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Is there a way to characterise (up to isomorphism) the cosimplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a point? ${}$

If the above class is not too general, can - at least - the standard cosimplicial space, given by $\Delta_n = \{(t_0, \dots, t_n) \in \mathbb{R}^{n+1} \: | \: t_i \geq 0, \sum t_i = 1 \} $ and the usual maps, be singled out in some way? (Or, if the class above is too general, may you suggest a more workable refinement using say 'convexity', 'connectedness', etc.?)

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  • $\begingroup$ cross-listed: math.stackexchange.com/questions/575382/… $\endgroup$ Nov 26, 2013 at 23:22
  • $\begingroup$ No bites on math.se. $\endgroup$ Nov 26, 2013 at 23:22
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    $\begingroup$ This question seems hopelessly general--for instance, it is at least as difficult as classifying arbitrary compact spaces that can embed in some $\mathbb{R}^n$. What you refer to as "the standard simplicial space" is not a simplicial space but a cosimplicial space. $\endgroup$ Nov 26, 2013 at 23:31
  • $\begingroup$ @Eric: yes, cosimplicial space! Thank you. Fixed now. $\endgroup$ Nov 26, 2013 at 23:38
  • $\begingroup$ @Eric: regarding the generality, I figured as much. But as per my backup question, can you suggest a less general class? $\endgroup$ Nov 26, 2013 at 23:43

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