Consider the *discrete* monoid $M$ of nondecreasing continuous maps from $[0,1]$ to itself preserving the extremities. Note that the monoid is right-cancellative ($x.z=y.z$ implies $x=y$ since $z$ is always onto).

> What do we know about the homotopy type of this monoid (viewed as a
> one-object category) ? In particular, about its homotopy groups ?

My background on this subject is very small. By a paper from Dusa McDuff ([On the classifying space of discrete monoids][1]), every path-connected space has the same homotopy type as the classifying space of some monoid. And the fondamental group of $BM$ is the groupification of $M$. 



  [1]: https://doi.org/10.1016/0040-9383(79)90022-3