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Finitely cocomplete categories of compact Hausdorff spaces
@Zhen Lin: That is a very good point. I will change the question accordingly. Thank you very much.
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Finitely cocomplete categories of compact Hausdorff spaces
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Is there a general theory of fiber theorems?
Two related results have to do with cell-like maps. A cell-like map is one in which the fibres are cell-like spaces. These are compact spaces which are contractible in a very strong sense. Result 1: A cell-like map is exactly a map $f:X\to Y$ such that any restriction $f:f^{-1}(U)\to U$ is a proper homotopy equivalence for any $U$ open in $Y$. Result 2: even more interestingly, cell-like maps between several types of manifold-like spaces are approximable by (and homotopic to) homeomorphisms. See mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0111.0128.ocr.pdf for a nice overview.
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Finitely cocomplete categories of compact Hausdorff spaces
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Multisimplicial geometric realization
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Multisimplicial geometric realization
I know the above example is not Hausdorff, so certainly not compactly generated Hausdorff, but it is the best I could come up with. However, it does not seem to me that assuming that X is compactly generated and Hausdorff would be sufficient to establish the NDR property either.
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Multisimplicial geometric realization
(continuation) Further, let $X=[0,1]^\delta$ be the unit interval (or any other set with at least two points, really) with the indiscrete topology: it has only two open subsets, so it is certainly not Hausdorff. Then $X$ deformation retracts to
$\{0\}$
, and the homotopy underlying one such deformation retraction gives a map $H:X\to\operatorname{Map}(I,X)$ such that $H(x)\in s(X)$ iff $x=0$. In conclusion, the composite $f\circ H: X\to I$ is a non-constant continuous map. This is not possible because $X$ is indiscrete and has more than two points. Please let me know if I made a mistake.
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Multisimplicial geometric realization
I absolutely apologize for prolonging this discussion into point-set topology. However, it seems to me that one requires non-trivial conditions on the space $X$ for the NDR condition that Peter states to hold ($X$ metrizable would certainly suffice). Perhaps that is what Peter means, but I acknowledge I am a bit dense. Let us consider the simplest case: the single degeneracy map $\newcommand{\Map}{\operatorname{Map}} s:X=\Map(\Delta^0,X)\to\Map(\Delta^1,X)=\Map(I,X)$. Assume there exists a map $f:\Map(I,X)\to I$ such that $f^{-1}(0)=s(X)$ is the image of the degeneracy. (to be continued)
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Multisimplicial geometric realization
@Peter: Thank you. I actually saw that answer two days ago. I certainly did not mean to imply you were unaware of the technicalities. My comment/answer above was mostly for my own benefit and, to a small extent, perhaps that of some future reader. Also, thank you very much for pointing out the error in my answer. I will correct it by reversing the arrows when I get the chance.
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Multisimplicial geometric realization
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Multisimplicial geometric realization
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Multisimplicial geometric realization
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Multisimplicial geometric realization
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Multisimplicial geometric realization
I'm pretty sure Tom meant "the $(n,\ldots,n)$-multisimplices" instead of "the $(n_1,\ldots,n_k)$-multisimplices" in his comment.
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Probing a manifold with closed curves
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What are some triangulations of Grassmannians?
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