0
$\begingroup$

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 )$?

$\endgroup$
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.