Given a connected topological space $X$, its space of path $PX$ is again a topological space. On the other hand, for a simplicial set $K\_{\bullet}$, its path space is given by 
$$
PK\_{n}:=\operatorname{Map}\_{sSet}(\Delta[1], K)\_{n}.
$$
Here my question: given a simplicial topological space $X\_{\bullet}$, what is its "path space" $PX\_{\bullet}$ (as a simplicicial topological space)? Do you know some candidates? 



Path space of a simplicial topological space?
It is sensible to restrict to simplicial {\em based} spaces $X\_\*$ and then apply the functor $P$ levelwise. This is used, for example, to compare $|\Omega X\_\*|$ with $\Omega |X\_\*|$ in The Geometry of iterated loop spaces <http://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf>


