nLab Delta-generated topological space




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

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


Basic concepts

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Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory




(Δ\Delta-Generated spaces)

A Δ\Delta-generated space (alias numerically 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 those 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,


(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).


(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} \,;

and 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.


(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”.


Coreflection into all topological spaces


(adjunction between topological spaces and diffeological spaces)

There is a pair of adjoint functors

(1)TopSpAAAACdfflgDtplgDifflgSp TopSp \underoverset{ \underset{ Cdfflg }{\longrightarrow} }{ \overset{ Dtplg }{\longleftarrow} }{\phantom{AA}\bot\phantom{AA}} DifflgSp

between the categories of TopologicalSpaces and of DiffeologicalSpaces, where


  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(Cdfflg(X))ϵ XX X \;\,\text{is}\;\Delta\text{-generated} \;\;\;\;\; \Leftrightarrow \;\;\;\;\; Dtplg(Cdfflg(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:

(2)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 was a gap in the original proof that DTopologicalSpaces QuillenDiffeologicalSpacesDTopologicalSpaces \simeq_{Quillen} DiffeologicalSpaces. The gap is claimed to be filled now, see the commented references here.

Essentially these adjunctions and their properties are observed in Shimakawa, Yoshida & Haraguchi 2010, Prop. 3.1, Prop. 3.2, Lem. 3.3, see also Christensen, Sinnamon & Wu 2014, Sec. 3.2. 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).


We spell out the existence of the idempotent adjunction (2):

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, it is sufficient to check (by this Prop.) 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


(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:

(3)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)


(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


(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 naturally\,weak homotopy equivalent to the homotopy type of XX presented by the ordinary singular simplicial set:

Sing diff(Cdfflg(X))W whSing(X). Sing_{diff} \big( Cdfflg(X) \big) \underoverset { \in \mathrm{W}_{wh} } {} {\longrightarrow} Sing(X) \,.

(Christensen & Wu 2013, Prop. 4.14)

Model category structure


(model structure on Delta-generated topological spaces)
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)

This Quillen equivalence factors through the model structure on compactly generated topological spaces (e.g. Gaucher 2007, p. 7):

Top Qu QukkTop Qu QuDDTop Qu. Top_{Qu} \underoverset { \underset{ k }{\longrightarrow} } { {\hookleftarrow} } { \;\;\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;\;\; } k Top_{Qu} \underoverset { \underset{ D }{\longrightarrow} } { {\hookleftarrow} } { \;\;\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;\;\; } D Top_{Qu} \,.

As a convenient category of topological spaces


The category of Euclidean-generated spaces/Δ\Delta-generated spaces (Def. ) is a Cartesian closed category.

Explicitly, the internal hom Maps DTopMaps_{DTop} is, equivalently:

  • the image under CdfflgCdfflg (1) of the mapping space Maps Top(X,Y)Maps_{Top}(X,Y) with the compact-open topology:

    Maps DTop(X,Y)Cdfflg(Maps Top(X,Y)), Maps_{DTop}(X,Y) \;\simeq\; Cdfflg \big( Maps_{Top} ( X ,\, Y ) \big) \,,
  • the image under Dtplg Dtplg (1) of the internal hom Maps DfflgMaps_{Dfflg} formed in diffeological spaces:

    Maps DTop(X,Y)Dtplg(Maps Dfflg(X,Y)). Maps_{DTop}(X,Y) \;\simeq\; Dtplg \big( Maps_{Dfflg} ( X ,\, Y ) \big) \,.

The first statement is a special case of Vogt 1971, Thm. 3.6, as highlighted in Gaucher 2007, Sec. 2. The second statement is the image of SYH 10, Prop. 4.7 under DtplgDtplg (using that ν=TD\nu = T \circ D, in their notation from p. 4).

In fact, SYH 10, Prop. 4.7 state something stronger, topologically characterizing Maps Dfflg(X,Y)Maps_{Dfflg}(X,Y) even before applying DtplgDtplg to it. This stronger statement has a nice form when specialized to CW-complexes:


The category of Euclidean-generated spaces/Δ\Delta-generated spaces (Def. ) contains all CW-complexes.

(SYH 10, Cor. 4.4)


(diffeological internal hom out of CW-complexes)

If XX is a CW-complex, regarded as an object in DTopSpDTopSp via Prop. , then for every ADTopSpA \,\in\, DTopSp their internal hom formed in diffeological spaces is isomorphic to the image under CdfflCdffl (1) of the mapping space Maps TopMaps_{Top} with its compact open topology:

XCWComplexDTopSpADTopSpMaps Dfflg(X,A)CdfflgMaps Top(X,A). X \,\in\, CWComplex \hookrightarrow DTopSp \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \underset{ A \,\in\, DTopSp }{\forall} Maps_{Dfflg} \big( X ,\, A \big) \;\; \simeq Cdfflg \, Maps_{Top} \left( X ,\, A \right) \,.


This is the following combination of statements from SYH 10:

  1. Prop. 4.7 there says that, in general:

    Maps Dfflg(X,A)Cdfflgl(smap(X,Y)) Maps_{Dfflg}\big( X ,\, A\big) \;\simeq\; Cdfflgl \big( \mathbf{smap}(X,Y) \big)

    (where “smap\mathbf{smap}” is defined on their p. 6),

  2. Prop. 4.3 there implies that, when XX is a CW-complex, as assumed here:

    Cdfflg(smap(X,Y))Cdfflg(Maps Top(X,Y)). Cdfflg \big( \mathbf{smap}(X,Y) \big) \;\simeq\; Cdfflg \big( Maps_{Top}(X,Y) \big) \,.

In summary:


(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:

(by Vogt 1971; SYH 10, Prop. 3.4)

Moreover, in further summary of the discussion further above, this convenient category of topological spaces is:

  1. a full subcategory of the quasi-topos of diffeological spaces (see there),

  2. which is in turn a full subcategory of the cohesive topos of smooth sets (see there);

  3. which in turn is a full sub- ( , 1 ) (\infty,1) -category of the cohesive ( , 1 ) (\infty,1) -topos of smooth \infty -groupoids

such that the canonical shape modality (the smooth path ∞-groupoid construction) still sees the correct underlying homotopy type of topological spaces (SS20, Ex. 3.18, see also at model structure on Delta-generated topological spaces):

From SS21



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)

Discussion in the generality of subcategory-generated spaces, including compactly generated topological spaces:

  • Philippe Gaucher, Section 2 of: Homotopical interpretation of globular complex by multipointed d-space, Theory and Applications of Categories, vol. 22, number 22, 588-621, 2009 (arXiv:0710.3553)

along the lines of

Discussion about the q-model structure, the m-model structure, the h-model structure and the notion of Δ\Delta-Hausdorff Δ\Delta-generated space (a natural separation condition for Δ\Delta-generated spaces) in:

Relation to diffeological spaces

Relation to diffeological spaces:

Last revised on November 4, 2022 at 08:19:58. See the history of this page for a list of all contributions to it.