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 dimension* of $f\colon X\to Y$ 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? 



Homotopy dimension of a mapping
Regarding Question 1: No, I do not think that's correct. In my opinion, the
definition should be one of the following:


The *relative homotopy dimension* of $f: X \to Y$ is $\le k$ if and only if there is a factorization
of $f$ as 
$$
X \overset{f'}\to Y' \overset{g} \to Y
$$ 
in which $f'$ is an inclusion, $Y'$ is obtained from $X$ by iterated cell attachments of dimension $\le k$, and $g$ is a weak homotopy equivalence. 


The notion of dimension I am describing is internal to the category of spaces under $X$, i.e,
$X\backslash\text{Top}$, where $f$ is to be regarded as an object of that category. In this scheme, the homotopy dimension of the identity map is $\le -1$.


There is another variant of though: let define us say that the *fiberwise homotopy dimension* of
$f: X\to Y$ is $\le k$ iff if there is a weak homotopy equivalence $X' \to X$ such that
$X'$ is a cell complex of dimension $\le k$. 


In particular $Y$ is homotopy equivalent to a cell complex of dimension $\le k$ if and only if the identity map of $Y$ has fiberwise dimension $\le k$. 


Regarding Question 2: To a certain extent, I have written about both of these notions in the paper: Poincaré duality embeddings and fiberwise homotopy theory, *Topology* **38**, 597$-$620 (1999), but this is by no means my concept, nor is my treatment to be regarded as definitive. 


**Added:** The above notions generalize to a single notion as follows: let $f: A \to Y$ be any map of spaces and define $\text{Top}\_f$ to be the category of factorizations of $f$
the objects of this category are factorizations $A \to X \to Y$ and morphisms are maps
$X \to X'$ commuting with the given structure maps. 


Then we can define dimension in this setting as follows: let's say that an object $X \in \text{Top}\_f$ has dimension $\le k$ iff it is built up from the initial object (represented by $A$) by attaching cells over $Y$ of dimension at most $k$.


It's easy to see that the case $f:\emptyset \to Y$ gives the notion of fiberwise dimension, whereas the case when $f: A \to \text{pt}$ gives the notion of relative dimension. 


