# contracting homotopy on simplicial sets

Let $X$ be a topological space and let $PX$ be its space of paths. Let $I=[0,1]$ with coordinate $s$. There is an homotopy $$F\: : \: I\times PM\to PM$$ Defined by $F(s,y)(t):=y(st)$. This map is an homotopy between $f_{0}\: : \: PX\to PX$ , $(f_{0}(y))(t)=y(0)$ and the identity. Let $K$ be a simplicial set. I would like to find a map between simplicial sets $$F\: : \: \Delta\times Map_{sSet}(\Delta,K )\to Map_{sSet}(\Delta,K )$$ which is an homotopy between the identity and the "constant path at the initial point". Do you have some ideas? Do I use a fibrant simplicial set $K$? Is it correct to define the path space of a simplicial set $K$ with $Map_{sSet}(\Delta,K )$?

$Map(\Delta,X)\to Map(\Delta,Map(\Delta,X)) = Map (\Delta\times \Delta,X)$
Then this comes from precomposition of the map $\Delta\times \Delta\to \Delta$ sending $(1,1)$ to $1$ and the other vertices to 0.
In general when you map something to a simplicial set you want the target to be fibrant (i.e. a Kan complex). In this case the naive'' mapping space $Map(\Delta,X)$ gives the correct homotopy type, but that's because the pathspace deformation retracts onto the constant paths.