The *homotopy dimension* $h \dim X$ of a space $X$ is the minimal dimension of a CW-complex $Z$ homotopy equivalent to $X$.

I am interested in the generalisation to maps $f\colon X\to Y$. Here's what I think it should be:

The

homotopy dimensionof $f\colon X\to Y$ is the smallest $k$ such that $f$ factorises through a $k$-dimensional CW-complex up to homotopy (meaning there is a $Z$ of dimension $k$ and maps $g\colon X\to Z$ and $h\colon Z\to Y$ with $f\simeq h\circ g$).

**Question 1:** Is this the "correct" generalisation? My hesitance stems from the fact that, with this definition, $h\dim 1_X$ (the homotopy dimension of the identity map $1_X\colon X\to X$) does not necessarily equal $h\dim X$. Indeed, the former is the smallest dimension of a CW-complex which dominates $X$, and (I believe) Wall has shown that there are spaces for which $h\dim 1_X$ $<$$\infty$ while $h\dim X=\infty$.

**Question 2:** I am sure that this is a well-known and well-studied notion, and that I am merely using the wrong search terms. Where should I look in the literature to learn more about this concept?