Contents

# Contents

## Definition

###### Definition

($\Delta$-Generated spaces)

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

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

###### Remark

(as colimits of topological simplices)

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

###### Remark

(as Euclidean-generated spaces)

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

$id \;\colon\; \Delta^n_{top} \overset{i_n}{\hookrightarrow} \mathbb{R}^n \overset{p_n}{\longrightarrow} \Delta^n_{top} \,.$

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

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

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

###### Remark

(as D-topological spaces)

By Prop. below, the Euclidean-generated spaces and hence, by Remark , the $\Delta$-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

### Coreflection into all topological spaces

###### Proposition

(adjunction between topological spaces and diffeological spaces)

There is a pair of adjoint functors

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

between the categories of TopologicalSpaces and of DiffeologicalSpaces, where

• $Cdfflg$ takes a topological space $X$ to the continuous diffeology, namely the diffeological space on the same underlying set $X_s$ whose plots $U_s \to X_s$ are the continuous functions (from the underlying topological space of the domain $U$).

• $Dtplg$ takes a diffeological space to the diffeological topology (D-topology), namely the topological space with the same underlying set $X_s$ and with the final topology that makes all its plots $U_{s} \to X_{s}$ into continuous functions: called the D-topology.

Hence a subset $O \subset \flat X$ is an open subset in the D-topology precisely if for each plot $f \colon U \to X$ the preimage $f^{-1}(O) \subset U$ is an open subset in the Cartesian space $U$.

Moreover:

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

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

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

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

Caution: There was a gap in the original proof that $DTopologicalSpaces \simeq_{Quillen} DiffeologicalSpaces$. 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).

###### Proof

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

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

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

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

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

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

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

given simply as the identity on the underlying functions of underlying sets. This makes it immediate that this hom-isomorphism is natural in $X$ and $Y$ 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

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

to find that the counit of the adjunction

$Dtplg(Cdfflg(X)) \overset{\epsilon_X}{\longrightarrow} X$

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

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

$Dtplg \circ Cdfflg \;\colon\; TopologicalSpaces \to TopologicalSpaces$

is an idempotent comonad, hence that

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

is a natural isomorphism. But, as before for the adjunction counit $\epsilon$, we have that also the adjunction unit $\eta$ 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

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

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

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

(3)$DiffeologicalSpaces \underoverset { \underset{Sing_{\mathrlap{diff}}}{\longrightarrow} } { \overset{ \left\vert - \right\vert_{\mathrlap{diff}} }{\longleftarrow} } { \phantom{AA}\bot\phantom{AA} } SimplicialSets \,,$

where the right adjoint is the diffeological singular simplicial set functor $Sing_{diff}$.

###### Remark

(diffeological singular simplicial set as path ∞-groupoid)

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

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

$Sing_{diff} \;\simeq\; Shp \circ i \;\;\colon\;\; DiffeologicalSpaces \overset{i}{\hookrightarrow} SmoothGroupoids_{\infty} \overset{Shape}{\longrightarrow} Groupoids_\infty$

###### Proposition

(topological homotopy type is cohesive shape of continuous diffeology)
For every $X \in$ 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 $X$ presented by the ordinary singular simplicial set:

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

### Model category structure

###### Proposition

(model structure on Delta-generated topological spaces)
The category of $\Delta$-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.

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

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

### As a convenient category of topological spaces

###### Proposition

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

Explicitly, the internal hom $Maps_{DTop}$ is, equivalently:

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 $Dtplg$ (using that $\nu = T \circ D$, in their notation from p. 4).

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

###### Proposition

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

(SYH 10, Cor. 4.4)

###### Proposition

(diffeological internal hom out of CW-complexes)

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

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

###### Proof

This is the following combination of statements from SYH 10:

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

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

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

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

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

In summary:

###### Proposition

(Euclidean-generated spaces are convenient)

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

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);

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):

### General

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

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

• 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 $\Delta$-Hausdorff $\Delta$-generated space (a natural separation condition for $\Delta$-generated spaces) in:

### Relation to diffeological spaces

Relation to diffeological spaces: