nLab
model structure on diffeological spaces

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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

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Contents

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

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(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).

Proof

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.

References

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

though Kihara 16 writes (p. 2) that

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

This gap is addressed in

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

See also

Last revised on November 26, 2020 at 06:33:30. See the history of this page for a list of all contributions to it.