pointed topological space in nLab

Context

Topology

• Idea
• Definition
• The category of pointed topological spaces
• Forgetting and adjoining basepoints
• Wedge sum and Smash product
• Mapping (co-)cones
• Properties
• General
• Relation to one-point compactification
• Smash-monoidal diagonals
• Related concepts
• References

Example

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

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

2. their coproduct in Top */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 TopTop (which is the disjoint union space), but by the pushout in TopTop of the form:

   * ⟶x X y↓ (po) ↓ Y ⟶ X∨Y. \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∨Y≃(X⊔Y)/(x∼y) X \vee Y \;\simeq\; (X \sqcup Y)/(x \sim y)

Generally for a set {(X i,x i)} i∈I\{(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∈I∈∏i∈IX i(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 TopTop over the diagram with a basepoint adjoined, and is called the wedge sum ∨ i∈IX i\vee_{i \in I} X_i, which is the quotient topological space of the disjoint union space with all the basepoints identified:

   ∨i∈IX i≃(⊔i∈IX i)/(x i∼x j) i,j∈I. \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 XX a CW-complex, then for every n∈ℕn \in \mathbb{N} the quotient of its nn-skeleton by its (n−1)(n-1)-skeleton is the wedge sum, def. , of nn-spheres, one for each nn-cell of XX:

X n/X n−1≃∨i∈I nS n. 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

(−)∧(−):Top */×Top */⟶Top */ (-)\wedge(-) \;\colon\; Top^{\ast/} \times Top^{\ast/} \longrightarrow Top^{\ast/}

given by

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

hence by the pushout in TopTop of the form

X⊔Y ⟶(id X,y),(x,id Y) X×Y ↓ (po) ↓ * ⟶ X∧Y. \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∧Y≃X×YX∨Y. X \wedge Y \simeq \frac{X\times Y}{X \vee Y} \,.

Example

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

• X +∨Y +≃(X⊔Y) +X_+ \vee Y_+ \simeq (X \sqcup Y)_+;

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

Proof

By example , 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

X +∧Y + ≃(X⊔*)×(X⊔*)(X⊔*)∨(Y⊔*) ≃X×Y⊔X⊔Y⊔*X⊔Y⊔* ≃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 I≔[0,1]⊂ℝI \coloneqq [0,1] \subset \mathbb{R} be the closed interval with its Euclidean metric topology.

Hence

I +∈Top */ I_+ \in Top^{\ast/}

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

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

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 cylinder over XX, and then identifying the interval over the basepoint of XX with the point.

(Generally, any construction in TopTop 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×IX\times I receives a canonical injection from the coproduct X⊔XX \sqcup X formed in TopTop, so the reduced cyclinder receives a canonical injection from the coproduct X⊔XX \sqcup X formed in Top */Top^{\ast/}, which is the wedge sum from example :

X∨X⟶X∧(I +). X \vee X \longrightarrow X \wedge (I_+) \,.

Definition

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

Cone(f)≔*⊔XCyl(X)⊔XY Cone(f) \coloneqq \ast \underset{X}{\sqcup} Cyl(X) \underset{X}{\sqcup} Y

in the colimiting diagram

X ⟶f Y ↓ i 1 ↓ i X ⟶i 0 Cyl(X) ↓ ↘ η ↓ * ⟶ ⟶ Cone(f), \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)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:

X ⟶f Y ↓ i 1 (po) ↓ i X ⟶i 0 Cyl(X) ⟶ Cyl(f) ↓ (po) ↓ (po) ↓ * ⟶ Cone(X) ⟶ Cone(f). \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)≔*⊔XCyl(X)Cone(X) \coloneqq \ast \underset{X}{\sqcup} Cyl(X);

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

Definition

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

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

ΣX=Cone(X→*). \Sigma X = Cone(X\to\ast)\,.

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

X ⟶ * ↓ i 1 (po) ↓ X ⟶i 0 Cyl(X) ⟶ Cone(X) ↓ (po) ↓ (po) ↓ * ⟶ Cone(X) ⟶ ΣX. \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)(i_0,i_1), hence (example ) of the wedge sum inclusion:

X∨X=X⊔X⟶(i 0,i 1)Cyl(X)⟶cofib(i 0,i 1)ΣX. 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 (−)∧(I +)(-)\wedge (I_+) of example are isomorphic to the smash product (def. ) with the circle] (the [[1-sphere?)

cofib(X∨X→X∧(I +))≃S 1∧X, cofib(X \vee X \to X \wedge (I_+)) \simeq S^1 \wedge X \,,

Proposition

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

Cone′(f)≃Cone(f +). Cone'(f) \simeq Cone(f_+) \,.

Proof

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

* ⟶ X⊔* ⟶(f,id) Y⊔* ↓ ↓ ↓ X⊔* ⟶ (X×I)⊔* ↓ ↓ * ⟶ ⟶ Cone(f +). \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

* ⟶ X⊔* ⟶(f,id) Y⊔* ↓ ↓ ↓ X⊔* ⟶ (X×I)⊔* ↓ ↓ *⊔* ⟶ ⟶ Cone′(f) + ↓ ↓ * ⟶ ⟶ Cone′(f). \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) +Cone'(f)_+ as shown. The remaining pushout then contracts the remaining copy of the point away.

Smash-monoidal diagonals

Write

(1)(PointedTopologicalSpaces,S 0,∧)∈SymmetricMonoidalCategories \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=* +S^0 \,=\, \ast_+.

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

(2)X 1×X 2=(𝒳 1,x 1)×(𝒳 2,x 2)≔(𝒳 1×𝒳 2,(x 1,x 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∧X 1≔X 1×X 2X 1∨X 2 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)𝒳 ⟶Δ 𝒳 𝒳×𝒳 x ↦ (x,x) \array{ \mathcal{X} &\overset{ \Delta_{\mathcal{X}} }{\longrightarrow}& \mathcal{X} \times \mathcal{X} \\ x &\mapsto& (x,x) }

a suitable notion of monoidal diagonals:

It is immediate that:

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