nLab D-topology

Idea

Given a diffeological space, the D-topology (“diffeological topology”) is a natural topology on its underlying set of points, namely the final topology which makes all its plots be continuous functions.

Definition

Definition

Given a diffeological space $(X, plots)$, with set of $plots \,\subset\, \underset{n \in \mathbb{N}}{\coprod} Set(\mathbb{R}^n,\, X)$, the D-topology $\tau_D$ on the underlying set $X$ is the final topology on $X$ with respect to the $plots$.

This means that $\tau_D \,\subset\, Sub(X)$ is the topology in which a subset $U \subset X$ is open if and only if under each $\big(\mathbb{R}^n \xrightarrow{\phi} X \big) \,\in\, plots$ its preimage $\phi^{-1}(U) \subset \mathbb{R}^n$ is open in the Euclidean topology on the plot’s domain.

(Iglesias-Zemmour 1985, Def. 1.2.3, review in IglesiasZemmour 2013, Sec. 2.8, CSW 2013, Sec. 3)

Properties

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:

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

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.

Related concepts

• D-topological space

References

The concept originates with:

• Patrick Iglesias-Zemmour, Def. 1.2.3 in: Fibrations difféologiques et Homotopie, st. doct. thesis (1985) (web, pdf, pdf)

• Patrick Iglesias-Zemmour, Section 2.8 of: Diffeology, Mathematical Surveys and Monographs, AMS (2013) (web, ISBN:978-0-8218-9131-5)

The relation to Delta-generated topological spaces:

• Kazuhisa Shimakawa, K. Yoshida, Tadayuki Haraguchi, Homology and cohomology via enriched bifunctors, Kyushu Journal of Mathematics, 2018 Volume 72 Issue 2 Pages 239-252 (arXiv:1010.3336, doi:10.2206/kyushujm.72.239)

Discussion of the D-topology for mapping spaces:

• J. Daniel Christensen, Gord Sinnamon, Enxin Wu, The $D$-topology for diffeological spaces, Pacific Journal of Mathematics 272 (1), 87-110, 2014 (arXiv:1302.2935, doi:10.2140/pjm.2014.272.87)