chatGPT provides a different answer : "Yes, mathematicians take Grothendieck's theory of "motives" very seriously. In fact, the theory of motives is one of the most active areas of research in algebraic geometry and number theory today. (...)"

ncatlab.org/nlab/show/HomePage

@AndreasBlass You're perfectly right. I use strictly increasing only to prove specific properties of the "topologification" of $f$.

@AlexanderCampbell In my case, $X:I\to M$ is already injective cofibrant indeed. Could you explain the reason please ?

@PeterTaylor In the language of directed homotopy, the associated map from $[0,1]^n$ to itself not only will take a directed path to a directed path, but also the $L_1$-arc length from $0^n$ will be preserved (here it coincides with the distance for the $d_1$ metric): different words for the same phenomenon.

@PeterTaylor I don't know what the Hamming distance is but all such $f$ have the property that $\epsilon_1+\dots+\epsilon_n=f(\epsilon_1)+\dots+f(\epsilon_n)$.