nLab pointed topological space

Idea

A pointed topological space (often pointed space, for short) is a topological space equipped with a choice of one of its points (elements). If the inclusion of that point is a Hurewicz cofibration then one speaks of a well-pointed topological space.

Although this concept may seem simple, pointed topological spaces play a central role for instance in algebraic topology as domains for reduced generalized (Eilenberg-Steenrod) cohomology theories and as an ingredient for the definition of spectra.

One reason why pointed topological spaces are important is that the category which they form is an intermediate stage in the stabilization of homotopy theory (the classical homotopy theory of topological spaces) to stable homotopy theory:

The category of pointed topological spaces has a zero object (the point space itself) and the canonical tensor product on pointed spaces is the smash product, which is non-cartesian monoidal category, in contrast to the plain product of topological space.

Definition

A pointed topological space is a topological space $(X,\tau)$ equipped with a choice of point $x \in X$. A homomorphism between pointed topological space $(X,x)$ $(Y,y)$ is a continuous function $f \colon X \to Y$ which preserves the chosen basepoints in that $f(x) = y$.

The category of pointed topological spaces

Stated in the language of category theory, this means that pointed topological spaces are the pointed objects in the category Top of topological spaces. This is the coslice category $Top^{\ast/}$ of topological spaces “under” the point space $\ast$:

an object in $Top^{\ast/}$ is equivalently a continuous function $x \colon \ast \to (X,\tau)$, which is equivalently just a choice of point in $X$, and a morphism in $Top^{\ast/}$ is a morphism $f \colon X \to Y$ in Top (hence a continuous function), such that this triangle diagram commutes

$$\array{ && \ast \\ & {}^{\mathllap{x}}\swarrow && \searrow^{\mathrlap{y}} && \\ X && \underset{f}{\longrightarrow} && Y }$$

which equivalently means that $f(x) = y$.

Forgetting and adjoining basepoints

Definition

The forgetful functor $Top^{\ast/} \to Top$ has a left adjoint given by forming the disjoint union space (coproduct in Top) with a point space (“adjoining a base point”), this is denoted by

$$(-)_+ \coloneqq (-) \sqcup \ast \;\colon \; Top \longrightarrow Top^{\ast/} \,.$$

Wedge sum and Smash product

Example

Given two pointed topological spaces $(X,x)$ and $(Y,y)$, then:

1. their Cartesian product in $Top^{\ast/}$ is simply their product topological space $X \times Y$ equipped with the pair of basepoints $(X\times Y, (x,y))$;

2. their coproduct in $Top^{\ast/}$ has to be computed using the second clause in this prop.: since the point $\ast$ has to be adjoined to the diagram, it is given not by the coproduct in $Top$ (which is the disjoint union space), but by the pushout in $Top$ of the form:

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

   This is called the wedge sum operation on pointed objects.

   This is the quotient topological space of the disjoint union space under the equivalence relation which identifies the two basepoints:

   $$X \vee Y \;\simeq\; (X \sqcup Y)/(x \sim y)$$

Generally for a set $\{(X_i,x_i)\}_{i \in I}$ of pointed topological spaces

1. their product is formed in Top, as the product topological space with the Tychonoff topology, with the tuple $(x_i)_{i \in I} \in \underset{i \in I}{\prod} X_i$ of basepoints being the new basepoint;

2. their coproduct is formed by the colimit in $Top$ over the diagram with a basepoint adjoined, and is called the wedge sum $\vee_{i \in I} X_i$, which is the quotient topological space of the disjoint union space with all the basepoints identified:

   $$\underset{i \in I}{\vee} X_i \;\simeq\; \left(\underset{i \in I}{\sqcup} X_i\right)/(x_i \sim x_j)_{i,j \in I} \,.$$

Example

For $X$ a CW-complex, then for every $n \in \mathbb{N}$ the quotient of its $n$-skeleton by its $(n-1)$-skeleton is the wedge sum, def. , of $n$-spheres, one for each $n$-cell of $X$:

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

Definition

The smash product of pointed topological spaces is the functor

$$(-)\wedge(-) \;\colon\; Top^{\ast/} \times Top^{\ast/} \longrightarrow Top^{\ast/}$$

given by

$$X \wedge Y \;\coloneqq\; \ast \underset{X\sqcup Y}{\sqcup} (X \times Y) \,,$$

hence by the pushout in $Top$ of the form

$$\array{ X \sqcup Y &\overset{(id_X,y),(x,id_Y) }{\longrightarrow}& X \times Y \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& X \wedge Y } \,.$$

In terms of the wedge sum from def. , this may be written concisely as the quotient space (this def) of the product topological space by the subspace constituted by the wedge sum

$$X \wedge Y \simeq \frac{X\times Y}{X \vee Y} \,.$$

t $\,$

symbol	name	category theory
$X \times Y$	product space	product in $Top^{\ast/}$
$X \vee Y$	wedge sum	coproduct in $Top^{\ast/}$
$X \wedge Y = \frac{X \times Y}{X \vee Y}$	smash product	tensor product in $Top^{\ast/}$

Example

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

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

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

Proof

By example , $X_+ \vee Y_+$ is given by the colimit in $Top$ over the diagram

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

This is clearly $A \sqcup \ast \sqcup B$. Then, by definition

$$\begin{aligned} X_+ \wedge Y_+ & \simeq \frac{(X \sqcup \ast) \times (X \sqcup \ast)}{(X\sqcup \ast) \vee (Y \sqcup \ast)} \\ & \simeq \frac{X \times Y \sqcup X \sqcup Y \sqcup \ast}{X \sqcup Y \sqcup \ast} \\ & \simeq X \times Y \sqcup \ast \,. \end{aligned}$$

Example

Let $I \coloneqq [0,1] \subset \mathbb{R}$ be the closed interval with its Euclidean metric topology.

Hence

$$I_+ \in Top^{\ast/}$$

is the interval with a disjoint basepoint adjoined, def. .

Now for $X$ any pointed topological space, then the smash product (def. )

$$X \wedge (I_+) = (X \times I)/(\{x_0\} \times I)$$

is the reduced cylinder over $X$: the result of forming the ordinary cylinder over $X$, and then identifying the interval over the basepoint of $X$ with the point.

(Generally, any construction in $Top$ properly adapted to pointed spaces is called the “reduced” version of the unpointed construction. Notably so for “reduced suspension” which we come to below.)

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

$$X \vee X \longrightarrow X \wedge (I_+) \,.$$

Mapping (co-)cones

Recall that the cone on a topological space $X$ is the quotient space of the product space with the closed interval

$$Cone(X) = (X \times [0,1])/( X \times \{0\} ) \,.$$

If $X$ is pointed with basepoint $x \in X$, then the reduced cone is the further quotient by the copy of the interval over the basepoint

$$Cone(X,x) = Cone(X) / ( \{x\} \times [0,1] ) \,.$$

For $f \colon X \to Y$ a continuous function, then

1. the mapping cylinder of $f$ is the attachment space

   $$Cyl(f) \coloneqq Y \cup_f Cyl(X)$$

2. the mapping cone of $f$ is the attachment space

   $$Cone(f) \coloneqq Y \cup_f Cone(X)$$

accordingly if $f \colon X \to Y$ is a continuous function between pointed spaces which preserves the basepoint, then the analogous construction with the reduced cylinder and the reduce cone, respectively, yield the reduced mapping cyclinder and the reduced mapping cone.

We now say this again in terms of pushouts:

Definition

For $f \colon X \longrightarrow Y$ a continuous function between pointed spces, its reduced mapping cone is the space

$$Cone(f) \coloneqq \ast \underset{X}{\sqcup} Cyl(X) \underset{X}{\sqcup} Y$$

in the colimiting diagram

$$\array{ && X &\stackrel{f}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{i_1}} && \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) \\ \downarrow && & \searrow^{\mathrlap{\eta}} & \downarrow \\ {*} &\longrightarrow& &\longrightarrow& Cone(f) } \,,$$

where $Cyl(X)$ is the reduced cylinder from def. .

Proposition

The colimit appearing in the definition of the reduced mapping cone in def. is equivalent to three consecutive pushouts:

$$\array{ && X &\stackrel{f}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cyl(f) \\ \downarrow &(po)& \downarrow & (po) & \downarrow \\ {*} &\longrightarrow& Cone(X) &\longrightarrow& Cone(f) } \,.$$

The two intermediate objects appearing here are called

• the plain reduced cone $Cone(X) \coloneqq \ast \underset{X}{\sqcup} Cyl(X)$;

• the reduced mapping cylinder $Cyl(f) \coloneqq Cyl(X) \underset{X}{\sqcup} Y$.

Definition

Let $X \in Top^{\ast/}$ be any pointed topological space.

The mapping cone, def. , of $X \to \ast$ is called the reduced suspension of $X$, denoted

$$\Sigma X = Cone(X\to\ast)\,.$$

Via prop. this is equivalently the coproduct of two copies of the cone on $X$ over their base:

$$\array{ && X &\stackrel{}{\longrightarrow}& \ast \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cone(X) \\ \downarrow &(po)& \downarrow & (po) & \downarrow \\ {*} &\longrightarrow& Cone(X) &\longrightarrow& \Sigma X } \,.$$

This is also equivalently the cofiberf of $(i_0,i_1)$, hence (example ) of the wedge sum inclusion:

$$X \vee X = X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \overset{cofib(i_0,i_1)}{\longrightarrow} \Sigma X \,.$$

Proposition

The reduced suspension objects (def. ) induced from the standard reduced cylinder $(-)\wedge (I_+)$ of example are isomorphic to the smash product (def. ) with the circle] (the [[1-sphere?)

$$cofib(X \vee X \to X \wedge (I_+)) \simeq S^1 \wedge X \,,$$

Proposition

For $f \colon X \longrightarrow Y$ a morphism in Top, then its unreduced mapping cone with respect to the standard cylinder object $X \times I$ def. , is isomorphic to the reduced mapping cone, of the morphism $f_+ \colon X_+ \to Y_+$ (with a basepoint adjoined) with respect to the standard reduced cylinder:

$$Cone'(f) \simeq Cone(f_+) \,.$$

Proof

By example , $Cone(f_+)$ is given by the colimit in $Top$ over the following diagram:

$$\array{ \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast \\ \downarrow && \downarrow && \downarrow \\ X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast \\ \downarrow && && \downarrow \\ \ast &\longrightarrow& &\longrightarrow& Cone(f_+) } \,.$$

We may factor the vertical maps to give

$$\array{ \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast \\ \downarrow && \downarrow && \downarrow \\ X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast \\ \downarrow && && \downarrow \\ \ast \sqcup \ast &\longrightarrow& &\longrightarrow& Cone'(f)_+ \\ \downarrow && && \downarrow \\ \ast &\longrightarrow& &\longrightarrow& Cone'(f) } \,.$$

This way the top part of the diagram (using the pasting law to compute the colimit in two stages) is manifestly a cocone under the result of applying $(-)_+$ to the diagram for the unreduced cone. Since $(-)_+$ is itself given by a colimit, it preserves colimits, and hence gives the partial colimit $Cone'(f)_+$ as shown. The remaining pushout then contracts the remaining copy of the point away.

Properties

General

Most of the relevant constructions on pointed topological spaces are immediate specializations of the general construction discussed at pointed object.

Relation to one-point compactification

Proposition

(one-point compactification intertwines Cartesian product with smash product)

On the subcategory $Top_{LCHaus}$ of Top on the locally compact Hausdorff spaces with proper maps between them, the functor of one-point compactification (Prop. )

$$(-)^{cpt} \;\colon\; Top_{LCHaus} \longrightarrow Top^{\ast/}$$

sends Cartesian products (product topological spaces) to smash products of pointed topological spaces, hence constitutes a strong monoidal functor, in that there is a natural homeomorphism:

$$\big( X \times Y \big)^{cpt} \;\simeq\; X^{cpt} \wedge Y^{cpt} \,.$$

This is briefly mentioned in, for instance, Bredon 93, p. 199. The argument may be found spelled out in: MO:a/1645794/, Cutler 20, Prop. 1.6.

Smash-monoidal diagonals

Write

(1)$$\big( PointedTopologicalSpaces, S^0, \wedge \big) \;\;\in\; SymmetricMonoidalCategories$$

• for the category of pointed topological spaces (with respect to some convenient category of topological spaces such as compactly generated topological spaces or D-topological spaces)

• regarded as a symmetric monoidal category with tensor product the smash product and unit the 0-sphere $S^0 \,=\, \ast_+$.

This category also has a Cartesian product, given on pointed spaces $X_i = (\mathcal{X}_i, x_i)$ with underlying $\mathcal{X}_i \in TopologicalSpaces$ by

(2)$$X_1 \times X_2 \;=\; (\mathcal{X}_1, x_1) \times (\mathcal{X}_2, x_2) \;\coloneqq\; \big( \mathcal{X}_1 \times \mathcal{X}_2 , (x_1, x_2) \big) \,.$$

But since this smash product is a non-trivial quotient of the Cartesian product

(3)$$X_1 \wedge X_1 \,\coloneqq\, \frac{X_1 \times X_2}{ X_1 \vee X_2 }$$

it is not itself cartesian, but just symmetric monoidal.

However, via the quotienting (3), it still inherits, from the diagonal morphisms on underlying topological spaces

(4)$$\array{ \mathcal{X} &\overset{ \Delta_{\mathcal{X}} }{\longrightarrow}& \mathcal{X} \times \mathcal{X} \\ x &\mapsto& (x,x) }$$

a suitable notion of monoidal diagonals:

Definition

[Smash monoidal diagonals]

For $X \,\in\, PointedTopologicalSpaces$, let $D_X \;\colon\; X \longrightarrow X \wedge X$ be the composite

of the Cartesian diagonal morphism (2) with the coprojection onto the defining quotient space (3).

It is immediate that:

Proposition

The smash monoidal diagonal $D$ (Def. ) makes the symmetric monoidal category (1) of pointed topological spaces with smash product a monoidal category with diagonals, in that

1. $D$ is a natural transformation;

2. $S^0 \overset{\;\;D_{S^0}\;\;}{\longrightarrow} S^0 \wedge S^0$ is an isomorphism.

While elementary in itself, this has the following profound consequence:

Remark

[Suspension spectra have diagonals]

Since the suspension spectrum-functor

$$\Sigma^\infty \;\colon\; PointedTopologicalSpaces \longrightarrow HighlyStructuredSpectra$$

is a strong monoidal functor from pointed topological spaces (1) to any standard category of highly structured spectra (by this Prop.) it follows that suspension spectra have monoidal diagonals, in the form of natural transformations

(5)$$\Sigma^\infty X \overset{ \;\; \Sigma^\infty(D_X) \;\; }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big)$$

to their respective symmetric smash product of spectra, which hence makes them into comonoid objects, namely coring spectra.

For example, given a Whitehead-generalized cohomology theory $\widetilde E$ represented by a ring spectrum

$$\big(E, 1^E, m^E \big) \;\; \in \; SymmetricMonoids \big( Ho(Spectra), \mathbb{S}, \wedge \big)$$

the smash-monoidal diagonal structure (5) on suspension spectra serves to define the cup product $(-)\cup (-)$ in the corresponding multiplicative cohomology theory structure:

$$\begin{aligned} & \big[ \Sigma^\infty X \overset{c_i}{\longrightarrow} \Sigma^{n_i} E \big] \,\in\, {\widetilde E}{}^{n_i}(X) \\ & \Rightarrow \;\; [c_1] \cup [c_2] \, \coloneqq \, \Big[ \Sigma^\infty X \overset{ \Sigma^\infty(D_X) }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big) \overset{ ( c_1 \wedge c_2 ) }{\longrightarrow} \big( \Sigma^{n_1} E \big) \wedge \big( \Sigma^{n_2} E \big) \overset{ m^E }{\longrightarrow} \Sigma^{n_1 + n_2}E \Big] \;\; \in \, {\widetilde E}{}^{n_1+n_2}(X) \,. \end{aligned}$$

Related concepts

• pointed simplicial set

• retractive space

• loop space, reduced suspension

• smash product, wedge sum

• pointed homotopy type

• spectrum

• classical model structure on pointed topological spaces

References

Textbook accounts:

• Pierre Gabriel, Michel Zisman, Chapters IV.4 and V.7 of Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) (pdf)

• Glen Bredon, Topology and Geometry, Graduate Texts in Mathematics 139, Springer 1993 (doi:10.1007/978-1-4757-6848-0, pdf)

Review:

• Tyrone Cutler, The category of pointed topological spaces (2020) [pdf, pdf]