(adjunction between topological spaces and diffeological spaces)
There is a pair of adjoint functors
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:
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):
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:
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.
These adjunctions and their properties are observed in Shimakawa-Yoshida-Haraguchi 10, Prop. 3.1, Prop. 3.2, Lemma 3.3. 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, but this may have a gap).
We spell out the existence of the idempotent adjunction (1):
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
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
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
to find that the counit of the adjunction
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, we check that the comonad
is an idempotent comonad, hence that
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.
Last revised on November 26, 2020 at 06:32:20. See the history of this page for a list of all contributions to it.