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If you mean by cofibrant: m-cofibrant. Yes they are. The category $\mathcal{P}$ has for objects the strictly positive real numbers and $\mathcal{P}(\ell,\ell')$ is the space of surjective nondecreasing maps from $[0,\ell]$ to $[0,\ell']$, space which is contractible. And your second condition holds as well. As for the calculation, I want to lift some model structure along a right adjoint and I want to apply the Quillen Path Object argument. Maybe I will ask another question if I can't use your result.
I don't know if you are interested by the opposite case: the left adjoint $\rm{SemiCat}\to \rm{Cat}$ from small semicategories to small categories which adds identity maps does not left induce on $\rm{SemiCat}$ a model structure by starting from the canonical model structure of $\rm{Cat}$.
@stackExchangeUser Your quantities $A$, $B$ etc... have no physical meaning. As I said, to obtain e.g. the Planck length, we observe that there is only one possibility to obtain a length using the Planck constant $ML^2T^{-1}$, the Gravitation constant $M^{-1}L^3T^{-2}$ and the speed of light $LT^{-1}$ by solving a $3\times 3$ linear system. This fact is interpreted as a mathematical formula which would be the consequence of an (unknown) physics under the Planck scale.
There are no physical reasons to consider a quantity like $LT^{-1}+M$. On the contrary, in physics, there are many speculations based on playing with dimensions. It is how the Planck time and the Planck length are introduced : by supposing that an unknown theory would produce as a byproduct the Planck time and the Planck length which, because of their values, are interpreted as the minimal time and length if the spacetime was discrete (which does not prove of course that the spacetime is discrete). Since physics under the Planck length is unknown, it is just speculation.
