nLab model structure on diffeological spaces

Idea

There may be different model category-structures on the category of diffeological spaces.

Of practical interest would be a model structure whose weak equivalences are the isomorphisms on standard smooth homotopy groups of diffeological spaces, i.e. the weak equivalences between the cohesive shapes of diffeological spaces regarded as the concrete objects in the cohesive (∞,1)-topos of smooth ∞-groupoids. By the discussion at shape via cohesive path ∞-groupoid these are detected by smooth functions out of simplices or cubes with their canonical diffeological structure (as discussed in Christensen-Wu 14).

A model category with this property has been claimed in Haraguchi-Shimakawa 13,, Haraguchi-Shimakawa 20

Another model category structure is discussed in Kihara 16, but this uses a non-standard diffeology on simplices (to enforce that all smooth singular simplicial complexes are fibrant objects).

Details

The approach of Haraguchi-Shimakawa 13, Haraguchi-Shimakawa 20 proceeds as follows:

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.

References

A proof of a model structure on diffeological spaces, with weak equivalences detected on standard smooth homotopy groups, is claimed in

• Tadayuki Haraguchi, Kazuhisa Shimakawa, A model structure on the category of diffeological spaces (arXiv:1311.5668)

though Kihara 16 writes (p. 2) that

there exists a gap in the proof of Haraguchi-Shimakawa 13, Theorem 5.6

This gap has been filled in

• Tadayuki Haraguchi, Homotopy structures of smooth CW complexes (arXiv:1811.06175)

• Tadayuki Haraguchi, Kazuhisa Shimakawa, A model structure on the category of diffeological spaces, I, Mathematical Journal of Okayama University (2024) [arXiv:2011.12842]

A different model structure, however not using the standard smooth homotopy groups(!), is claimed (and published) in

• Hiroshi Kihara, Model category of diffeological spaces, Journal of Homotopy and Related Structures, (2018), 1-40 (arXiv:1605.06794)

See also

• J. Daniel Christensen, Enxin Wu, The homotopy theory of diffeological spaces, I. Fibrant and cofibrant objects, New York J. Math. 20 (2014), 1269-1303 (arXiv:1311.6394)

• Hiroshi Kihara, Quillen equivalences between the model categories of smooth spaces, simplicial sets, and arc-gengerated spaces (arXiv:1702.04070)

• Tadayuki Haraguchi, Kazuhisa Shimakawa, On homotopy types of diffeological cell complexes (arXiv:1912.05359)