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Stub for Delta-generated space.
I added several recent references.
I have cross-linked with directed homotopy theory
I corrected a Unicode problem in Jiří Rosický’s name.
Curiously, his name does not redirect to the article Jiří Rosický, even though it is hyperlinked.
Instead, clicking on the hyperlinks attempts to create a new page with the same name (!).
Thanks for the alert. I have now copy-and-pasted the name from the entry title into the entry. The rendering looks the same, but now the link works, so probably there was some unicode ambiguity at play, or the like. (?) A similar issue still happens with hyphens here and there.
Re #5: I think the nLab does not normalize Unicode characters. As a result, the same letter ř, say, can be encoded in two different ways: as a single code point and as r followed by a combining character.
I suggest that we add a Unicode normalization step during the CGI processing of submitted forms.
I have added (here) statement of the proposition that Delta-generated spaces are the fixed points of the adjunction between topological spaces and diffeological spaces
Would it add anything worthwhile to describe Delta-generated spaces as those diffeological spaces for which the unit is an isomorphism?
I have added a remark (here) that $\Delta$-generated is the same as “Euclidean generated”:
For each $n$ the topological simplex $\Delta^n$ is a retract of the ambient Euclidean space/Cartesian space $\mathbb{R}^n$ (as a non-empty convex subset of a Euclidean space it is in fact an absolute retract). Hence the identity function on $\Delta^n$ factors as
$id \;\colon\; \Delta^n_{top} \overset{i_n}{\hookrightarrow} \mathbb{R}^n \overset{p_n}{\longrightarrow} \Delta^n_{top} \,.$It follows that every continuous function $f$ with domain the topological simplex extends as a continuous function to Euclidean space:
$\array{ \Delta^m_{top} &\overset{f}{\longrightarrow}& X \\ \mathllap{{}^{i_n}}\big\downarrow & \nearrow _{\mathrlap{\exists}} \\ \mathbb{R}^n }$Therefore the condition that a topological space $X$ be $\Delta$-generated (Def. \ref{DeltaGeneratedSpace}) is equivalent to saying that its topology is final with respect to all continuous functions $\mathbb{R}^n \to X$ out of Euclidean/Cartesian spaces.
We might thus equivalently speak of Euclidean-generated spaces.
I have expanded the statement about convenience (here). Currently it reads as follows:
The category of $\Delta$-generated spaces (Def. \ref{DeltaGeneratedSpace}) is a convenient category of topological spaces in that:
it contains all CW-complexes (SYH 10, Cor. 4.4),
it is complete and cocomplete category (SYH 10, Prop. 3.4),
it is cartesian closed (SYH 10, Cor. 4.6):
its internal homs $[X,Y]$ are given by the D-topology of the internal homs in diffeological spaces between continuous diffeologies (SYH 10, Prop. 4.7):
$[X,Y] \;\coloneqq\; Dtplg\big( [ Cdfflg(X), Cdfflg(Y) ] \big)$it is locally presentable (FR 08, Cor. 3.7)
Do we have an example that shows that Δ-generated spaces are not locally cartesian closed?
The quasitopos of diffeological spaces is locally cartesian closed, and Δ-generated spaces are reflective inside all diffeological spaces. Does this tell us anything about local cartesian closedness of Δ-generated spaces?
added this pointer, for relation to k-spaces:
I have moved the Properties-section “As a convenient category of topological spaces” from the first to the last in the list of subsections (now here), so that all the statements about diffeological homotopy type can be referred to, and then I added the remark that things become ever more convenient by further embedding into diffeological spaces, then smooth sets, then smooth $\infty$-groupoids, and in such a way that the canonical shape modality of the latter still sees the correct homotopy type of all topological spaces.
(This addition is copied over from part of a similar edit that I just made at convenient category of topological spaces as announced here.)
I have enhanced the discussion of the cartesian closure (here):
Made explicit the pleasing fact that we have two equivalent formulations of the internal hom, corresponding to the left and to the right side of the defining idempotent adjunction, namely
$Maps_{DTop}(X,Y) \;\simeq\; Cdfflg \big( Maps_{Top} ( X ,\, Y ) \big) \,,$(which follows already from Vogt 1971) but also
$Maps_{DTop}(X,Y) \;\simeq\; Dtplg \big( Maps_{Dfflg} ( X ,\, Y ) \big) \,.$Shimakawa & Haraguchi actually prove something stronger, part of which says that at least if the domain is a CW-complex, then the correct homotopy mapping space is also given by the internal hom formed in diffeological spaces (this might also want to go there, but now I have stated it here):
$X \,\in\, CWComplex \hookrightarrow DTopSp \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \underset{ A \,\in\, DTopSp }{\forall} Maps_{Dfflg} \big( X ,\, A \big) \;\; \simeq Cdfflg \, Maps_{Top} \left( X ,\, A \right) \,.$This follows by combining two separate statements they make (as I have tried to make explicit in the entry now).
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