

**(δ 0,δ 1)I!* \ast \sqcup \ast \stackrel{(\delta_0,\delta_1)}{\longrightarrow} I \stackrel{\exists !}{\longrightarrow} \ast

this is the standard interval object in Top.

For XTopX \in Top, the product topological space X×IX\times I is called the standard cylinder object over XX. The endpoint inclusions of the interval make it factor the codiagonal on XX

X:XX((id,δ 0),(id,δ 1))X×IX. \nabla_X \;\colon\; X \sqcup X \stackrel{((id,\delta_0),(id,\delta_1))}{\longrightarrow} X \times I \longrightarrow X \,.
Definition

For f,g:XYf,g\colon X \longrightarrow Y two continuous functions between topological spaces X,YX,Y, then a left homotopy f Lgf \Rightarrow_L g is a continuous function

η:X×IY \eta \;\colon\; X \times I \longrightarrow Y

out of the product topological space of XX with the standard interval of def. 1, such that this fits into a commuting diagram of the form

X (id,δ 0) f X×I η Y (id,δ 1) g X. \array{ X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ X \times I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ X } \,.
Definition

A continuous function f:XYf \;\colon\; X \longrightarrow Y is called a homotopy equivalence if there exists a continuous function XY:gX \longleftarrow Y \;\colon\; g and left homotopies, def. 2

η 1:fg Lid Y \eta_1 \;\colon\; f\circ g \Rightarrow_L id_Y

and

η 2:gf Lid X. \eta_2 \;\colon\; g\circ f \Rightarrow_L id_X \,.

If here η 2\eta_2 is constant along II, ff is is said to exhibit XX as a deformation retract of YY.

Another key application of the concept of left homotopy is to the definition of homotopy groups.

Definition

For XX a topological space, then its set π 0(X)\pi_0(X) of connected components, also called the 0-th homotopy set, is the set of left homotopy-equivalence classes of points *X\ast \to X, def.2. By composition this extends to a functor

π 0:TopSet. \pi_0 \colon Top \longrightarrow Set \,.

For nn \in \mathbb{N}, n1n \geq 1 and for x:*Xx \colon \ast \to X any point, then the nnth homotopy group π n(X)\pi_n(X) of XX at xx is the group

  • whose underlying set is the set of left homotopy-equivalence classes of maps I nXI^n \longrightarrow X that take the boundary of I nI^n to xx and where the left homotopies η\eta are constrained to be constant on the boundary;

  • whose group product operation takes [α:I nX][\alpha \colon I^n \to X] and [β:I nX][\beta \colon I^n \to X] to [αβ][\alpha \cdot \beta] with

αβ:I nI nI n1I n(α,β)X, \alpha \cdot \beta \;\colon\; I^n \stackrel{\simeq}{\longrightarrow} I^n \underset{I^{n-1}}{\sqcup} I^n \stackrel{(\alpha,\beta)}{\longrightarrow} X \,,

where the first morphism is any homeomorphism from the unit nn-cube to the nn-cube with one side twice the unit length (see also at positive dimension spheres are H-cogroup objects).

By composition, this construction extends to a functor

π 1:Top */Grp 1 \pi_{\bullet \geq 1} \;\colon\; Top^{\ast/} \longrightarrow Grp^{\mathbb{N}_{\geq 1}}

from pointed topological spaces to graded group.

Definition

A continuous function f:XYf \colon X \longrightarrow Y is called a weak homotopy equivalence if its image under all the homotopy group functors of def. 4 is an isomorphism, hence if

π 0(f):π 0(X)π 0(X) \pi_0(f) \;\colon\; \pi_0(X) \stackrel{\simeq}{\longrightarrow} \pi_0(X)

and for all xXx \in X and all n1n \geq 1

π n(f):π n(X,x)π n(Y,f(y)). \pi_n(f) \;\colon\; \pi_n(X,x) \stackrel{\simeq}{\longrightarrow} \pi_n(Y,f(y)) \,.
Proposition

Every homotopy equivalence, def. 3, is a weak homotopy equivalence, def. 5.

In particular a deformation retraction, def. 3, is a weak homotopy equivalence.

Proof

First observe that for all XX\in Top the inclusion maps

X(id,δ 0)X×I X \overset{(id,\delta_0)}{\longrightarrow} X \times I

into the standard cylinder object, def. 1, are weak homotopy equivalences: by postcomposition with the contracting homotopy of the interval from example \ref{StandardContractionOfStandardInterval} all homotopy groups of X×IX \times I have representatives that factor through this inclusion.

Then given a general homotopy equivalence, apply the homotopy groups functor to the corresponding homotopy diagrams (where for the moment we notationally suppress the choice of basepoint for readability) to get two commuting diagrams

π (X) π (id,δ 0) π (f)π (g) π (X×I) π (η) π (Y) π (id,δ 1) π (id) π (X),π (Y) π (id,δ 0) π (g)π (f) π (Y×I) π (η) π (X) π (id,δ 1) π (id) π (Y). \array{ \pi_\bullet(X) \\ {}^{\mathllap{\pi_\bullet(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{\pi_\bullet(f)\circ \pi_\bullet(g)}} \\ \pi_\bullet(X \times I) &\stackrel{\pi_\bullet(\eta)}{\longrightarrow}& \pi_\bullet(Y) \\ {}^{\mathllap{\pi_\bullet(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{\pi_\bullet(id)}} \\ \pi_\bullet(X) } \;\;\;\;\;\;\; \,, \;\;\;\;\;\;\; \array{ \pi_\bullet(Y) \\ {}^{\mathllap{\pi_\bullet(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{\pi_\bullet(g)\circ \pi_\bullet(f)}} \\ \pi_\bullet(Y \times I) &\stackrel{\pi_\bullet(\eta)}{\longrightarrow}& \pi_\bullet(X) \\ {}^{\mathllap{\pi_\bullet(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{\pi_\bullet(id)}} \\ \pi_\bullet(Y) } \,.

By the previous observation, the vertical morphisms here are isomorphisms, and hence these diagrams exhibit π (f)\pi_\bullet(f) as the inverse of π (g)\pi_\bullet(g), hence both as isomorphisms.

Example

For XTopX\in Top, the projection X×IXX\times I \longrightarrow X from the cylinder object of XX, def. 1, is a weak homotopy equivalence, def. 5.

This means that the factorization

X:XXX×I \nabla_X \;\colon\; X \sqcup X \stackrel{}{\longrightarrow} X\times I \stackrel{\simeq}{\longrightarrow}

of the codiagonal X\nabla_X in def. 1, which in general is far from being a monomorphism, may be thought of as factoring it through a monomorphism after replacing XX, up to weak homotopy equivalence, by X×IX\times I.

In fact XXX×IX \sqcup X \to X \times I has better properties than the generic monomorphism has, in particular better homotopy invariant properties: it has the left lifting property against all Serre fibrations EpB E \stackrel{p}{\longrightarrow} B (def. 14) that are also weak homotopy equivalences.

For YY a topological space, the set Hom Top(I,Y)Hom_{Top}(I,Y) of continuous functions from the standard interval II, def. 1, to YY is the set of continuous paths in XX. Every such path may be though of as a left homotopy between its endpoints. Hence a function XHom Top(I,Y)X \longrightarrow Hom_{Top}(I,Y) is an XX-parameterized collection of such paths. In order for that to also give a concept of homotopy, we need to impose a continuity condition on how the paths may vary, hence we need to put a suitable topology on Hom Top(I,X)Hom_{Top}(I,X). This is the compact-open topology:

Definition

For XX a topological space and YY a locally compact Hausdorff topological space, the mapping space

X YTop X^Y \in Top

is the topological space

Accordingly this is called the compact-open topology on the set of functions.

The construction extends to a functor

() ():Top lcH op×TopTop. (-)^{(-)} \;\colon\; Top_{lcH}^{op} \times Top \longrightarrow Top \,.
Proposition

For XX a topological space and YY a locally compact Hausdorff topological space, the topological mapping space X YX^Y from def. 6 is an exponential object: there is a natural bijection

Hom Top(Z×Y,X)Hom Top(Z,X Y) Hom_{Top}(Z \times Y, X) \simeq Hom_{Top}(Z, X^Y)

between continuous functions out of any product topological space of YY with any ZTopZ \in Top and continuous functions from ZZ into the mapping space.

Remark

Proposition 2 fails if YY is not locally compact and Hausdorff. Therefore the plain category Top of all topological spaces is not a Cartesian closed category.

This is no problem for the construction of the homotopy theory of topological spaces as such, but it becomes a technical nuisance when comparing it for instance to the simplicial homotopy theory via the singular nerve and realization adjunction, since it implies that geometric realization of simplicial sets does not necessarily preserve finite limits.

On the other hand, without changing any of the following discussion one may just pass to a more convenient category of topological spaces such as notably the full subcategory of compactly generated topological spaces Top cgTopTop_{cg} \hookrightarrow Top which is Cartesian closed.

Definition

For XX a topological space, its path space object is the topological mapping space X IX^I, def. 2, out of the standard interval II of def. 1.

Example

The endpoint inclusion into the standard interval, def. 1, makes the path space X IX^I of def. 7 factor the diagonal on XX through the inclusion of constant paths and the endpoint evaluation of paths:

Δ X:XX I*X IX **IX×X. \Delta_X \;\colon\; X \stackrel{X^{I \to \ast}}{\longrightarrow} X^I \stackrel{X^{\ast \sqcup \ast \to I}}{\longrightarrow} X \times X \,.

Here

  1. X I*X^{I \to \ast} is a weak homotopy equivalence;

  2. X **IX^{\ast \sqcup \ast \to I} is a Serre fibration.

So while in general the diagonal Δ X\Delta_X is far from being an epimorphism or even just a Serre fibration, the factorization through the path space object may be thought of as replacing XX, up to weak homotopy equivalence, by its path space, such as to turn its diagonal into a Serre fibration after all.

Definition

For f,g:XYf,g\colon X \longrightarrow Y two continuous functions between topological spaces X,YX,Y, then a right homotopy f Rgf \Rightarrow_R g is a continuous function

η:XY I \eta \;\colon\; X \longrightarrow Y^I

into the path space object of XX, def. 7, such that this fits into a commuting diagram of the form

Y f X δ 0 X η Y I g Y δ 1 Y. \array{ && Y \\ & {}^{\mathllap{f}}\nearrow & \uparrow^{\mathrlap{X^{\delta_0}}} \\ X &\stackrel{\eta}{\longrightarrow}& Y^I \\ & {}_{\mathllap{g}}\searrow & \downarrow^{\mathrlap{Y^{\delta_1}}} \\ && Y } \,.

Cell complexes

Definition

For nn \in \mathbb{N} write

  • D n{x n||x|1} nD^n \coloneqq \{ \vec x\in \mathbb{R}^n | {\vert \vec x \vert \leq 1}\} \hookrightarrow \mathbb{R}^n for the standard topological n-disk;

  • S n1=D n{x n||x|=1} nS^{n-1} = \partial D^n \coloneqq \{ \vec x\in \mathbb{R}^n | {\vert \vec x \vert = 1}\} \hookrightarrow \mathbb{R}^n for the standard topological n-sphere;

Write

I Top{S n1ι nD n} nMor(Top) I_{Top} \coloneqq \{S^{n-1} \stackrel{\iota_n}{\hookrightarrow} D^{n}\}_{n \in \mathbb{N}} \subset Mor(Top)

for the set of canonical boundary inclusion maps. This going to be called the set of standard topological generating cofibrations.

Notice that S 1=S^{-1} = \emptyset and that S 0=**S^0 = \ast \sqcup \ast.

Definition

For XTopX \in Top and for nn \in \mathbb{N}, an nn-cell attachment to XX is the pushout of a generatic cofibration, def. 9

S n1 ϕ X ι n D n X ϕS n1D n \array{ S^{n-1} &\stackrel{\phi}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow && \downarrow \\ D^n &\longrightarrow& X {}_\phi\underset{S^{n-1}}{\sqcup} D^n }

along some continuous function ϕ\phi.

A continuous function f:XYf \colon X \longrightarrow Y is called a topological relative cell complex if it is exhibited by a (possibly infinite) sequence of cell attachments to XX, hence if it is a transfinite composition of pushouts

iS n i1 X k1 iι n i (po) iD n i X k \array{ \underset{i}{\coprod} S^{n_i - 1} &\longrightarrow& X_{k-1} \\ {}^{\mathllap{\underset{i}{\coprod}\iota_{n_i}}}\downarrow &(po)& \downarrow \\ \underset{i}{\coprod} D^{n_i} &\longrightarrow& X_{k} }

of coproducts of generating cofibrations.

A topological space is a cell complex if X\emptyset \longrightarrow X is a relative cell complex.

A relative cell complex is called a finite relative cell complex if it is obtained from a finite number of cell attachments.

A (relative) cell complex is called a (relative) CW-complex if in the above transfinite composition is countable

X=X 0 X 1 X 2 f Y=limX \array{ X = X_0 &\longrightarrow& X_1 &\longrightarrow& X_2 &\longrightarrow& \cdots \\ & {}_{\mathllap{f}}\searrow & \downarrow & \swarrow && \cdots \\ && Y = \underset{\longrightarrow}{\lim} X_\bullet }

and if X kX_k is obtained from X k1X_{k-1} by attaching cells precisely only of dimension kk.

Remark

Strictly speaking a relative cell complex, def. 10, is a function f:XYf\colon X \to Y, together with its cell structure, hence together with the information of the pushout diagrams and the transfinite composition of the pushout maps that exhibit it.

In many applications, however, all that matters is that there is some (relative) cell decomosition, and then one tends to speak loosely and mean by a (relative) cell complex only a (relative) topological space that admits some cell decomposition.

Definition

For CMor(Top)C \subset Mor(Top) any class of morphisms, the concept of relative CC-cell complexes is defined as in def. 10, with the boundary inclusions ι nI Top\iota_n \in I_{Top} replaced by the maps in CC:

a relative CC-cell complex is a transfinite composition of pushouts of coproducts of the maps in CMor(Top)C \hookrightarrow Mor(Top).

Given a relative CC-cell complex ι:XY\iota \colon X \to Y, def. 11, it is typically interesting to study the extension problem along ff, i.e. to ask which topological spaces EE are such that every continuous function f:XEf\colon X \longrightarrow E has an extension f˜\tilde f along ι\iota

X f E ι f˜ Y. \array{ X &\stackrel{f}{\longrightarrow}& E \\ {}^{\mathllap{\iota}}\downarrow & \nearrow_{\mathrlap{\exists \tilde f}} \\ Y } \,.

If so, then this means that EE is sufficiently “spread out” with respect to the maps in CC. More generally one considers this extension problem fiberwise, i.e. with both EE and YY (hence also XX) equipped with a map to some base space BB.

Definition

Given a category 𝒞\mathcal{C} and a sub-class CMor(𝒞)C \subset Mor(\mathcal{C}) of its morphisms, then a morphism p:EBp \colon E \longrightarrow B in 𝒞\mathcal{C} is said to have the right lifting property against the morphisms in CC if every commuting diagram in 𝒞\mathcal{C} of the form

X E c p Y B, \array{ X &\longrightarrow& E \\ {}^{\mathllap{c}}\downarrow && \downarrow^{\mathrlap{p}} \\ Y &\longrightarrow& B } \,,

with cCc \in C, has a lift ww, in that it may be completed to a commuting diagram of the form

X E c w p Y B. \array{ X &\longrightarrow& E \\ {}^{\mathllap{c}}\downarrow &{}^{\mathllap{w}}\nearrow& \downarrow^{\mathrlap{p}} \\ Y &\longrightarrow& B } \,.

We will also say that ff is a CC-injective morphism if it satisfies the right lifting property against CC.

Fibrations

Definition

Write

J Top{D n(id,δ 0)D n×I} nMor(Top) J_{Top} \coloneqq \{D^n \stackrel{(id,\delta_0)}{\hookrightarrow} D^n \times I\}_{n \in \mathbb{N}} \subset Mor(Top)

for the set of inclusions of the topological n-disks, def. 9, into their cylinder objects, def. 1, along (for definiteness) the left endpoint inclusion.

These inclusions are similar to the standard topological generating cofibrations of def. 9, but in contrast to these they are “acyclic” (meaning: trivial on homotopy classes of maps from “cycles” given by n-spheres) in that they are weak homotopy equivalences (example 1).

Accordingly, JJ is to be called the set of standard topological generating acyclic cofibrations.

Lemma

The maps D nD n×ID^n \hookrightarrow D^n \times I in def. 13 are finite relative cell complexes, def. 10.

Proof

There is a homeomorphism

D n = D n (id,δ 0) D n×I D n+1 \array{ D^n & = & D^n \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow \\ D^n \times I &\simeq& D^{n+1} }

such that the map on the right is the inclusion of one hemisphere into the boundary n-sphere of D n+1D^{n+1}. This inclusion is the result of attaching two cells:

S n1 ι n D n ι n (po) D n S n = S n id S n ι n+1 (po) D n+1 id D n+1. \array{ S^{n-1} &\overset{\iota_n}{\longrightarrow}& D^n \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^{n} \\ && \downarrow^{=} \\ S^n &\overset{id}{\longrightarrow}& S^n \\ {}^{\mathllap{\iota_{n+1}}}\downarrow &(po)& \downarrow \\ D^{n+1} &\underset{id}{\longrightarrow}& D^{n+1} } \,.
Definition

A continuous function p:EBp \colon E \longrightarrow B is called a Serre fibration if it is a J TopJ_{Top}-injective morphisms; i.e. if it has the right lifting property, def. 12, against all topological generating acylic cofibrations, def. 13; hence if for every commuting diagram of continuous functions of the form

D n E (id,δ 0) p D n×I B, \array{ D^n &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow^{\mathrlap{p}} \\ D^n\times I &\longrightarrow& B } \,,

has a lift ww, in that it may be completed to a commuting diagram of the form

D n E (id,δ 0) w p D n×I B. \array{ D^n &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow &{}^{\mathllap{w}}\nearrow& \downarrow^{\mathrlap{p}} \\ D^n\times I &\longrightarrow& B } \,.
Remark

Def. 14 says, in view of the definition of left homotopy, that a Serre fibration pp is a map with the property that given a left homotopy, def. 2, between two functions into its codomain, and given a lift of one the two functions through pp, then also the homotopy between the two lifts, in particular the second function lifts, too, and both lifts are related by left homotopy.

Therefore the condition on a Serre fibration is also called the homotopy lifting property for maps whose domain is an n-disk.

More generally one may ask functions pp to have such homotopy lifting property for functions with arbitrary domain. These are called Hurewicz fibrations.

Remark

The precise shape of D nD^n and D n×ID^n \times I in def. 14 turns out not to actually matter much for the nature of Serre fibrations. We will eventually find below (prop. 8) that what actually matters here is only that the inclusions D nD n×ID^n \hookrightarrow D^n \times I are relative cell complexes and weak homotopy equivalences and that all of these may be generated from them in a suitable way.

But for simple special cases this is readily seen directly, too. Notably it is trivial, but nevertheless important in applications, that we could replace the n-disks in def. 14 with any homeomorphic topological space. A choice that becomes important in the comparison to the classical model structure on simplicial sets is to instead take the topological n-simplices Δ n\Delta^n. Hence a Serre fibration is equivalently characterized as having lifts in all diagrams of the form

Δ n E (id,δ 0) p Δ n×I B. \array{ \Delta^n &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow^{\mathrlap{p}} \\ \Delta^n\times I &\longrightarrow& B } \,.

Other deformations of the nn-disks are useful in computations, too. For instance there is a homeomorphism from the nn-disk to its “cylinder with interior and end removed”, formally:

(D n×{0})(D n×I) D n D n×I×I D n×I \array{ (D^n \times \{0\})\cup (\partial D^n \times I) &\simeq& D^n \\ \downarrow && \downarrow \\ D^n \times I \times I &\simeq& D^n\times I }

and hence ff is a Serre fibration equivalently also if it admits lifts in all diagrams of the form

(D n×{0})(D n×I) E (id,δ 0) p Δ n×I B. \array{ (D^n \times \{0\})\cup (\partial D^n \times I) &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow^{\mathrlap{p}} \\ \Delta^n\times I &\longrightarrow& B } \,.
Proposition

Let f:XYf\colon X \longrightarrow Y be a Serre fibration, def. 14, let y:*Yy \colon \ast \to Y be any point and write

F yιXfY F_y \overset{\iota}{\hookrightarrow} X \overset{f}{\longrightarrow} Y

for the fiber inclusion over that point. Then for every choice x:*Xx \colon \ast \to X of lift of the point yy through ff, the induced sequence of homotopy groups

π (F y,x)ι *π (X,x)f *π (Y) \pi_{\bullet}(F_y, x) \overset{\iota_\ast}{\longrightarrow} \pi_\bullet(X, x) \overset{f_\ast}{\longrightarrow} \pi_\bullet(Y)

is exact, in that the kernel of f *f_\ast is canonically identified with the image of ι *\iota_\ast:

ker(f *)im(ι *). ker(f_\ast) \simeq im(\iota_\ast) \,.
Proof

It is clear that the image of ι *\iota_\ast is in the kernel of f *f_\ast (every sphere in F yXF_y\hookrightarrow X becomes constant on yy, hence contractible, when sent forward to YY).

For the converse, let [α]π (X,x)[\alpha]\in \pi_{\bullet}(X,x) be represented by some α:S n1X\alpha \colon S^{n-1} \to X. Assume that [α][\alpha] is in the kernel of f *f_\ast. This means equivalently that α\alpha fits into a commuting diagram of the form

S n1 α X f D n κ Y, \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ D^n &\overset{\kappa}{\longrightarrow}& Y } \,,

where κ\kappa is the contracting homotopy witnessing that f *[α]=0f_\ast[\alpha] = 0.

Now since xx is a lift of yy, there exists a left homotopy

η:κconst y \eta \;\colon\; \kappa \Rightarrow const_y

as follows:

S n1 α X ι n f D n κ Y (id,1) id D n (id,0) D n×I η Y * y Y \array{ && S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ && {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f}} \\ && D^n &\overset{\kappa}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{(id,1)}} && \downarrow^{\mathrlap{id}} \\ D^n &\overset{(id,0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y \\ \downarrow && && \downarrow \\ \ast && \overset{y}{\longrightarrow} && Y }

(for instance: regard D nD^n as embedded in n\mathbb{R}^n such that 0 n0 \in \mathbb{R}^n is identified with the basepoint on the boundary of D nD^n and set η(v,t)κ(tv)\eta(\vec v,t) \coloneqq \kappa(t \vec v)).

The pasting of the top two squares that have appeared this way is equivalent to the following commuting square

S n1 α X (id,1) f S n1×I (ι n,id) D n×I η Y. \array{ S^{n-1} &\longrightarrow& &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{(id,1)}}\downarrow && && \downarrow^{\mathrlap{f}} \\ S^{n-1} \times I &\overset{(\iota_n, id)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y } \,.

Because S n1S n1×IS^{n-1} \to S^{n-1}\times I is a J TopJ_{Top}-relative cell complex and ff is a Serre fibraiton? (see there), this has a lift

η˜:S n1×IX. \tilde \eta \;\colon\; S^{n-1} \times I \longrightarrow X \,.

Notice that η˜\tilde \eta is a basepoint preserving left homotopy from α=η˜| 1\alpha = \tilde \eta|_1 to some αη˜| 0\alpha' \coloneqq \tilde \eta|_0. Being homotopic, they represent the same element of π n1(X,x)\pi_{n-1}(X,x):

[α]=[α]. [\alpha'] = [\alpha] \,.

But the new representative α\alpha' has the special property that its image in YY is not just trivializable, but trivialized: combining η˜\tilde \eta with the previous diagram shows that it sits in the following commuting diagram

α: S n1 (id,0) S n1×I η˜ X ι n (ι n,id) f D n (id,0) D n×I η Y * y Y. \array{ \alpha' \colon & S^{n-1} &\overset{(id,0)}{\longrightarrow}& S^{n-1}\times I &\overset{\tilde \eta}{\longrightarrow}& X \\ & \downarrow^{\iota_n} && \downarrow^{\mathrlap{(\iota_n,id)}} && \downarrow^{\mathrlap{f}} \\ & D^n &\overset{(id,0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y \\ & \downarrow && && \downarrow \\ & \ast && \overset{y}{\longrightarrow} && Y } \,.

The commutativity of the outer square says that f *αf_\ast \alpha' is constant, hence that α\alpha' is entirely contained in the fiber F yF_y. Said more abstractly, the universal property of fibers gives that α\alpha' factors through F yιXF_y\overset{\iota}{\hookrightarrow} X, hence that [α]=[α][\alpha'] = [\alpha] is in the image of ι *\iota_\ast.

Background from model category theory

This section recalls some standard arguments in model category theory.

Remark

As usual, by a retract of a morphism XfYX \stackrel{f}{\longrightarrow} Y in some category 𝒞\mathcal{C} we mean a retract in the arrow category 𝒞 Δ[1]\mathcal{C}^{\Delta[1]}, hence a morphism AgBA \stackrel{g}{\longrightarrow} B such that in 𝒞 Δ[1]\mathcal{C}^{\Delta[1]} there is a factorization of the identity on gg through ff

id g:gfg. id_g \;\colon\; g \longrightarrow f \longrightarrow g \,.

This means equivalently that in 𝒞\mathcal{C} there is a commuting diagram of the form

id A: A X A g f g id A: A Y A. \array{ id_A \colon & A &\longrightarrow& X &\longrightarrow& A \\ & \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} \\ id_A \colon & A &\longrightarrow& Y &\longrightarrow& A } \,.

Lifting

Definition

Let 𝒞\mathcal{C} be any category. Given a diagram in 𝒞\mathcal{C} of the form

X f Y p B \array{ X &\stackrel{f}{\longrightarrow}& Y \\ {}^{\mathllap{p}}\downarrow \\ B }

then an extension of the morphism ff along the morphism pp is a completion to a commuting diagram of the form

X f Y p f˜ B. \array{ X &\stackrel{f}{\longrightarrow}& Y \\ {}^{\mathllap{p}}\downarrow & \nearrow_{\mathrlap{\tilde f}} \\ B } \,.

Dually, given a diagram of the form

A p X f Y \array{ && A \\ && \downarrow^{\mathrlap{p}} \\ X &\stackrel{f}{\longrightarrow}& Y }

then a lift of ff through pp is a completion to a commuting diagram of the form

A f˜ p X f Y. \array{ && A \\ &{}^{\mathllap{\tilde f}}\nearrow& \downarrow^{\mathrlap{p}} \\ X &\stackrel{f}{\longrightarrow}& Y } \,.

Combining these cases: given a square commuting diagram

X 1 f 1 Y 1 p l p r X 2 f 1 Y 2 \array{ X_1 &\stackrel{f_1}{\longrightarrow}& Y_1 \\ {}^{\mathllap{p_l}}\downarrow && \downarrow^{\mathrlap{p_r}} \\ X_2 &\stackrel{f_1}{\longrightarrow}& Y_2 }

then a lifting in the diagram is a completion to a commuting diagram of the form

X 1 f 1 Y 1 p l p r X 2 f 1 Y 2. \array{ X_1 &\stackrel{f_1}{\longrightarrow}& Y_1 \\ {}^{\mathllap{p_l}}\downarrow &\nearrow& \downarrow^{\mathrlap{p_r}} \\ X_2 &\stackrel{f_1}{\longrightarrow}& Y_2 } \,.

Given a sub-class of morphhisms CMor(𝒞)C \subset Mor(\mathcal{C}), then a morphism p rp_r as above is said to have the right lifting property against CC if in all square diagrams with p rp_r on the right and any p lCp_l \in C on the left a lift exists. Dually, a fixed p lp_l is said to have the left lifting property against CC if in all square diagrams with p lp_l on the left and any p rCp_r \in C on the left a lift exists.

Lemma

Let 𝒞\mathcal{C} be a category with all small colimits, and let CMor(𝒞)C\subset Mor(\mathcal{C}) be a sub-class of its morphisms.

Then every CC-injective morphism, def. 12, has the right lifting property, def. 15, against all CC-relative cell complexes, def. 11 and their retracts, remark 5.

Proof

This is an immediate consequence of the general fact (here) that classes of morphisms characterized by a left lifting property are closed under the operations of coproducts, pushouts, retracts and transfinite composition.

Lemma

(retract argument)

If in a composite morphism

g:ip g \;\colon\; \stackrel{i}{\longrightarrow} \stackrel{p}{\longrightarrow}

the factor pp has the right lifting property, def. 12, against the total morphism gg, then gg is a retract (rem. 5) of ii.

The small object argument

Given a set CMor(𝒞)C \subset Mor(\mathcal{C}) of morphisms in some category 𝒞\mathcal{C}, a natural question is how to factor any given morphism f:XYf\colon X \longrightarrow Y through a relative CC-cell complex, def. 11, followed by a CC-injective morphism, def. 12

f:XCcellX^CfibY. f \;\colon\; X \stackrel{\in C cell}{\longrightarrow} \hat X \stackrel{\in C fib}{\longrightarrow} Y \,.

A first approximation to such a factorization turns out to be given simply by forming X^=X 1\hat X = X_1 by attaching all possible CC-cells to XX. Namely let

(C/f){dom(c) X cC f cod(c) Y} (C/f) \coloneqq \left\{ \array{ dom(c) &\stackrel{}{\longrightarrow}& X \\ {}^{\mathllap{c\in C}}\downarrow && \downarrow^{\mathrlap{f}} \\ cod(c) &\longrightarrow& Y } \right\}

be the set of all ways to find a CC-cell attachment in ff, and consider the pushout X^\hat X of the coproduct of morphisms in CC over all these:

c(C/f)dom(c) X c(C/f)c (po) c(C/f)cod(c) X 1. \array{ \underset{c\in(C/f)}{\coprod} dom(c) &\longrightarrow& X \\ {}^{\mathllap{\underset{c\in(C/f)}{\coprod} c}}\downarrow &(po)& \downarrow^{\mathrlap{}} \\ \underset{c\in(C/f)}{\coprod} cod(c) &\longrightarrow& X_1 } \,.

This gets already close to producing the intended factorization:

First of all the resulting map XX 1X \to X_1 is a CC-relative cell complex, by construction.

Second, by the fact that the coproduct is over all commuting squres to ff, the morphism ff itself makes a commuting diagram

c(C/f)dom(c) X c(C/f)c f c(C/f)cod(c) Y \array{ \underset{c\in(C/f)}{\coprod} dom(c) &\longrightarrow& X \\ {}^{\mathllap{\underset{c\in(C/f)}{\coprod} c}}\downarrow && \downarrow^{\mathrlap{f}} \\ \underset{c\in(C/f)}{\coprod} cod(c) &\longrightarrow& Y }

and hence the universal property of the colimit means that ff is indeed factored through that CC-cell complex X 1X_1; we may suggestively arrange that factorizing diagram like so:

c(C/f)dom(c) X id c(C/f)dom(c) X 1 c(C/f)c c(C/f)cod(c) Y. \array{ \underset{c\in(C/f)}{\coprod} dom(c) &\longrightarrow& X \\ {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{}} \\ \underset{c\in(C/f)}{\coprod} dom(c) && X_1 \\ {}^{\mathllap{\underset{c\in(C/f)}{\coprod} c}}\downarrow &\nearrow& \downarrow \\ \underset{c\in(C/f)}{\coprod} cod(c) &\longrightarrow& Y } \,.

This shows that, finally, the colimiting co-cone map – the one that now appears diagonally – almost exhibits the desired right lifting of X 1YX_1 \to Y against the cCc\in C. The failure of that to hold on the nose is only the fact that a horizontal map in the middle of the above diagram is missing: the diagonal map obtained above lifts not all commuting diagrams of cCc\in C into ff, but only those where the top morphism dom(c)X 1dom(c) \to X_1 factors through XX 1X \to X_1.

The idea of the small object argument now is to fix this only remaining problem by iterating the construction: next factor X 1YX_1 \to Y in the same way into

X 1X 2Y X_1 \longrightarrow X_2 \longrightarrow Y

and so forth. Since relative CC-cell complexes are closed under composition, at stage nn the resulting XX nX \longrightarrow X_n is still a CC-cell complex, getting bigger and bigger. But accordingly, the failure of the accompanying X nYX_n \longrightarrow Y to be a CC-injective morphism becomes smaller and smaller, for it now lifts against all diagrams where dom(c)X ndom(c) \longrightarrow X_n factors through X n1X nX_{n-1}\longrightarrow X_n, which intuitively is less and less of a condition as the X n1X_{n-1} grow larger and larger.

The concept of small object is just what makes this intuition precise and finishes the small object argument. For the present purpose we just need the following simple version:

Definition

For 𝒞\mathcal{C} a category and CMor(𝒞)C \subset Mor(\mathcal{C}) a sub-class of its morphisms, say that these have small domains if for every cCc\in C and for every CC-relative cell complex f:XX^f\colon X \longrightarrow \hat X every morphism dom(c)X^dom(c)\longrightarrow \hat X factors through a finite relative subcomplex.

Proposition

(small object argument)

Let 𝒞\mathcal{C} be a locally small category with all small colimits. If a set CMor(𝒞)C\subset Mor(\mathcal{C}) of morphisms has all small domains in the sense of def. 16, then every morphism f:Xf\colon X\longrightarrow in 𝒞\mathcal{C} factors through a CC-relative cell complex, def. 11, followed by a CC-injective morphism, def. 12

f:XCcellX^CfibY. f \;\colon\; X \stackrel{\in C cell}{\longrightarrow} \hat X \stackrel{\in C fib}{\longrightarrow} Y \,.

The classical model structure Top QuillenTop_{Quillen}

Definition

Say that a continuous function, hence a morphism in Top is

and as usual:

  • an acyclic cofibration if it is a cofibration and a weak equivalence;

  • an acyclic fibration if it is a fibration and a weak equivalence.

Technical lemmas

The proof (below) that def. 17 defines a model category structure involves two technical lemmas which concern the special nature of topological spaces (“point-set topology”). With these two lemmas in hand, the rest of the proof is a routine argument in model category theory.

Lemma

Assuming the axiom of choice and the law of excluded middle, every compact subspace of a topological cell complex, def. 10, intersects the interior of a finite number of cells.

(e.g. Hirschhorn 15, section 3.1)

Proof

So let YY be a topological cell complex and CYC \hookrightarrow Y a compact subspace. Define a subset

PY P \subset Y

by choosing one point in the interior of the intersection with CC of each cell of YY that intersects CC.

It is now sufficient to show that PP has no accumulation point. Because, by the compactness of XX, every non-finite subset of CC does have an accumulation point, and hence the lack of such shows that PP is a finite set and hence that CC intersects the interior of finitely many cells of YY.

To that end, let cCc\in C be any point. If cc is a 0-cell in YY, write U c{c}U_c \coloneqq \{c\}. Otherwise, write e ce_c for the unique cell of YY that contains cc in its interior. By construction, there is exactly one point of PP in the interior of e ce_c. Hence there is an open neighbourhood cU ce cc \in U_c \subset e_c containing no further points of PP beyond possibly cc itself, if cc happens to be that single point of PP in e ce_c.

It is now sufficient to show that U cU_c may be enlarged to an open subset U˜ c\tilde U_c of YY containing no point of PP, except for possibly cc itself, for that means that cc is not an accumulation point of PP.

To that end, let α c\alpha_c be the ordinal that labels the stage Y α cY_{\alpha_c} of the transfinite composition in the cell complex-presentation of YY at which the cell e ce_c above appears. Let γ\gamma be the ordinal of the full cell complex. Then define the set

T{(β,U)|α cβγ,UopenY β,UY α=U c,UP{,{c}}}, T \coloneqq \left\{ \; (\beta, U) \;|\; \alpha_c \leq \beta \leq \gamma \;\,,\; U \underset{open}{\subset} Y_\beta \;\,,\; U \cap Y_\alpha = U_c \;\,,\; U \cap P \in \{ \emptyset, \{c\} \} \; \right\} \,,

and regard this as a partially ordered set by declaring a partial ordering via

(β 1,U 1)<(β 2,U 2)β 1<β 2,U 2Y β 1=U 1. (\beta_1, U_1) \lt (\beta_2, U_2) \;\;\;\; \Leftrightarrow \;\;\;\; \beta_1 \lt \beta_2 \;\,,\; U_2 \cap Y_{\beta_1} = U_1 \,.

This is set up such that every element (β,U)(\beta, U) of TT with β\beta the maximum value β=γ\beta = \gamma is an extension U˜ c\tilde U_c that we are after.

Observe then that for (β s,U s) sS(\beta_s, U_s)_{s\in S} a chain in (T,<)(T,\lt) (a subset on which the relation <\lt restricts to a total order), it has an upper bound in TT given by the union ( sβ s, sU s)({\cup}_s \beta_s ,\cup_s U_s). Therefore Zorn's lemma applies, saying that (T,<)(T,\lt) contains a maximal element (β max,U max)(\beta_{max}, U_{max}).

Hence it is now sufficient to show that β max=γ\beta_{max} = \gamma. We argue this by showing that assuming β max<γ\beta_{\max}\lt \gamma leads to a contradiction.

So assume β max<γ\beta_{max}\lt \gamma. Then to construct an element of TT that is larger than (β max,U max)(\beta_{max},U_{max}), consider for each cell dd at stage Y β max+1Y_{\beta_{max}+1} its attaching map h d:S n1Y β maxh_d \colon S^{n-1} \to Y_{\beta_{max}} and the corresponding preimage open set h d 1(U max)S n1h_d^{-1}(U_{max})\subset S^{n-1}. Enlarging all these preimages to open subsets of D nD^n (such that their image back in X β max+1X_{\beta_{max}+1} does not contain cc), then (β max,U max)<(β max+1, dU d)(\beta_{max}, U_{max}) \lt (\beta_{max}+1, \cup_d U_d ). This is a contradiction. Hence β max=γ\beta_{max} = \gamma, and we are done.

Lemma

Every J TopJ_{Top}-relative cell complex (def. 13, def. 11) is a weak homotopy equivalence, def. 5.

Proof

Let XX^X \longrightarrow \hat X be a J TopJ_{Top}-relative cell complex.

Notice that with the elements D nD n×ID^n \hookrightarrow D^n \times I of J TopJ_{Top} themselves, also each stage X αX α+1X_{\alpha} \to X_{\alpha+1} in the transfinite composition defining X^\hat X is a homotopy equivalence, hence, by prop. 1, a weak homotopy equivalence.

This means that all morphisms in the following diagram (notationally suppressing basepoints and showing only the finite stages)

π n(X) π n(X 1) π n(X 2) π n(X 3) lim απ n(X α) \array{ \pi_n(X) &\overset{\simeq}{\longrightarrow}& \pi_n(X_1) &\overset{\simeq}{\longrightarrow}& \pi_n(X_2) &\overset{\simeq}{\longrightarrow}& \pi_n(X_3) &\overset{\simeq}{\longrightarrow}& \cdots \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\simeq}} & \swarrow_{\mathrlap{\simeq}} & \cdots \\ && \underset{\longleftarrow}{\lim}_\alpha \pi_n(X_\alpha) }

are isomorphisms.

Moreover, lemma 4 gives that every representative and every null homotopy of elements in π n(X^)\pi_n(\hat X) already exists at some finite stage X kX_k. This means that also the universally induced morphism

lim απ n(X α)π n(X^) \underset{\longleftarrow}{\lim}_\alpha \pi_n(X_\alpha) \overset{\simeq}{\longrightarrow} \pi_n(\hat X)

is an isomorphism. Hence the composite π n(X)π n(X^)\pi_n(X) \overset{\simeq}{\longrightarrow} \pi_n(\hat X) is an isomorphism.

Lemma

The continuous functions with the right lifting property, def. 12 against the set I Top={S n1D n}I_{Top} = \{S^{n-1}\hookrightarrow D^n\} of topological generating cofibrations, def. 9, are precisely those which are both weak homotopy equivalences, def. 5 as well as Serre fibrations, def. 14.

Proof

We break this up into three sub-statements:

A) I TopI_{Top}-injective morphisms are in particular weak homotopy equivalences

Let p:X^Xp \colon \hat X \to X have the right lifting property against I TopI_{Top}

S n1 X^ ι n p D n X \array{ S^{n-1} &\longrightarrow & \hat X \\ {}^{\mathllap{\iota_n}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{p}} \\ D^n &\longrightarrow& X }

We check that the lifts in these diagrams exhibit π (f)\pi_\bullet(f) as being an isomorphism on all homotopy groups, def. 4:

For n=0n = 0 the existence of these lifts says that every point of XX is in the image of pp, hence that π 0(X^)π 0(X)\pi_0(\hat X) \to \pi_0(X) is surjective. Let then S 0=**X^S^0 = \ast \coprod \ast \longrightarrow \hat X be a map that hits two connected components, then the existence of the lift says that if they have the same image in π 0(X)\pi_0(X) then they were already the same connected component in X^\hat X. Hence π 0(X^)π 0(X)\pi_0(\hat X)\to \pi_0(X) is also injective and hence is a bijection.

Similarly, for n1n \geq 1, if S nX^S^n \to \hat X represents an element in π n(X^)\pi_n(\hat X) that becomes trivial in π n(X)\pi_n(X), then the existence of the lift says that it already represented the trivial element itself. Hence π n(X^)π n(X)\pi_n(\hat X) \to \pi_n(X) has trivial kernel and so is injective.

Finally, to see that π n(X^)π n(X)\pi_n(\hat X) \to \pi_n(X) is also surjective, hence bijective, observe that every element in π n(X)\pi_n(X) is equivalently represented by a commuting diagram of the form

S n1 * X^ D n X = X, \array{ S^{n-1} &\longrightarrow& \ast &\longrightarrow& \hat X \\ \downarrow && \downarrow && \downarrow \\ D^n &\longrightarrow& X &=& X } \,,

and so here the lift gives a representative of a preimage in π n(X^)\pi_{n}(\hat X).

B) I TopI_{Top}-injective morphisms are in particular Serre fibrations

By lemma 2 an I TopI_{Top}-injective morphisms has also the right lifting property against all relative cell complexes, and hence by lemma 1 it is also a J TopJ_{Top}-injective morphism, hence a Serre fibration.

C) Acyclic Serre fibrations are in particular I TopI_{Top}-injective morphisms

Let f:XYf\colon X \to Y be a Serre fibration that induces isomorphisms on homotopy groups. In degree 0 this means that ff is an isomorphism on connected components, and this means that there is a lift in every commuting square of the form

S 1= X f D 0=* Y \array{ S^{-1} = \emptyset &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ D^0 = \ast &\longrightarrow& Y }

(this is π 0(f)\pi_0(f) being surjective) and in every commuting square of the form

S 0 X ι 0 f D 1=* Y \array{ S^0 &\longrightarrow& X \\ {}^{\mathllap{\iota_0}}\downarrow && \downarrow^{\mathrlap{f}} \\ D^1 = \ast &\longrightarrow& Y }

(this is π 0(f)\pi_0(f) being injective). Hence we are reduced to showing that for n2n \geq 2 every diagram of the form

S n1 α X ι n f D n κ Y \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f}} \\ D^n &\overset{\kappa}{\longrightarrow}& Y }

has a lift.

To that end, pick a basepoint on S n1S^{n-1} and write xx and yy for its images in XX and YY, respectively

Then the diagram above expresses that f *[α]=0π n1(Y,y)f_\ast[\alpha] = 0 \in \pi_{n-1}(Y,y) and hence by assumption on ff it follows that [α]=0π n1(X,x)[\alpha] = 0 \in \pi_{n-1}(X,x), which in turn mean that there is κ\kappa' making the upper triangle of our lifting problem commute:

S n1 α X ι n κ D n. \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow & \nearrow_{\mathrlap{\kappa'}} \\ D^n } \,.

It is now sufficient to show that any such κ\kappa' may be deformed to a ρ\rho' which keeps making this upper triangle commute but also makes the remaining lower triangle commute.

To that end, notice that by the commutativity of the original square, we already have at least this commuting square:

S n1 ι n D n ι n fκ D n κ Y. \array{ S^{n-1} &\overset{\iota_n}{\longrightarrow}& D^n \\ {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f \circ \kappa'}} \\ D^n &\underset{\kappa}{\longrightarrow}& Y } \,.

This induces the universal map (κ,fκ)(\kappa,f \circ \kappa') from the pushout of its cospan in the top left, which is the n-sphere (see this example):

S n1 ι n D n ι n (po) fκ D n κ S n (κ,fκ) Y. \array{ S^{n-1} &\overset{\iota_n}{\longrightarrow}& D^n \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow^{\mathrlap{f \circ \kappa'}} \\ D^n &\underset{\kappa}{\longrightarrow}& S^n \\ && & \searrow^{(\kappa,f \circ \kappa')} \\ && && Y } \,.

This universal morphism represents an element of the nnth homotopy group:

[(κ,fκ)]π n(Y,y). [(\kappa,f \circ \kappa')] \in \pi_n(Y,y) \,.

By assumption that ff is a weak homotopy equivalence, there is a [ρ]π n(X,x)[\rho] \in \pi_{n}(X,x) with

f *[ρ]=[(κ,fκ)] f_\ast [\rho] = [(\kappa,f \circ \kappa')]

hence on representatives there is a lift up to homotopy

X ρ f S n (κ,fκ) Y. \array{ && X \\ &{}^{\mathllap{\rho}}\nearrow_{\mathrlap{\Downarrow}} & \downarrow^{\mathrlap{f}} \\ S^n &\underset{(\kappa,f\circ \kappa')}{\longrightarrow}& Y } \,.

Morever, we may always find ρ\rho of the form (ρ,κ)(\rho', \kappa') for some ρ:D nX\rho' \colon D^n \to X. (“Paste κ\kappa' to the reverse of ρ\rho.”)

Consider then the map

S n(fρ,κ)Y S^n \overset{(f\circ \rho', \kappa)}{\longrightarrow} Y

and observe that this represents the trivial class:

[(fρ,κ)] =[(fρ,fκ)]+[(fκ,κ)] =f *[(ρ,κ)]=[ρ]+[(fκ,κ)] =[(κ,fκ)]+[(fκ,κ)] =0. \begin{aligned} [(f \circ \rho', \kappa)] & = [(f\circ \rho', f\circ \kappa')] + [(f\circ \kappa', \kappa)] \\ & = f_\ast \underset{= [\rho]}{\underbrace{[(\rho',\kappa')]}} + [(f\circ \kappa', \kappa)] \\ & = [(\kappa,f \circ \kappa')] + [(f\circ \kappa', \kappa)] \\ & = 0 \end{aligned} \,.

This means equivalently that there is a homotopy

ϕ:fρκ \phi \; \colon \; f\circ \rho' \Rightarrow \kappa

fixing the boundary of the nn-disk.

Hence if we denote homotopy by double arrows, then we have now achieved the following situation

S n1 α X ι n ρ ϕ f D n Y \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow & {}^{\rho'}\nearrow_{\Downarrow^{\phi}} & \downarrow^{\mathrlap{f}} \\ D^n &\longrightarrow& Y }

and it now suffices to show that ϕ\phi may be lifted to a homotopy of just ρ\rho', fixing the boundary, for then the resulting homotopic ρ\rho'' is the desired lift.

To that end, notice that the condition that ϕ:D n×IY\phi \colon D^n \times I \to Y fixes the boundary of the nn-disk means equivalently that it extends to a morphism

S n1S n1×ID n×I(fα,ϕ)Y S^{n-1} \underset{S^{n-1}\times I}{\sqcup} D^n \times I \overset{(f\circ \alpha,\phi)}{\longrightarrow} Y

out of the pushout that identifies in the cylinder over D nD^n all points lying over the boundary. Hence we are reduced to finding a lift in

D n ρ X f S n1S n1×ID n×I (fα,ϕ) Y. \array{ D^n &\overset{\rho'}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ S^{n-1}\underset{S^{n-1}\times I}{\sqcup} D^n \times I &\overset{(f\circ \alpha,\phi)}{\longrightarrow}& Y } \,.

But inspection of the left map reveals that it is homeomorphic again to D nD n×ID^n \to D^n \times I, and hence the lift does indeed exist.

Verification of the axioms

We use the lemmas above to prove that the classes of morphisms in def. 17 satify the conditions for a model category structure on the category Top.

Proposition

The classical weak equivalences, def. 17, satify two-out-of-three.

Proof

Since isomorphisms (of homotopy groups) satisfy 2-out-of-3, this property is directly inherited via the very definition of weak homotopy equivalence, def. 5.

Proposition

Every morphism f:XYf\colon X \longrightarrow Y in Top factors as a classical cofibration followed by an acyclic fibration, def. 17:

f:XCofX^WFibY. f \;\colon\; X \stackrel{\in Cof}{\longrightarrow} \hat X \stackrel{\in W \cap Fib}{\longrightarrow} Y \,.
Proof

By lemma 4 the set I Top={S n1D n}I_{Top} = \{S^{n-1}\hookrightarrow D^n\} of topological generating cofibrations, def. 9, has small domains, in the sense of def. 16 (the n-spheres are compact). Hence by the small object argument, prop. 4, ff factors as an I TopI_{Top}-relative cell complex, hence just a plain relative cell complex, followed by an I TopI_{Top}-injective morphisms, def. 12.

f:XCofX^I topInjY. f \;\colon\; X \stackrel{\in Cof}{\longrightarrow} \hat X \stackrel{\in I_{top} Inj}{\longrightarrow} Y \,.

By lemma 6 the map X^Y\hat X \to Y is both a weak equivalence as well as a Serre fibration.

Proposition

Every morphism f:XYf\colon X \longrightarrow Y in Top factors as an acyclic classical cofibration followed by a classical fibration, def. 17:

f:XWCofX^FibY. f \;\colon\; X \stackrel{\in W \cap Cof}{\longrightarrow} \hat X \stackrel{\in Fib}{\longrightarrow} Y \,.
Proof

By lemma 4 the set J Top={D nD n×I}J_{Top} = \{D^n \hookrightarrow D^n\times I\} of topological generating acyclic cofibrations, def. 13, has small domains, in the sense of def. 16 (the n-disks are compact). Hence by the small object argument, prop. 4, ff factors as an J TopJ_{Top}-relative cell complex, followed by an J topJ_{top}-injective morphisms, def. 12:

f:XJ TopCellX^J TopInjY. f \;\colon\; X \stackrel{\in J_{Top} Cell}{\longrightarrow} \hat X \stackrel{\in J_{Top} Inj}{\longrightarrow} Y \,.

By definition this makes X^Y\hat X \to Y a Serre fibration, hence a fibration.

By lemma 1 a relative J TopJ_{Top}-cell complex is in particular a relative I TopI_{Top}-cell complex. Hence XX^X \to \hat X is a cofibration. By lemma 5 it is also a weak equivalence.

Proposition

Every commuting square in Top with the left morphism a classical cofibration and the right morphism a fibration, def. 17

gCof fFib \array{ &\longrightarrow& \\ {}^{\mathllap{{g \in} \atop { Cof}}}\downarrow && \downarrow^{\mathrlap{{f \in }\atop Fib}} \\ &\longrightarrow& }

admits a lift as soon as one of the two is also a weak equivalence.

Proof

A) If the fibration ff is also a weak equivalence, then lemma 6 says that it has the right lifting property against the generating cofibrations I TopI_{Top}, and lemma 2 implies the claim.

B) If the cofibration gg on the left is also a weak equivalence, consider any factorization into a relative J TopJ_{Top}-cell complex, def. 13, def. 11, followed by a fibration,

g:J TopCellFib, g \;\colon\; \stackrel{\in J_{Top} Cell}{\longrightarrow} \stackrel{\in Fib}{\longrightarrow} \,,

as in the proof of prop. 7. Now by two-out-of-three, prop. 5, the factorizing fibration is actually an acyclic fibration. By case A), this acyclic fibration has the right lifting property against the cofibration gg itself, and so the retract argument, lemma 3 gives that gg is a retract of a relative J TopJ_{Top}-cell complex. With this, finally lemma 2 implies that ff has the right lifting property against gg.

Finally:

Proposition

The systems (Cof,WFib)(Cof , W \cap Fib) and (WCof,Fib)(\W \cap Cof, Fib) are weak factorization systems.

Proof

We have already seen the factorization and the lifting property, it remains to see that the given left/right classes exhaust the class of morphisms with the given lifting property.

For the fibrations this is by definition, for the the acyclic fibrations this is by lemma 6.

The remaining statement for CofCof and WCofW\cap Cof follows from a general argument (here) for cofibrantly generated model categories:

So let f:XYf \colon X \longrightarrow Y be in (I TopInj)Proj(I_{Top} Inj) Proj, we need to show that then ff is a retract of a relative cell complex. To that end, apply the small object argument to factor ff as

f:XY^IInjY. f \;\colon \; X \overset{}{\longrightarrow} \hat Y \overset{\in I Inj}{\longrightarrow} Y \,.

It follows that ff has the left lefting property again Y^Y\hat Y \to Y, and hence by the retract argument it is a retract of XY^X \to \hat Y, which proves the claim for CofCof.

The argument for WCofW \cap Cof is analogous, using the small object argument now for J TopJ_{Top}.

In conclusion:

Theorem

The classes of morphisms in Mor(Top)Mor(Top) of def. 17,

define a model category structure, Top QuillenTop_{Quillen}.

The homotopy category

We may now pass to the homotopy category of a model category and find Ho(Top) the “classical homotopy category” (or maybe “Quillen-Serre homotopy category”). For discussion of the Quillen equivalence to the classical model structure on simplicial sets (the “homotopy hypothesis”), see there.

Remark

Theorem 1 in itself implies only that every topological space is weakly equivalent to a retract of a cell complex, def. 10. But by the existence of CW approximations, every topological space is weakly homotopy equivalent even to a CW complex. In particular, by the Quillen equivalence to the Quillen model structure on simplicial sets, every topological space is weakly homotopy equivalent to the geometric realization of its singular simplicial complex (and every geometric realization of a simplicial set is (by this proposition) a CW-complex, def. 10.

Properties

Proposition

The model categories Top QuillenTop_{Quillen} and (Top */) Quillen(Top^{\ast/})_{Quillen} are proper model categories.

(Hirschhorn 02,theorem 13.1.10)

Right properness is immediate from the fact that all objects are fibrant. Left properness needs an argument. First check that weak equivalences are preserved under pushout of inclusion maps along cell attachments. Then use that a general cofibration is a retract a relative cell complex inclusion. Observe that if weak equivalences are preserved under pushout along some class of morphisms, then also under pushout along retracts of these. Hence reduce to pushout along relative cell complexes. By the first statement these are a transfinite pasting composite along pushouts that preserve weak equivalences.

Related model structures

We discuss various further model category structures whose existence follows by immediate variation of the above proof of theorem 1:

Classical model structure on pointed topological spaces

Every coslice category 𝒞 X/\mathcal{C}^{X/} of a model category 𝒞\mathcal{C} inherits the coslice model structure, whose classes of morphisms are those of 𝒞\mathcal{C} as seen by the forgetful functor U:𝒞 X/𝒞U \colon \mathcal{C}^{X/}\longrightarrow \mathcal{C}.

Accordingly there is the induced classical model structure on pointed topological spaces Top Quillen */Top^{\ast/}_{Quillen}.

Definition

Let 𝒞\mathcal{C} be a category with terminal object and finite colimits. Then the forgetful functor U:𝒞 */𝒞U \colon \mathcal{C}^{\ast/} \to \mathcal{C} from its category of pointed objects, def. \ref{CategoryOfPointedObjects}, has a left adjoint

𝒞 */U() +𝒞 \mathcal{C}^{\ast/} \underoverset {\underset{U}{\longrightarrow}} {\overset{(-)_+}{\longleftarrow}} {\bot} \mathcal{C}

given by forming the disjoint union (coproduct) with a base point (“adjoining a base point”).

Proposition

Let 𝒞\mathcal{C} be a category with all limits and colimits. Then also the category of pointed objects 𝒞 */\mathcal{C}^{\ast/}, def. \ref{CategoryOfPointedObjects}, has all limits and colimits.

Moreover:

  1. the limits are the limits of the underlying diagrams in 𝒞\mathcal{C}, with the base point of the limit induced by its universal property in 𝒞\mathcal{C};

  2. the colimits are the limits in 𝒞\mathcal{C} of the diagrams with the basepoint adjoined.

Proof

It is immediate to check the relevant universal property. For details see at slice category – limits and colimits.

Example

Given two pointed objects (X,x)(X,x) and (Y,y)(Y,y), then:

  1. their product in 𝒞 */\mathcal{C}^{\ast/} is simply (X×Y,(x,y))(X\times Y, (x,y));

  2. their coproduct in 𝒞 */\mathcal{C}^{\ast/} has to be computed using the second clause in prop. 11: since the point *\ast has to be adjoined to the diagram, it is given not by the coproduct in 𝒞\mathcal{C}, but by the pushout in 𝒞\mathcal{C} of the form:

    * x X y (po) Y XY. \array{ \ast &\overset{x}{\longrightarrow}& X \\ {}^{\mathllap{y}}\downarrow &(po)& \downarrow \\ Y &\longrightarrow& X \vee Y } \,.

    This is called the wedge sum operation on pointed objects.

Generally for a set {X i} iI\{X_i\}_{i \in I} in Top */Top^{\ast/}

  1. their product is formed in TopTop as in example \ref{ProductTopologicalSpace}, with the new basepoint canonically induced;

  2. their coproduct is formed by the colimit in TopTop over the diagram with a basepoint adjoined, and is called the wedge sum iIX i\vee_{i \in I} X_i.

Example

For XX a CW-complex, def. 10 then for every nn \in \mathbb{N} the quotient (example \ref{QuotientSpaceAsPushout}) of its nn-skeleton by its (n1)(n-1)-skeleton is the wedge sum, def. 3, of nn-spheres, one for each nn-cell of XX:

X n/X n1iI nS n. X^n / X^{n-1} \simeq \underset{i \in I_n}{\vee} S^n \,.
Definition

For 𝒞 */\mathcal{C}^{\ast/} a category of pointed objects with finite limits and finite colimits, the smash product is the functor

()():𝒞 */×𝒞 */𝒞 */ (-)\wedge(-) \;\colon\; \mathcal{C}^{\ast/} \times \mathcal{C}^{\ast/} \longrightarrow \mathcal{C}^{\ast/}

given by

XY*XY(X×Y), X \wedge Y \;\coloneqq\; \ast \underset{X\sqcup Y}{\sqcup} (X \times Y) \,,

hence by the pushout in 𝒞\mathcal{C}

XY (id X,y),(x,id Y) X×Y * XY. \array{ X \sqcup Y &\overset{(id_X,y),(x,id_Y) }{\longrightarrow}& X \times Y \\ \downarrow && \downarrow \\ \ast &\longrightarrow& X \wedge Y } \,.

In terms of the wedge sum from def. 3, this may be written concisely as

XY=X×YXY. X \wedge Y = \frac{X\times Y}{X \vee Y} \,.
Remark

For a general category 𝒞\mathcal{C} in def. 19, the smash product need not be associative, namely it fails to be associative if the functor ()×Z(-)\times Z does not preserve the quotients involved in the definition.

In particular this may happen for 𝒞=\mathcal{C} = Top.

A sufficient condition for ()×Z(-) \times Z to preserve quotients is that it is a left adjoint functor. This is the case in the smaller subcategory of compactly generated topological spaces, we come to this in prop. \ref{SmashProductInTopcgIsAssociative} below.

These two operations are going to be ubiquituous in stable homotopy theory:

symbolnamecategory theory
XYX \vee Ywedge sumcoproduct in 𝒞 */\mathcal{C}^{\ast/}
XYX \wedge Ysmash producttensor product in 𝒞 */\mathcal{C}^{\ast/}
Example

For X,YTopX, Y \in Top, with X +,Y +Top */X_+,Y_+ \in Top^{\ast/}, def. 18, then

  • X +Y +(XY) +X_+ \vee Y_+ \simeq (X \sqcup Y)_+;

  • X +Y +(X×Y) +X_+ \wedge Y_+ \simeq (X \times Y)_+.

Proof

By example 3, X +Y +X_+ \vee Y_+ is given by the colimit in TopTop over the diagram

* X * * Y. \array{ && && \ast \\ && & \swarrow && \searrow \\ X &\,\,& \ast && && \ast &\,\,& Y } \,.

This is clearly A*BA \sqcup \ast \sqcup B. Then, by definition 19

X +Y + (X*)×(X*)(X*)(Y*) X×YXY*XY* X×Y*. \begin{aligned} X_+ \wedge Y_+ & \simeq \frac{(X \sqcup \ast) \times (X \sqcup \ast)}{(X\sqcup \ast) \vee (Y \sqcup \ast)} \\ & \simeq \frac{X \times Y \sqcup X \sqcup Y \sqcup \ast}{X \sqcup Y \sqcup \ast} \\ & \simeq X \times Y \sqcup \ast \,. \end{aligned}
Example

Let 𝒞 */=Top */\mathcal{C}^{\ast/} = Top^{\ast/} be pointed topological spaces. Then

I +Top */ I_+ \in Top^{\ast/}

denotes the standard interval object I=[0,1]I = [0,1] from def. 1, with a djoint basepoint adjoined, def. 18. Now for XX any pointed topological space, then

X(I +)=(X×I)/({x 0}×I) X \wedge (I_+) = (X \times I)/(\{x_0\} \times I)

is the reduced cylinder over XX: the result of forming the ordinary cyclinder over XX as in def. 1, and then identifying the interval over the basepoint of XX with the point.

(Generally, any construction in 𝒞\mathcal{C} properly adapted to pointed objects 𝒞 */\mathcal{C}^{\ast/} is called the “reduced” version of the unpointed construction. Notably so for “reduced suspension” which we come to below.)

Just like the ordinary cylinder X×IX\times I receives a canonical injection from the coproduct XXX \sqcup X formed in TopTop, so the reduced cyclinder receives a canonical injection from the coproduct XXX \sqcup X formed in Top */Top^{\ast/}, which is the wedge sum from example 3:

XXX(I +). X \vee X \longrightarrow X \wedge (I_+) \,.
Example

For (X,x),(Y,y)(X,x),(Y,y) pointed topological spaces with YY a locally compact topological space, then the pointed mapping space is the topological subspace of the mapping space of def. 6

Maps((Y,y),(X,x)) *(X Y,const x) Maps((Y,y),(X,x))_\ast \hookrightarrow (X^Y, const_x)

on those maps which preserve the basepoints, and pointed by the map constant on the basepoint of XX.

In particular, the standard topological pointed path space object on some pointed XX (the pointed variant of def. 7) is the pointed mapping space Maps(I +,X) *Maps(I_+,X)_\ast.

The pointed consequence of prop. 2 then gives that there is a natural bijection

Hom Top */((Z,z)(Y,y),(X,x))Hom Top */((Z,z),Maps((Y,y),(X,x)) *) Hom_{Top^{\ast/}}((Z,z) \wedge (Y,y), (X,x)) \simeq Hom_{Top^{\ast/}}((Z,z), Maps((Y,y),(X,x))_\ast)

between basepoint-preserving continuous functions out of a smash product, def. 19, with pointed continuous functions of one variable into the pointed mapping space.

Definition

Write

I Top */={S + n1(ι n) +D + n}Mor(Top */) I_{Top^{\ast/}} = \left\{ S^{n-1}_+ \overset{(\iota_n)_+}{\longrightarrow} D^n_+ \right\} \;\; \subset Mor(Top^{\ast/})

and

J Top */={D + n(id,δ 0) +(D n×I) +}Mor(Top */), J_{Top^{\ast/}} = \left\{ D^n_+ \overset{(id, \delta_0)_+}{\longrightarrow} (D^n \times I)_+ \right\} \;\;\; \subset Mor(Top^{\ast/}) \,,

respectively, for the classes of morphisms obtained from the classical generating cofibrations, def. 9, and the classical generating acyclic cofibrations, def. 13, under adjoining of basepoints.

Theorem

The classes in def. 20 exhibit the classical model structure on pointed topological spaces, Top Quillen */Top^{\ast/}_{Quillen} as a cofibrantly generated model category.

This is a special case of a general statement about cofibrant generation of coslice model structures, see this proposition. But it also follows by a proof directly analogous to that of theorem 1:

Proof

Due to the fact that in J Top */J_{Top^{\ast/}} a basepoint is freely adjoined, lemma 6 goes through verbatim for the pointed case, with J TopJ_{Top} replaced by J Top */J_{Top^{\ast/}}, as do the other two lemmas above that depend on point-set topology, lemma 4 and lemma 5. With this, the rest of the proof follows by the same general abstract reasoning as above in the proof of theorem 1.

Model structure on compactly generated topological spaces

The category Top has the technical inconvenience that mapping spaces X YX^Y (def. 6) exist only for YY a locally compact topological space but fail to exist more generally. In other words: Top is not cartesian closed. But cartesian closure is necessary for some purposes of homotopy theory, for instance it ensures that

  1. the smash product on pointed topological spaces is associative;

  2. there is a concept of topologically enriched functors with values in topological spaces, to which we turn below.

  3. geometric realization of simplicial sets preserves products.

Now, since, by the above, the homotopy theory of topological spaces only cares about the CW approximation to any topological space, it is plausible to ask for a full subcategory of Top which still contains all CW-complexes, still has all limits and colimits, still supports a model category structure constructed in the same way as above, but which in addition is cartesian closed, and preferably such that the model structure interacts well with the cartesian closure.

Such a full subcategory exists, the category of compactly generated topological spaces. This we briefly describe now.

Definition

Let XX be a topological space.

A subset AXA \subset X is called kk-closed if for every continuous function ϕ:KX\phi \colon K \longrightarrow X out of a compact Hausdorff KK, then the preimage ϕ 1(A)\phi^{-1}(A) is a closed subset of KK.

XX is called compactly generated if its closed subsets exhaust (hence coincide with) the kk-closed subsets.

Write

Top cgTop Top_{cg} \hookrightarrow Top

for the full subcategory of Top on the compactly generated topological spaces.

Definition

Write

TopkTop cgTop Top \overset{k}{\longrightarrow} Top_{cg} \hookrightarrow Top

for the functor which sends any topological space X=(S,τ)X = (S,\tau) to the topological space with the same underlying set SS, but with open subsets kτk \tau the collection of all kk-open subsets.

Lemma

Let XTop cgTopX \in Top_{cg} \hookrightarrow Top (def. 21) and let YTopY\in Top. Then continuous functions

XY X \longrightarrow Y

are also continuous when regarded as a function

Xk(Y) X \longrightarrow k(Y)

with kk from def. 22.

Proof

We need to show that for AXA \subset X a kk-closed subset, then f 1(A)Xf^{-1}(A) \subset X is closed subset.

Let ϕ:KX\phi \colon K \longrightarrow X be any continuous function out of a compact Hausdorff space KK. Since AA is kk-closed by assumption, we have that (fϕ) 1(A)=ϕ 1(f 1(A))K(f \circ \phi)^{-1}(A) = \phi^{-1}(f^{-1}(A))\subset K is closed in KK. This means that f 1(A)f^{-1}(A) is kk-closed in XX. But by the assumption that XX is compactly generated, it follows that f 1(A)f^{-1}(A) is already closed.

Corollary

For XTop cgX \in Top_{cg} there is a natural bijection

Hom Top(X,Y)Hom Top cg(X,k(Y)), Hom_{Top}(X,Y) \simeq Hom_{Top_{cg}}(X, k(Y)) \,,

which means equivalently that the functor kk (def. 22) together with the inclusion from def. 21 forms an pair of adjoint functors

Top cgkTop. Top_{cg} \stackrel{\hookrightarrow}{\underoverset{k}{\bot}{\longleftarrow}} Top \,.

This in turn means equivalently that Top cgTopTop_{cg} \hookrightarrow Top is a coreflective subcategory with coreflector kk. In particular kk is idemotent in that there are natural homeomorphisms

k(k(X))k(X). k(k(X))\simeq k(X) \,.

Hence colimits in Top cgTop_{cg} exists and are computed as in Top. Also limits in Top cgTop_{cg} exists, these are obtained by computing the limit in Top and then applying the functor kk to the result.

Definition

For X,YTop cgX, Y \in Top_{cg} (def. 21) the compactly generated mapping space X YTop cgX^Y \in Top_{cg} is the compactly generated topological space whose underlying set is the set C(X,Y)C(X,Y) of continuous functions f:XYf \colon X \to Y, and for which a subbase for its topology has elements U κU^\kappa, for UYU \subset Y any open subset and κ:KX\kappa \colon K \to X a continuous function out of a compact Hausdorff space KK given by

U κ{fC(X,Y)|f(κ(K))U}. U^\kappa \coloneqq \left\{ f\in C(X,Y) | f(\kappa(K)) \subset U \right\} \,.
Proposition

The category Top cgTop_{cg} (def. 21) is cartesian closed:

for every XTop cgX \in Top_{cg} then the operation X×()×()×XX\times (-) \times (-)\times X of forming the Cartesian product in Top cgTop_{cg} (which by cor. 1 is kk applied to the usual product topological space) together with the operation () X(-)^X of forming the compactly generated mapping space (def. 23) forms a pair of adjoint functors

Top cg() XX×()Top cg. Top_{cg} \stackrel{\overset{X \times (-)}{\longleftarrow}}{\underset{(-)^X}{\longrightarrow}} Top_{cg} \,.

e.g. (Strickland 09, prop. 2.12)

Due to the idempotency kkkk \circ k \simeq k (cor. 1) it is useful to know plenty of conditions under which a given topological space is already compactly generated, for then applying kk to it does not change it.

Example

Every CW-complex is compactly generated.

Proof

Since a CW-complex is a Hausdorff space, by prop. 16 and prop. 17 its kk-closed subsets are precisely those whose intersection with every compact subspace is closed.

Since a CW-complex XX is a colimit in Top over attachments of standard n-disks D n iD^{n_i} (its cells), by the characterization of colimits in TopTop (prop.) a subset of XX is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the nn-disks are compact, this implies one direction: if a subset AA of XX intersected with all compact subsets is closed, then AA is closed.

For the converse direction, since a CW-complex is a Hausdorff space and since compact subspaces of Hausdorff spaces are closed, the intersection of a closed subset with a compact subset is closed.

(Lewis 78, p. 148)

Recall that by corollary 1, all colimits of compactly generated spaces are again compactly generated.

Example

The product topological space of a CW-complex with a compact CW-complex is compactly generated.

(Hatcher “Topology of cell complexes”, theorem A.6)

More generally:

Proposition

The product topological space of a compactly generated topological space with a locally compact? Hausdorff topological space is itself compactly generated.

(Strickland 09, prop. 2.6)

Proposition

For every topological space XX, the canonical function k(X)Xk(X) \longrightarrow X is a weak homotopy equivalence.

Proof

By example 8, example 10 and lemma 7, continuous functions S nk(X)S^n \to k(X) and their left homotopies S n×Ik(X)S^n \times I \to k(X) are in bijection with functions S nXS^n \to X and their homotopies S n×IXS^n \times I \to X.

Theorem

The restriction of the model category structure on Top QuillenTop_{Quillen} from theorem 1 along the inclusion Top cgTopTop_{cg} \hookrightarrow Top of def. 21 is still a model category structure, which is cofibrantly generated by the same sets I TopI_{Top} (def. 9) and J TopJ_{Top} (def. 13) The coreflection of cor. 1 is a Quillen equivalence

Top cg,QuillenkTop Quillen. Top_{cg, Quillen} \stackrel{\hookrightarrow}{\underoverset{k}{\bot}{\longleftarrow}} Top_{Quillen} \,.
Proof

By example 8, the sets I TopI_{Top} and J TopJ_{Top} are indeed in Mor(Top cg)Mor(Top_{cg}). By example 10 all arguments above about left homotopies between maps out of these basic cells go through verbatim in Top cgTop_{cg}. Hence the three technical lemmas above depending on actual point-set topology, topology, lemma 4, lemma 5 and lemma 6, go through verbatim as before. Accordingly, since the remainder of the proof of theorem 1 of Top QuillenTop_{Quillen} follows by general abstract arguments from these, it also still goes through verbatim for (Top cg) Quillen(Top_{cg})_{Quillen}.

Hence the (acyclic) cofibrations in (Top cg) Quillen(Top_{cg})_{Quillen} are identified with those in Top QuillenTop_{Quillen}, and so the inclusion is a part of a Quillen adjunction. To see that this is a Quillen equivalence, it is sufficient to check that for XX a compactly generated space then a continuous function f:XYf \colon X \longrightarrow Y is a weak homotopy equivalence (def. 5) precisely if the adjunct f˜:Xk(Y)\tilde f \colon X \to k(Y) is a weak homotopy equivalence. But, by lemma 7, f˜\tilde f is the same function as ff, just considered with different codomain. Hence the result follows with prop. 14.

Pointed compactly generated topological spaces

Moreover:

Proposition

Write Top cg */Top_{cg}^{\ast/} for the category of pointed compactly generated topological spaces (def. 21). Then the smash product

()():Top cg */×Top cg */Top cg */ (-)\wedge (-) \;\colon\; Top_{cg}^{\ast/} \times Top_{cg}^{\ast/} \longrightarrow Top_{cg}^{\ast/}

is associative and the 0-sphere is a tensor unit for it, hence (Top cg */,,S 0)(Top_{cg}^{\ast/}, \wedge, S^0) is a symmetric monoidal category.

Moreover together with the pointed mapping space version () * X(-)_\ast^X of the compactly generated mapping space of def. 23, Top cg */Top_{cg}^{\ast/} becomes a closed monoidal category:

for every XTop cg */X \in Top_{cg}^{\ast/} then the operations of forming the smash product X()X\wedge (-) and of forming the pointed mapping space () * X(-)_\ast^X constitute a pair of adjoint functors

Top cg */() * XX()Top cg */. Top_{cg}^{\ast/} \stackrel{\overset{X \wedge (-)}{\longleftarrow}}{\underset{(-)_\ast^X}{\longrightarrow}} Top_{cg}^{\ast/} \,.
Proof

For the first statement, since ()×X(-)\times X is a left adjoint by prop. 12, it presevers colimits and in particular quotient space projections. Therefore with X,Y,ZTop cg */X, Y, Z \in Top_{cg}^{\ast/} then

(XY)Z =X×YX×{y}{x}×Y×Z(XY)×{z}{[x]=[y]}×Z X×Y×ZX×{y}×Z{x}×Y×ZX×Y×{z} X×Y×ZXYZ. \begin{aligned} (X \wedge Y) \wedge Z & = \frac{ \frac{X\times Y}{X \times\{y\} \sqcup \{x\}\times Y} \times Z }{ (X \wedge Y)\times \{z\} \sqcup \{[x] = [y]\} \times Z} \\ & \simeq \frac{\frac{X \times Y \times Z}{X \times \{y\}\times Z \sqcup \{x\}\times Y \times Z}}{ X \times Y \times \{z\} } \\ &\simeq \frac{X\times Y \times Z}{ X \vee Y \vee Z} \end{aligned} \,.

The analogous reasoning applies to yield also X(YZ)X×Y×ZXYZX \wedge (Y\wedge Z) \simeq \frac{X\times Y \times Z}{ X \vee Y \vee Z}.

Compactly generated weakly Hausdorff spaces

While the inclusion Top cgTopTop_{cg} \hookrightarrow Top above does satisfy the requirement that it gives a cartesian closed category with all limits and colimits and containing all CW-complexes, one may ask for yet smaller subcategories that still share all these properties but potentially exhibit further convenient properties still.

One may in addition demand all compactly generated spaces to be Hausdorff topological spaces (e. g. Hirschhorn 15, top of p. 2) and use Hausdorff reflection (in addition to reflection onto compactly generated spaces) to make colimits land again in Hausdorff space.

A morre popular choice introduced in (McCord 69) is weak Hausdorffness, i.e. to add the further restriction to topopological spaces which are not only compactly generated but also weakly Hausdorff. This was motivated from (Steenrod 67) where compactly generated Hausdorff spaces were used by the observation (McCord 69, section 2) that Hausdorffness is not preserved my many colimit operations, notably not by forming quotient spaces.

On the other hand, in above we wouldn’t have imposed Hausdorffness in the first place. Possibly more intrinsic advantages of Top cgwHTop_{cgwH} over Top cgTop_{cg} are the following:

Definition

A topological space XX is called weakly Hausdorff if for every continuous function

f:KX f \;\colon\; K \longrightarrow X

out of a compact Hausdorff space KK, its image f(K)Xf(K) \subset X is a closed subset of XX.

Proposition

Every Hausdorff space is a weakly Hausdorff space, def. 24.

Proposition

For XX a weakly Hausdorff topological space, def. 24, then a subset AXA \subset X is kk-closed, def. 21, precisely if for every subset KXK \subset X that is compact Hausdorff with respect to the subspace topology, then the intersection KAK \cap A is a closed subset of XX.

e.g. (Strickland 09, lemma 1.4 (c))

Monoidal and topologically enriched model structure

We discuss that (Top cg) Quillen(Top_{cg})_{Quillen} and (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen} are monoidal model categories and enriched model categories over themselves, the former with respect to Cartesian product and the latter with respect to the induced smash product.

Definition

Let i 1:X 1Y 1i_1 \colon X_1 \to Y_1 and i 2:X 2Y 2i_2 \colon X_2 \to Y_2 be morphisms in Top cgTop_{cg}, def. 21. Their pushout product

i 1i 2((id,i 2),(i 1,id)) i_1\Box i_2 \coloneqq ((id, i_2), (i_1,id))

is the universal morphism in the following diagram

X 1×X 2 (i 1,id) (id,i 2) Y 1×X 2 (po) X 1×Y 2 (Y 1×X 2)X 1×X 2(X 1×Y 2) ((id,i 2),(i 1,id)) Y 1×Y 2 \array{ && X_1 \times X_2 \\ & {}^{\mathllap{(i_1,id)}}\swarrow && \searrow^{\mathrlap{(id,i_2)}} \\ Y_1 \times X_2 && (po) && X_1 \times Y_2 \\ & {}_{\mathllap{}}\searrow && \swarrow \\ && (Y_1 \times X_2) \underset{X_1 \times X_2}{\sqcup} (X_1 \times Y_2) \\ && \downarrow^{\mathrlap{((id, i_2), (i_1,id))}} \\ && Y_1 \times Y_2 }
Example

If i 1:X 1Y 1i_1 \colon X_1 \hookrightarrow Y_1 and i 2:X 2Y 2i_2 \colon X_2 \hookrightarrow Y_2 are inclusions, then their pushout product i 1i 2i_1 \Box i_2 from def. 25 is the inclusion

(X 1×Y 2Y 1×X 2)Y 1×Y 2. \left( X_1 \times Y_2 \;\cup\; Y_1 \times X_2 \right) \hookrightarrow Y_1 \times Y_2 \,.

For instance

({0}I)({0}I) \left( \{0\} \hookrightarrow I \right) \Box \left( \{0\} \hookrightarrow I \right)

is the inclusion of two adjacent edges of a square into the square.

Example

The pushout product with an initial morphism is just the ordinary Cartesian product functor

(X)()X×(), (\emptyset \to X) \Box (-) \simeq X \times (-) \,,

i.e.

(X)(AfB)(X×AX×fX×B). (\emptyset \to X) \Box (A \overset{f}{\to} B) \simeq (X\times A \overset{X \times f}{\longrightarrow} X \times B ) \,.
Proof

The product topological space with the empty space is the empty space, hence the map ×A(id,f)×B\emptyset \times A \overset{(id,f)}{\longrightarrow} \emptyset \times B is an isomorphism, and so the pushout in the pushout product is X×AX \times A. From this one reads off the universal map in question to be X×fX \times f:

×A X×A (po) ×B X×A ((id,f),!) X×B. \array{ && \emptyset \times A \\ & {}^{\mathllap{}}\swarrow && \searrow^{\mathrlap{\simeq}} \\ X \times A && (po) && \emptyset \times B \\ & {}_{\mathllap{}}\searrow && \swarrow \\ && X \times A \\ && \downarrow^{\mathrlap{((id, f), \exists !)}} \\ && X \times B } \,.
Example

With

I Top:{S n1i nD n}andJ Top:{D nj nD n×I} I_{Top} \colon \{ S^{n-1} \overset{i_n}{\hookrightarrow} D^n\} \;\;\; and \;\;\; J_{Top} \colon \{ D^n \overset{j_n}{\hookrightarrow} D^n \times I\}

the generating cofibrations (def. 9) and generating acyclic cofibrations (def. 13) of (Top cg) Quillen(Top_{cg})_{Quillen} (theorem 3), then their pushout-products (def. 25) are

i n 1i n 2 i n 1+n 2 i n 1j n 2 j n 1+n 2. \begin{aligned} i_{n_1} \Box i_{n_2} & \simeq i_{n_1 + n_2} \\ i_{n_1} \Box j_{n_2} & \simeq j_{n_1 + n_2} \end{aligned} \,.
Proof

To see this, it is profitable to model n-disks and n-spheres, up to homeomorphism, as nn-cubes D n[0,1] n nD^\n \simeq [0,1]^n \subset \mathbb{R}^n and their boundaries S n1[0,1] nS^{n-1} \simeq \partial [0,1]^n . For the idea of the proof, consider the situation in low dimensions, where one readily sees pictorially that

i 1i 1:(=||) i_1 \Box i_1 \;\colon\; \left(\;\; = \;\;\cup\;\; \vert\vert\;\;\right) \hookrightarrow \Box

and

i 1j 0:(=|). i_1 \Box j_0 \;\colon\; \left(\;\; = \;\;\cup\;\; \vert \;\; \right) \hookrightarrow \Box \,.

Generally, D nD^n may be represented as the space of nn-tuples of elements in [0,1][0,1], and S nS^n as the suspace of tuples for which at least one of the coordinates is equal to 0 or to 1.

Accordingly, S n 1×D n 2D n 1+n 2S^{n_1} \times D^{n_2} \hookrightarrow D^{n_1 + n_2} is the subspace of (n 1+n 2)(n_1+n_2)-tuples, such that at least one of the first n 1n_1 coordinates is equal to 0 or 1, while D n 1×S n 2D n 1+n 2D^{n_1} \times S^{n_2} \hookrightarrow D^{n_1+ n_2} is the subspace of (n 1+n 2)(n_1 + n_2)-tuples such that east least one of the last n 2n_2 coordinates is equal to 0 or to 1. Therefore

S n 1×D n 2D n 1×S n 2S n 1+n 2. S^{n_1} \times D^{n_2} \cup D^{n_1} \times S^{n_2} \simeq S^{n_1 + n_2} \,.

And of course it is clear that D n 1×D n 2D n 1+n 2D^{n_1} \times D^{n_2} \simeq D^{n_1 + n_2}. This shows the first case.

For the second, use that S n 1×D n 2×IS^{n_1} \times D^{n_2} \times I is contractible to S n 1×D n 2S^{n_1} \times D^{n_2} in D n 1×D n 2×ID^{n_1} \times D^{n_2} \times I, and that S n 1×D n 2S^{n_1} \times D^{n_2} is a subspace of D n 1×D n 2D^{n_1} \times D^{n_2}.

Definition

Let i:ABi \colon A \to B and p:XYp \colon X \to Y be two morphisms in Top cgTop_{cg}, def. 21. Their pullback powering is

p i(p B,X i) p^{\Box i} \coloneqq (p^B, X^i)

being the universal morphism in

X B (p B,X i) Y B×Y AX A Y B (pb) X A Y i p A Y A \array{ && X^B \\ && \downarrow^{\mathrlap{(p^B, X^i)}} \\ && Y^B \underset{Y^A}{\times} X^A \\ & \swarrow && \searrow \\ Y^B && (pb) && X^A \\ & {}_{\mathllap{Y^i}}\searrow && \swarrow_{\mathrlap{p^A}} \\ && Y^A }
Proposition

Let i 1,i 2,pi_1, i_2 , p be three morphisms in Top cgTop_{cg}, def. 21. Then for their pushout-products (def. 25) and pullback-powerings (def. 26) the following lifting properties are equivalent (“Joyal-Tierney calculus”):

i 1i 2 has LLP against p i 1 has LLP against p i 2 i 2 has LLP against p i 1. \array{ & i_1 \Box i_2 & \text{has LLP against} & p \\ \Leftrightarrow & i_1 & \text{has LLP against} & p^{\Box i_2} \\ \Leftrightarrow & i_2 & \text{has LLP against} & p^{\Box i_1} } \,.
Proof

We claim that by the cartesian closure of Top cgTop_{cg}, and carefully collecting terms, one finds a natural bijection between commuting squares and their lifts as follows:

Q f X B i 1 p i 2 P (g 1,g 2) Y B×Y AX AQ×BQ×AP×A (f˜,g˜ 2) X i 1i 2 p P×B g˜ 1 Y, \array{ Q &\overset{f}{\longrightarrow}& X^B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{p^{\Box i_2}}} \\ P &\underset{(g_1,g_2)}{\longrightarrow}& Y^B \underset{Y^A}{\times} X^A } \;\;\;\;\;\;\; \leftrightarrow \;\;\;\;\;\;\; \array{ Q \times B \underset{Q \times A}{\sqcup} P \times A &\overset{(\tilde f, \tilde g_2)}{\longrightarrow}& X \\ {}^{\mathllap{i_1 \Box i_2}}\downarrow && \downarrow^{\mathrlap{p}} \\ P \times B & \underset{\tilde g_1}{\longrightarrow} & Y } \,,

where the tilde denotes product/hom-adjuncts, for instance

Pg 1Y BP×Bg˜ 1Y \frac{ P \overset{g_1}{\longrightarrow} Y^B }{ P \times B \overset{\tilde g_1}{\longrightarrow} Y }

etc.

To see this in more detail, observe that both squares above each represent two squares from the two components into the fiber product and out of the pushout, respectively, as well as one more square exhibiting the compatibility condition on these components:

Q f X B i 1 p i 2 P (g 1,g 2) Y B×Y AX A {Q f X B i 1 p B P g 1 Y B,Q f X B i 1 X i 2 P g 1 X A,P g 2 X A g 1 p A Y B Y i 2 Y A} {Q×B f˜ X (i 1,id) p P×B g˜ 2 Y,Q×A (id,i 2) Q×B (i 1,id) f˜ P×A g˜ 2 X,P×A g˜ 2 X (id,i 2) p P×B g˜ 1 Y} Q×BQ×AP×A (f˜,g˜ 2) X i 1i 2 p P×B g˜ 1 Y. \begin{aligned} & \;\;\;\; \array{ Q &\overset{f}{\longrightarrow}& X^B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{p^{\Box i_2}}} \\ P &\underset{(g_1,g_2)}{\longrightarrow}& Y^B \underset{Y^A}{\times} X^A } \\ \simeq & \;\;\;\; \left\{ \;\;\;\; \array{ Q &\overset{f}{\longrightarrow}& X^B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{p^B}} \\ P &\underset{g_1}{\longrightarrow}& Y^B } \;\;\;\;\; \,, \;\;\;\;\; \array{ Q &\overset{f}{\longrightarrow}& X^B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{X^{i_2}}} \\ P &\underset{g_1}{\longrightarrow}& X^A } \;\;\;\;\; \,, \;\;\;\;\; \array{ P &\overset{g_2}{\longrightarrow}& X^A \\ {}^{\mathllap{g_1}}\downarrow && \downarrow^{\mathrlap{p^A}} \\ Y^B &\underset{Y^{i_2}}{\longrightarrow}& Y^A } \;\;\;\;\; \right\} \\ \leftrightarrow & \;\;\;\; \left\{ \;\;\;\;\; \array{ Q \times B &\overset{\tilde f}{\longrightarrow}& X \\ {}^{\mathllap{(i_1,id)}}\downarrow && \downarrow^{\mathrlap{p}} \\ P \times B &\underset{\tilde g_2}{\longrightarrow}& Y } \;\;\;\;\; \,, \;\;\;\;\; \array{ Q \times A &\overset{(id,i_2)}{\longrightarrow}& Q \times B \\ {}^{\mathllap{(i_1,id)}}\downarrow && \downarrow^{\mathrlap{\tilde f}} \\ P \times A &\underset{\tilde g_2}{\longrightarrow}& X } \;\;\;\;\; \,, \;\;\;\;\; \array{ P \times A &\overset{\tilde g_2}{\longrightarrow}& X \\ {}^{\mathllap{(id,i_2)}}\downarrow && \downarrow^{\mathrlap{p}} \\ P \times B &\underset{\tilde g_1}{\longrightarrow}& Y } \;\;\;\;\; \right\} \\ \simeq & \;\;\;\; \array{ Q \times B \underset{Q \times A}{\sqcup} P \times A &\overset{(\tilde f, \tilde g_2)}{\longrightarrow}& X \\ {}^{\mathllap{i_1 \Box i_2}}\downarrow && \downarrow^{\mathrlap{p}} \\ P \times B & \underset{\tilde g_1}{\longrightarrow} & Y } \end{aligned} \,.
Proposition

The pushout-product in Top cgTop_{cg} (def. 21) of two classical cofibrations is a classical cofibration:

Cof clCof clCof cl. Cof_{cl} \Box Cof_{cl} \subset Cof_{cl} \,.

If one of them is acyclic, then so is the pushout-product:

Cof cl(W clCof cl)W clCof cl. Cof_{cl} \Box (W_{cl} \cap Cof_{cl}) \subset W_{cl}\cap Cof_{cl} \,.
Proof

Regarding the first point:

By example 13 we have

I TopI TopI Top I_{Top} \Box I_{Top} \subset I_{Top}

Hence

I TopI Top has LLP against W clFib cl I Top has LLP against (W clFib cl) I Top Cof cl has LLP against (W clFib cl) I Top I TopCof cl has LLP against W clFib cl I Top has LLP against (W clFib cl) Cof cl Cof cl has LLP against (W clFib cl) Cof cl Cof clCof cl has LLP against W clFib cl, \array{ & I_{Top} \Box I_{Top} & \text{has LLP against} & W_{cl} \cap Fib_{cl} \\ \Leftrightarrow & I_{Top} & \text{has LLP against} & (W_{cl} \cap Fib_{cl})^{\Box I_{Top}} \\ \Rightarrow & Cof_{cl} & \text{has LLP against} & (W_{cl} \cap Fib_{cl})^{\Box I_{Top}} \\ \Leftrightarrow & I_{Top} \Box Cof_{cl} & \text{has LLP against} & W_{cl} \cap Fib_{cl} \\ \Leftrightarrow & I_{Top} & \text{has LLP against} & (W_{cl} \cap Fib_{cl})^{Cof_{cl}} \\ \Rightarrow & Cof_{cl} & \text{has LLP against} & (W_{cl} \cap Fib_{cl})^{Cof_{cl}} \\ \Leftrightarrow & Cof_{cl} \Box \Cof_{cl} & \text{has LLP against} & W_{cl} \cap Fib_{cl} } \,,

where all logical equivalences used are those of prop. 18 and where all implications appearing are by the closure property of lifting problems (prop.).

Regarding the second point: By example 13 we moreover have

I TopJ TopJ Top I_{Top} \Box J_{Top} \subset J_{Top}

and the conclusion follows by the same kind of reasoning.

Remark

In model category theory the property in proposition 19 is referred to as saying that the model category (Top cg) Quillen(Top_{cg})_{Quillen} from theorem 3

  1. is a monoidal model category with respect to the Cartesian product on Top cgTop_{cg};

  2. is an enriched model category, over itself.

A key point of what this entails is the following:

Proposition

For X(Top cg) QuillenX \in (Top_{cg})_{Quillen} cofibrant (a retract of a cell complex) then the product-hom-adjunction for YY (prop. 12) is a Quillen adjunction

(Top cg) Quillen() XX×()(Top cg) Quillen. (Top_{cg})_{Quillen} \underoverset \underset{(-)^X}{\longrightarrow} \overset{X \times (-)}{\longleftarrow} {\bot} (Top_{cg})_{Quillen} \,.
Proof

By example 12 we have that the left adjoint functor is equivalently the pushout product functor with the initial morphism of XX:

X×()(X)(). X \times (-) \simeq (\emptyset \to X) \Box (-) \,.

By assumption (X)(\emptyset \to X) is a cofibration, and hence prop. 19 says that this is a left Quillen functor.

The statement and proof of prop. 20 has a direct analogue in pointed topological spaces

Proposition

For X(Top cg */) QuillenX \in (Top^{\ast/}_{cg})_{Quillen} cofibrant with respect to the classical model structure on pointed compactly generated topological spaces (theorem \ref{ClassicalModelStructureOnCompactlyGeneratedTopologicalSpaces}, prop. \ref{ModelStructureOnSliceCategory}) (hence a retract of a cell complex with non-degenerate basepoint, remark \ref{NonDegenerateBasepointAsCofibrantObjects}) then the pointed product-hom-adjunction from corollary \ref{SmashHomAdjunctionOnPointedCompactlyGeneratedTopologicalSpaces} is a Quillen adjunction (def. \ref{QuillenAdjunction}):

(Top cg */) Quillen() XX×()(Top cg */) Quillen. (Top^{\ast/}_{cg})_{Quillen} \underoverset \underset{(-)^X}{\longrightarrow} \overset{X \times (-)}{\longleftarrow} {\bot} (Top^{\ast/}_{cg})_{Quillen} \,.
Proof

Let now \Box_\wedge denote the smash pushout product and () ()(-)^{\Box(-)} the smash pullback powering defined as in def. 25 and def. 26, but with Cartesian product replaced by smash product (def. 19) and compactly generated mapping space replaced by pointed mapping spaces (def. 7).

By theorem 2 (Top cg */) Quillen(Top_{cg}^{\ast/})_{Quillen} is cofibrantly generated by I Top */=(I Top) +I_{Top^{\ast/}} = (I_{Top})_+ and J Top */=(J Top) +J_{Top^{\ast/}}= (J_{Top})_+. Example 5 gives that for i nI Topi_n \in I_{Top} and j nJ Topj_n \in J_{Top} then

(i n 1) + (i n 2) +(i n 1+n 2) + (i_{n_1})_+ \Box_\wedge (i_{n_2})_+ \simeq (i_{n_1 + n_2})_+

and

(i n 1) + (i n 2) +(i n 1+n 2) +. (i_{n_1})_+ \wedge_\wedge (i_{n_2})_+ \simeq (i_{n_1 + n_2})_+ \,.

Hence the pointed analog of prop. 19 holds and therefore so does the pointed analog of the conclusion in prop. 20.

The model structure on topologically enriched functors

The projective model structure on enriched functors, enriched over the classical model structure on topological spaces above, is an immediate corollary of the above proof (Piacenza 91).

In the following we say Top-enriched category and Top-enriched functor etc. for what often is referred to as “topological category” and “topological functor” etc. As discussed there, these latter terms are ambiguous.

Definition

A topologically enriched category 𝒞\mathcal{C} is a Top-enriched category, hence:

  1. a class Obj(𝒞)Obj(\mathcal{C}), called the class of objects;

  2. for each a,bObj(𝒞)a,b\in Obj(\mathcal{C}) a topological space

    𝒞(a,b)Top, \mathcal{C}(a,b)\in Top \,,

    called the space of morphisms or the hom-space between aa and bb;

  3. for each a,b,cObj(𝒞)a,b,c\in Obj(\mathcal{C}) a continuous function

    a,b,c:𝒞(a,b)×𝒞(b,c)𝒞(a,c) \circ_{a,b,c} \;\colon\; \mathcal{C}(a,b)\times \mathcal{C}(b,c) \longrightarrow \mathcal{C}(a,c)

    out of the product topological space, called the composition operation

  4. for each aObj(𝒞)a \in Obj(\mathcal{C}) a point Id a𝒞(a,a)Id_a\in \mathcal{C}(a,a), called the identity morphism on aa

such that the composition is associative and unital.

Similarly a pointed topologically enriched category is such a structire with Top kTop_k replaced by pointed topological spaces and with the Cartesian product replaced by the smash product of pointed topological spaces.

Remark

Given a (pointed) topologically enriched category as in def. 27, then forgetting the topology on the hom-spaces (along the forgetful functor U:Top kSetU \colon Top_k \to Set) yields an ordinary locally small category with

Hom 𝒞(a,b)=U(𝒞(a,b)). Hom_{\mathcal{C}}(a,b) = U(\mathcal{C}(a,b)) \,.

It is in this sense that 𝒞\mathcal{C} is a category with extra structure, and hence “enriched”.

The archetypical example is the following:

Example

Write

Top kTop Top_k \hookrightarrow Top

for the full subcategory of Top on the compactly generated topological spaces. Under forming Cartesian product

()×():Top k×Top kTop k (-)\times (-) \;\colon\; Top_k \times Top_k \longrightarrow Top_k

and mapping spaces

() ():Top k op×Top kTop k (-)^{(-)} \;\colon\; Top_k^{op}\times Top_k \longrightarrow Top_k

this is a cartesian closed category (see at convenient category of topological spaces). As such it canonically obtains the structure of a topologically enriched category, def. 27, with hom-spaces given by mapping spaces

Top k(X,Y)Y X Top_k(X,Y) \coloneqq Y^X

and with composition

Y X×Z YZ X Y^X \times Z^Y \longrightarrow Z^X

given by the (product\dashv mapping-space)-adjunct of the evaluation morphism

X×Y X×Z Y(ev,id)Y×Z YevZ. X \times Y^X \times Z^Y \overset{(ev, id)}{\longrightarrow} Y \times Z^Y \overset{ev}{\longrightarrow} Z \,.

Similarly, pointed compactly generated topological spaces Top k */Top_k^{\ast/} form a pointed topologically enriched category.

Definition

A topologically enriched functor between two topologically enriched categories

F:𝒞𝒟 F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}

is a Top-enriched functor, hence:

  1. a function

    F 0:Obj(𝒞)Obj(𝒟) F_0 \colon Obj(\mathcal{C}) \longrightarrow Obj(\mathcal{D})

    of objects;

  2. for each a,bObj(𝒞)a,b \in Obj(\mathcal{C}) a continuous function

    F a,b:𝒞(a,b)𝒟(F 0(a),F 0(b)) F_{a,b} \;\colon\; \mathcal{C}(a,b) \longrightarrow \mathcal{D}(F_0(a), F_0(b))

    of hom-spaces

such that this preserves composition and identity morphisms in the evident sense.

A homomorphism of topologically enriched functors

η:FG \eta \;\colon\; F \Rightarrow G

is a Top-enriched natural transformation: for each cObj(𝒞)c \in Obj(\mathcal{C}) a choice of morphism η c𝒟(F(c),G(c))\eta_c \in \mathcal{D}(F(c),G(c)) such that for each pair of objects c,d𝒞c,d \in \mathcal{C} the two continuous functions

η dF():𝒞(c,d)𝒟(F(c),G(d)) \eta_d \circ F(-) \;\colon\; \mathcal{C}(c,d) \longrightarrow \mathcal{D}(F(c), G(d))

and

G()η c:𝒞(c,d)𝒟(F(c),G(d)) G(-) \circ \eta_c \;\colon\; \mathcal{C}(c,d) \longrightarrow \mathcal{D}(F(c), G(d))

agree.

We write [𝒞,𝒟][\mathcal{C}, \mathcal{D}] for the resulting category of topologically enriched functors. This itself naturally obtains the structure of topologically enriched category, see at enriched functor category.

Example

For 𝒞\mathcal{C} any topologically enriched category, def. 27 then a topologically enriched functor

F:𝒞Top k F \;\colon\; \mathcal{C} \longrightarrow Top_k

to the archetical topologically enriched category from example 14 may be thought of as a topologically enriched copresheaf, at least if 𝒞\mathcal{C} is small (in that its class of objects is a proper set).

Hence the category of topologically enriched functors

[𝒞,Top k] [\mathcal{C}, Top_k]

according to def. 28 may be thought of as the (co-)presheaf category over 𝒞\mathcal{C} in the realm of topological enriched categories.

A funcotor F[𝒞,Top k]F \in [\mathcal{C}, Top_k] is equivalently

such that composition is respected, in the evident sense.

For every object c𝒞c \in \mathcal{C}, there is a topologically enriched representable functor, denoted y(c)or𝒞(c,)y(c) or \mathcal{C}(c,-) which sends objects to

y(c)(d)=𝒞(c,d)Top y(c)(d) = \mathcal{C}(c,d) \in Top

and whose action on morphisms is, under the above identification, just the composition operation in 𝒞\mathcal{C}.

There is a full blown Top-enriched Yoneda lemma. The following records a slightly simplified version which is all that is needed here:

Proposition

(topologically enriched Yoneda-lemma)

Let 𝒞\mathcal{C} be a topologically enriched category, def. 27, write [𝒞,Top k][\mathcal{C}, Top_k] for its category of topologically enriched (co-)presheaves, and for cObj(𝒞)c\in Obj(\mathcal{C}) write y(c)=𝒞(c,)[𝒞,Top k]y(c) = \mathcal{C}(c,-) \in [\mathcal{C}, Top_k] for the topologically enriched functor that it represents, all according to example 15. Recall also the Top-tensored functors FXF \cdot X from that example.

For cObj(𝒞)c\in Obj(\mathcal{C}), XTopX \in Top and F[𝒞,Top k]F \in [\mathcal{C}, Top_k], there is a natural bijection between

  1. morphisms y(c)XFy(c) \cdot X \longrightarrow F in [𝒞,Top k][\mathcal{C}, Top_k];

  2. morphisms XF(c)X \longrightarrow F(c) in Top.

Proof

Given a morphism η:y(c)XF\eta \colon y(c) \cdot X \longrightarrow F consider its component

η c:𝒞(c,c)×XF(c) \eta_c \;\colon\; \mathcal{C}(c,c)\times X \longrightarrow F(c)

and restrict that to the identity morphism id c𝒞(c,c)id_c \in \mathcal{C}(c,c) in the first argument

η c(id c,):XF(c). \eta_c(id_c,-) \;\colon\; X \longrightarrow F(c) \,.

We claim that just this η c(id c,)\eta_c(id_c,-) already uniquely determines all components

η d:𝒞(c,d)×XF(d) \eta_d \;\colon\; \mathcal{C}(c,d)\times X \longrightarrow F(d)

of η\eta, for all dObj(𝒞)d \in Obj(\mathcal{C}): By definition of the transformation η\eta (def. 28), the two functions

F()η c:𝒞(c,d)F(d) 𝒞(c,c)×X F(-) \circ \eta_c \;\colon\; \mathcal{C}(c,d) \longrightarrow F(d)^{\mathcal{C}(c,c) \times X}

and

η d𝒞(c,)×X:𝒞(c,d)F(d) 𝒞(c,c)×X \eta_d \circ \mathcal{C}(c,-) \times X \;\colon\; \mathcal{C}(c,d) \longrightarrow F(d)^{\mathcal{C}(c,c) \times X}

agree. This means that they may be thought of jointly as a function with values in commuting squares in TopTop of this form:

f𝒞(c,c)×X η c F(c) 𝒞(c,f) F(f) 𝒞(c,d)×X η d F(d) f \;\;\;\; \mapsto \;\;\;\; \array{ \mathcal{C}(c,c) \times X &\overset{\eta_c}{\longrightarrow}& F(c) \\ {}^{\mathllap{\mathcal{C}(c,f)}}\downarrow && \downarrow^{\mathrlap{F(f)}} \\ \mathcal{C}(c,d) \times X &\underset{\eta_d}{\longrightarrow}& F(d) }

For any f𝒞(c,d)f \in \mathcal{C}(c,d), consider the restriction of

η d𝒞(c,f)F(d) 𝒞(c,c)×X \eta_d \circ \mathcal{C}(c,f) \in F(d)^{\mathcal{C}(c,c) \times X}

to id c𝒞(c,c)id_c \in \mathcal{C}(c,c), hence restricting the above commuting squares to

f{id c}×X η c F(c) 𝒞(c,f) F(f) {f}×X η d F(d) f \;\;\;\; \mapsto \;\;\;\; \array{ \{id_c\} \times X &\overset{\eta_c}{\longrightarrow}& F(c) \\ {}^{\mathllap{\mathcal{C}(c,f)}}\downarrow && \downarrow^{\mathrlap{F}(f)} \\ \{f\} \times X &\underset{\eta_d}{\longrightarrow}& F(d) }

This shows that η d\eta_d is fixed to be the function

η d(f,x)=F(f)η c(id c,x) \eta_d(f,x) = F(f)\circ \eta_c(id_c,x)

and this is a continuous function since all the operations it is built from are continuous.

Conversely, given a continuous function α:XF(c)\alpha \colon X \longrightarrow F(c), define for each dd the function

η d:(f,x)F(f)α. \eta_d \colon (f,x) \mapsto F(f) \circ \alpha \,.

Running the above analysis backwards shows that this determines a transformation η:y(c)×XF\eta \colon y(c)\times X \to F.

Definition

For 𝒞\mathcal{C} a small topologically enriched category, def. 27, write

I Top 𝒞{y(c)(S n1ι nD n)} ncObj(𝒞) I_{Top}^{\mathcal{C}} \;\coloneqq\; \left\{ y(c)\cdot (S^{n-1} \overset{\iota_n}{\longrightarrow} D^n) \right\}_{{n \in \mathbb{N}} \atop {c \in Obj(\mathcal{C})}}

and

J Top 𝒞{y(c)(D n(id,δ 0)D n×I)} ncObj(𝒞) J_{Top}^{\mathcal{C}} \;\coloneqq\; \left\{ y(c)\cdot (D^n \overset{(id, \delta_0)}{\longrightarrow} D^n \times I) \right\}_{{n \in \mathbb{N}} \atop {c \in Obj(\mathcal{C})}}

for the classes (here: sets) of morphisms given by tensoring the representable functors, example 15 with the generating cofibrations (def.9) and acyclic generating cofibrations (def. 13) of Top kTop_k, respectively.

These are going to be called the generating cofibrations and acyclic generating cofibrations for the projective model structure on topologically enriched functors on 𝒞\mathcal{C}.

Similarly, for 𝒞\mathcal{C} a pointed topologically enriched category, write

I Top */ 𝒞{y(c)(S + n1(ι n) +D + n)} ncObj(𝒞) I_{Top^{\ast/}}^{\mathcal{C}} \;\coloneqq\; \left\{ y(c)\cdot (S^{n-1}_+ \overset{(\iota_n)_+}{\longrightarrow} D^n_+) \right\}_{{n \in \mathbb{N}} \atop {c \in Obj(\mathcal{C})}}

and

J Top */ 𝒞{y(c)(D + n(id,δ 0) +(D n×I) +)} ncObj(𝒞) J_{Top^{\ast/}}^{\mathcal{C}} \;\coloneqq\; \left\{ y(c)\cdot (D^n_+ \overset{(id, \delta_0)_+}{\longrightarrow} (D^n \times I)_+) \right\}_{{n \in \mathbb{N}} \atop {c \in Obj(\mathcal{C})}}

for the same construction applied to the pointed generating (acyclic) cofibrations of def. 20.

Remark

By the Top-enriched Yoneda lemma, prop. 22, and the defining property of tensoring over Top kTop_k, there are natural bijections

y(c)XFXF(c) \frac{ y(c)\cdot X \longrightarrow F }{ X \longrightarrow F(c) }

between

  1. natural transformations from y(c)Xy(c)\cdot X (the tensoring with XTop kX \in Top_k of the representable functor of cObj(𝒞)c\in Obj(\mathcal{C})) to some topologically enriched functor FF ,

  2. continuous functions from XX to the value of that topological functor on the object cc.

Definition

Given a small (pointed) topologically enriched category 𝒞\mathcal{C}, def. 27, say that a morphism in the category of (pointed) topologically enriched copresheaves [𝒞,Top k][\mathcal{C}, Top_k] ([𝒞,Top k */][\mathcal{C},Top_k^{\ast/}]), example 15, hence a natural transformation between topologically enriched functors, η:FG\eta \colon F \to G is

  • a projective weak equivalence, if for all cObj(𝒞)c\in Obj(\mathcal{C}) the component η c:F(c)G(c)\eta_c \colon F(c) \to G(c) is a weak homotopy equivalence (def. 5);

  • a projective fibration if for all cObj(𝒞)c\in Obj(\mathcal{C}) the component η c:F(c)G(c)\eta_c \colon F(c) \to G(c) is a Serre fibration (def. 14);

  • a projective cofibration if it is a retract (rmk. 5) of an I Top 𝒞I_{Top}^{\mathcal{C}}-relative cell complex (def. 11, def. 29).

Write

[𝒞,Top Quillen] proj [\mathcal{C}, Top_{Quillen}]_{proj}

for the category of topologically enriched functors equipped with these classes of morphisms.

Theorem

The classes of morphisms in def. 30 constitute a model category structure on [𝒞,Top][\mathcal{C}, Top], called the projective model structure on enriched functors [𝒞,Top Quillen] proj[\mathcal{C}, Top_{Quillen}]_{proj}.

(Piacenza 91, theorem 5.4)

Proof

Via remark 10, the statement essentially reduces objectwise to the proof of theorem 1:

In particular, the technical lemmas 4, 5 and 6 generalize immediately to the present situation, with the evident small change of wording.

For instance the fact that a morphism of topologically enriched functors η:FG\eta \colon F \to G that has the right lifting property against the elements of I Top 𝒞I_{Top}^{\mathcal{C}} is a projective weak equivalence, follows by noticing that remark 10 gives a natural bijection of commuting diagrams (and their fillers) of the form

(y(c)S n1 F (idι n) η y(c)D n G)(S n1 F(c) η c D n G(c)), \left( \array{ y(c) \cdot S^{n-1} &\longrightarrow& F \\ {}^{\mathllap{(id\cdot \iota_n)}}\downarrow && \downarrow^{\mathrlap{\eta}} \\ y(c) \cdot D^n &\longrightarrow& G } \right) \;\;\;\leftrightarrow\;\;\; \left( \array{ S^{n-1} &\longrightarrow& F(c) \\ \downarrow && \downarrow^{\mathrlap{\eta_c}} \\ D^n &\longrightarrow& G(c) } \right) \,,

and hence the statement follows with part A) of the proof of lemma 6.

With these three lemmas in hand, the remaining formal part of the proof goes through verbatim as above: repeatedly use the small object argument and the retract argument to establish the two weak factorization systems. (While again the structure of a category with weak equivalences is evident.)

Remark

The same argument applies to functors with values in the classical model structure on pointed topological spaces

[𝒞,Top Quillen */]. [\mathcal{C}, Top_{Quillen}^{\ast/}] \,.
Example

The strict Bousfield-Friedlander model structure on sequential spectra is equivalently the projective model structure on functors on the non-full subcategory of Top */Top^{\ast/} on the “standard spheres” (see at sequential spectrum – As diagram spectra)

The actual stable Bousfield-Friedlander model structure is then the left Bousfield localization of that at the stable weak homotopy equivalences.

SeqSpec(Top) stableBousf.loc.SeqSpec(Top) strict[StdSpheres,Top Quillen */] proj. SeqSpec(Top)_{stable} \stackrel{\longleftarrow}{\overset{Bousf.\; loc.}{\longrightarrow}} SeqSpec(Top)_{strict} \simeq [StdSpheres, Top^{\ast/}_{Quillen}]_{proj} \,.

References

The original article is

  • Dan Quillen, chapter II, section 3 of Homotopical algebra, Lecture Notes in Mathematics 43, Springer-Verlag 1967, iv+156 pp.

An expository, concise and comprehensive writeup of the proof of the model category axioms is in

Useful discussion of the issue of compactly generated topological spaces in the context of homotopy theory is in

  • Gaunce Lewis, Compactly generated spaces (pdf), appendix A of The Stable Category and Generalized Thom Spectra PhD thesis Chicago, 1978

  • Neil Strickland, The category of CGWH spaces, 2009 (pdf)

The observation that the proof directly extends to give the projective model structures on enriched functors, enriched over Top QuillenTop_{Quillen}, is due to

  • Robert Piacenza section 5 of Homotopy theory of diagrams and CW-complexes over a category, Can. J. Math. Vol 43 (4), 1991 (pdf)

    also chapter VI of Peter May et al., Equivariant homotopy and cohomology theory, 1996 (pdf)

The quick way to see the topological enrichment is indicated for instance in

  • Stefan Schwede, pages 5-6 of Orbispaces, orthogonal spaces, and the universal compact Lie group (pdf)

Revised on July 25, 2017 13:21:44 by Anonymous (129.132.9.56)