nLab
Delta-generated topological space

Contents

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

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

Definition

(Δ\Delta-Generated spaces)

A Δ\Delta-generated space (Smith, Dugger 03) is a topological space XX whose topology is the final topology induced by all continuous functions of the form Δ top nX\Delta^n_{top} \to X, hence whose domain Δ top n\Delta^n_{top} is one of the standard topological simplices, for nn \in \mathbb{N}.

A morphism between Δ\Delta-generated spaces is just a continuous function, hence the category of Δ\Delta-generated spaces is the full subcategory on these spaces inside all TopologicalSpaces,

Remark

(as colimits of topological simplices)

Equivalently, the class of Δ\Delta-generated spaces is the closure of the set of topological simplices Δ top n\Delta^n_{top} under small colimits in topological spaces (see at Topuniversal constructions).

Remark

(as Euclidean-generated spaces)

For each nn the topological simplex Δ n\Delta^n is a retract of the ambient Euclidean space/Cartesian space n\mathbb{R}^n (as a non-empty convex subset of a Euclidean space it is in fact an absolute retract). Hence the identity function on Δ n\Delta^n factors as

id:Δ top ni n np nΔ top n. id \;\colon\; \Delta^n_{top} \overset{i_n}{\hookrightarrow} \mathbb{R}^n \overset{p_n}{\longrightarrow} \Delta^n_{top} \,.

It follows that every continuous function ff with domain the topological simplex extends as a continuous function to Euclidean space:

Δ top m f X i n n \array{ \Delta^m_{top} &\overset{f}{\longrightarrow}& X \\ \mathllap{{}^{i_n}}\big\downarrow & \nearrow _{\mathrlap{\exists}} \\ \mathbb{R}^n }

Therefore the condition that a topological space XX be Δ\Delta-generated (Def. ) is equivalent to saying that its topology is final with respect to all continuous functions nX\mathbb{R}^n \to X out of Euclidean/Cartesian spaces.

Remark

(as D-topological spaces)

By Prop. below, the Euclidean-generated spaces and hence, by Remark , the Δ\Delta-generated spaces, are equivalently those that arise from equipping a diffeological space with its D-topology, hence are, in this precise sense, the D-topological spaces. Luckily, “D-topological space” may also serve as an abbreviation for “Delta-generated topological space”.

Properties

As a convenient category of topological spaces

Proposition

(Euclidean-generated spaces are convenient)

The category of Euclidean-generated spaces/Δ\Delta-generated spaces (Def. ) is a convenient category of topological spaces in that:

Coreflection into all topological spaces

Proposition

(adjunction between topological spaces and diffeological spaces)

There is a pair of adjoint functors

TopologicalSpacesAAAACdfflgDtplgDiffeologicalSpaces TopologicalSpaces \underoverset{ \underset{ Cdfflg }{\longrightarrow} }{ \overset{ Dtplg }{\longleftarrow} }{\phantom{AA}\bot\phantom{AA}} DiffeologicalSpaces

between the categories of TopologicalSpaces and of DiffeologicalSpaces, where

Moreover:

  1. the fixed points of this adjunction XX \inTopologicalSpaces (those for which the counit is an isomorphism, hence here: a homeomorphism) are precisely the Delta-generated topological spaces (i.e. D-topological spaces):

    XisΔ-generatedDtplg(Cdffg(X))ϵ XX X \;\,\text{is}\;\Delta\text{-generated} \;\;\;\;\; \Leftrightarrow \;\;\;\;\; Dtplg(Cdffg(X)) \underoverset{\simeq}{\;\;\epsilon_X\;\;}{\longrightarrow} X
  2. this is an idempotent adjunction, which exhibits Δ\Delta-generated/D-topological spaces as a reflective subcategory inside diffeological spaces and a coreflective subcategory inside all topological spaces:

(1)TopologicalSpacesAAAACdfflgDTopologicalSpacesAAAADtplgDiffeologicalSpaces TopologicalSpaces \underoverset { \underset{ Cdfflg }{\longrightarrow} } { \overset{ }{\hookleftarrow} } {\phantom{AA}\bot\phantom{AA}} DTopologicalSpaces \underoverset { \underset{ }{\hookrightarrow} } { \overset{ Dtplg }{\longleftarrow} } {\phantom{AA}\bot\phantom{AA}} DiffeologicalSpaces

Finally, these adjunctions are a sequence of Quillen equivalences with respect to the:

classical model structure on topological spacesmodel structure on D-topological spacesmodel structure on diffeological spaces

Caution: There may be a gap in the proof that DTopologicalSpaces QuillenDiffeologicalSpacesDTopologicalSpaces \simeq_{Quillen} DiffeologicalSpaces, see the commented references here.

These adjunctions and their properties are observed in Shimakawa-Yoshida-Haraguchi 10, Prop. 3.1, Prop. 3.2, Lemma 3.3. The model structures and Quillen equivalences are due to Haraguchi 13, Thm. 3.3 (on the left) and Haraguchi-Shimakawa 13, Sec. 7 (on the right, but this may have a gap).

Proof

We spell out the existence of the idempotent adjunction (1):

First, to see we have an adjunction DtplgCdfflgDtplg \dashv Cdfflg, we check the hom-isomorphism (here).

Let XDiffeologicalSpacesX \in DiffeologicalSpaces and YTopologicalSpacesY \in TopologicalSpaces. Write () s(-)_s for the underlying sets. Then a morphism, hence a continuous function of the form

f:Dtplg(X)Y, f \;\colon\; Dtplg(X) \longrightarrow Y \,,

is a function f s:X sY sf_s \colon X_s \to Y_s of the underlying sets such that for every open subset AY sA \subset Y_s and every smooth function of the form ϕ: nX\phi \colon \mathbb{R}^n \to X the preimage (f sϕ s) 1(A) n(f_s \circ \phi_s)^{-1}(A) \subset \mathbb{R}^n is open. But this means equivalently that for every such ϕ\phi, fϕf \circ \phi is continuous. This, in turn, means equivalently that the same underlying function f sf_s constitutes a smooth function f˜:XCdfflg(Y)\widetilde f \;\colon\; X \longrightarrow Cdfflg(Y).

In summary, we thus have a bijection of hom-sets

Hom(Dtplg(X),Y) Hom(X,Cdfflg(Y)) f s (f˜) s=f s \array{ Hom( Dtplg(X), Y ) &\simeq& Hom(X, Cdfflg(Y)) \\ f_s &\mapsto& (\widetilde f)_s = f_s }

given simply as the identity on the underlying functions of underlying sets. This makes it immediate that this hom-isomorphism is natural in XX and YY and this establishes the adjunction.

Next, to see that the D-topological spaces are the fixed points of this adjunction, we apply the above natural bijection on hom-sets to the case

Hom(Dtplg(Cdfflg(Z)),Y) Hom(Cdfflg(Z),Cdfflg(Y)) (ϵ Z) s (id) s \array{ Hom( Dtplg(Cdfflg(Z)), Y ) &\simeq& Hom(Cdfflg(Z), Cdfflg(Y)) \\ (\epsilon_Z)_s &\mapsto& (\mathrm{id})_s }

to find that the counit of the adjunction

Dtplg(Cdfflg(X))ϵ XX Dtplg(Cdfflg(X)) \overset{\epsilon_X}{\longrightarrow} X

is given by the identity function on the underlying sets (ϵ X) s=id (X s)(\epsilon_X)_s = id_{(X_s)}.

Therefore η X\eta_X is an isomorphism, namely a homeomorphism, precisely if the open subsets of X sX_s with respect to the topology on XX are precisely those with respect to the topology on Dtplg(Cdfflg(X))Dtplg(Cdfflg(X)), which means equivalently that the open subsets of XX coincide with those whose pre-images under all continuous functions ϕ: nX\phi \colon \mathbb{R}^n \to X are open. This means equivalently that XX is a D-topological space.

Finally, to see that we have an idempotent adjunction, we check that the comonad

DtplgCdfflg:TopologicalSpacesTopologicalSpaces Dtplg \circ Cdfflg \;\colon\; TopologicalSpaces \to TopologicalSpaces

is an idempotent comonad, hence that

DtplgCdfflgDtplgηCdfflgDtplgCdfflgDtplgCdfflg Dtplg \circ Cdfflg \overset{ Dtplg \cdot \eta \cdot Cdfflg }{\longrightarrow} Dtplg \circ Cdfflg \circ Dtplg \circ Cdfflg

is a natural isomorphism. But, as before for the adjunction counit ϵ\epsilon, we have that also the adjunction unit η\eta is the identity function on the underlying sets. Therefore, this being a natural isomorphism is equivalent to the operation of passing to the D-topological refinement of the topology of a topological space being an idempotent operation, which is clearly the case.

Topological homotopy type and diffeological shape

Definition

(diffeological singular simplicial set)

Consider the simplicial diffeological space

Δ Δ diff DiffeologicalSpaces [n] Δ diff n{x n+1|ix i=1} \array{ \Delta & \overset{ \Delta^\bullet_{diff} }{ \longrightarrow } & DiffeologicalSpaces \\ [n] &\mapsto& \Delta^n_{diff} \mathrlap{ \coloneqq \big\{ \vec x \in \mathbb{R}^{n+1} \;\vert\; \underset{i}{\sum} x^i = 1 \big\} } }

which in degree nn is the standard extended n-simplex inside Cartesian space n+1\mathbb{R}^{n+1}, equipped with its sub-diffeology.

This induces a nerve and realization adjunction between diffeological spaces and simplicial sets:

DiffeologicalSpacesAAAASing diff|| diffSimplicialSets, DiffeologicalSpaces \underoverset { \underset{Sing_{\mathrlap{diff}}}{\longrightarrow} } { \overset{ \left\vert - \right\vert_{\mathrlap{diff}} }{\longleftarrow} } { \phantom{AA}\bot\phantom{AA} } SimplicialSets \,,

where the right adjoint is the diffeological singular simplicial set functor Sing diffSing_{diff}.

(e.g. Christensen-Wu 13, Def. 4.3)

Remark

(diffeological singular simplicial set as path ∞-groupoid)

Regarding simplicial sets as presenting ∞-groupoids, we may think of Sing diff(X)Sing_{diff}(X) (Def. ) as the path ∞-groupoid of the diffeological space XX.

In fact, by the discussion at shape via cohesive path ∞-groupoid we have that Sing diffSing_{diff} is equvialent to the shape of diffeological spaces regarded as objects of the cohesive (∞,1)-topos of smooth ∞-groupoids:

Sing diffShpi:DiffeologicalSpacesiSmoothGroupoids ShapeGroupoids Sing_{diff} \;\simeq\; Shp \circ i \;\;\colon\;\; DiffeologicalSpaces \overset{i}{\hookrightarrow} SmoothGroupoids_{\infty} \overset{Shape}{\longrightarrow} Groupoids_\infty
Proposition

(topological homotopy type is cohesive shape of continuous diffeology)

For every XX \in TopologicalSpaces, the cohesive shape/path ∞-groupoid presented by its diffeological singular simplicial set (Def. , Remark ) of its continuous diffeology is equivalent to the homotopy type of XX presented by the ordinary singular simplicial set:

Sing diff(Cdfflg(X)) wheSing(X). Sing_{diff} \big( Cdfflg(X) \big) \;\simeq_{whe}\; Sing(X) \,.

(Christensen-Wu 13, Prop. 4.14)

Model category structure

Proposition

The category of Δ\Delta-generated spaces carries the structure of a cofibrantly generated model category with the same generating (acyclic) cofibrations as for the classical model structure on topological spaces and such that the coreflection into all TopologicalSpaces (Prop. ) is a Quillen equivalence to the classical model structure on topological spaces.

(Haraguchi 13, Theorem 3.3)

References

General

Δ\Delta-generated spaces were originally proposed by Jeff Smith as a nice category of spaces for homotopy theory.

A proof that the category of Δ\Delta-generated spaces is locally presentable is in:

See also at directed homotopy theory.

The model structure on Delta-generated topological spaces Quillen equivalent to the classical model structure on topological spaces is due to

  • Tadayuki Haraguchi, On model structure for coreflective subcategories of a model category, Math. J. Okayama Univ.57(2015), 79–84 (arXiv:1304.3622, MR3289294, Zbl 1311.55027)

Relation to diffeological spaces

Relation to diffeological spaces:

Last revised on June 8, 2020 at 04:58:33. See the history of this page for a list of all contributions to it.