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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.
Given a diffeological space , with set of , the D-topology on the underlying set is the final topology on with respect to the .
This means that is the topology in which a subset is open if and only if under each its preimage 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)
(adjunction between topological spaces and diffeological spaces)
There is a pair of adjoint functors
between the categories of TopologicalSpaces and of DiffeologicalSpaces, where
takes a topological space to the continuous diffeology, namely the diffeological space on the same underlying set whose plots are the continuous functions (from the underlying topological space of the domain ).
takes a diffeological space to the diffeological topology (D-topology), namely the topological space with the same underlying set and with the final topology that makes all its plots into continuous functions: called the D-topology.
Hence a subset is an open subset in the D-topology precisely if for each plot the preimage is an open subset in the Cartesian space .
Moreover:
the fixed points of this adjunction 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 -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 . 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).
We spell out the existence of the idempotent adjunction (2):
First, to see we have an adjunction , we check the hom-isomorphism (here).
Let and . Write for the underlying sets. Then a morphism, hence a continuous function of the form
is a function of the underlying sets such that for every open subset and every smooth function of the form the preimage is open. But this means equivalently that for every such , is continuous. This, in turn, means equivalently that the same underlying function constitutes a smooth function .
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 and 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 .
Therefore is an isomorphism, namely a homeomorphism, precisely if the open subsets of with respect to the topology on are precisely those with respect to the topology on , which means equivalently that the open subsets of coincide with those whose pre-images under all continuous functions are open. This means equivalently that 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
is an idempotent comonad, hence that
is a natural isomorphism. But, as before for the adjunction counit , we have that also the adjunction unit 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.
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:
Discussion of the D-topology for mapping spaces:
Last revised on December 21, 2021 at 10:51:44. See the history of this page for a list of all contributions to it.