contracting homotopy on simplicial sets

Question

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 $$ F\: : \: \Delta[1]\times Map_{sSet}(\Delta[1],K )\to Map_{sSet}(\Delta[1],K ) $$ 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[1],K )$?

Answer 1

It is easier to define the map you're looking for if you adjoint it over and define a map

$Map(\Delta[1],X)\to Map(\Delta[1],Map(\Delta[1],X)) = Map (\Delta[1]\times \Delta[1],X)$

Then this comes from precomposition of the map $\Delta[1]\times \Delta[1]\to \Delta[1]$ 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 $Map(\Delta[1],X)$ gives the correct homotopy type, but that's because the pathspace deformation retracts onto the constant paths.