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.?)