Product CW-complexes are defined via characteristic maps rather than from attaching maps, so via maps from $\mathbb D^n$ rather than from $\mathbb S^{n-1}$, because we have the propriety that $\mathbb D^n\times\mathbb D^m=\mathbb D^{m+n}$. I want to define products in a synthetic way, only manipulating objetcs up to homotopy (so no discs, anywhere, even during the construction as a provisional step, because there are nothing more than trivial things up to homotopy) so I need to use only the spheres. So basically if $X$ and $Y$ are CW-complexes and $Z$ is the product, to attach one new cell to $Z\_{p-1}$ means attaching a product of cells $\mathbb S^{n-1}\to X\_{n-1}$ and $\mathbb S^{m-1}\to Y\_{m-1}$, where $n+m=p$, and this should give a function $\mathbb S^{n+m-1}\to Z\_{p-1}$. We actually already have such a map when the CW-complexes $X$ and $Y$ are the spheres $\mathbb S^{n}$ and $\mathbb S^{m}$, which is the Whitehead map. But note that there I mean the Whitehead product not defined as the attaching map of the product of two cells, because it would be a construction using discs in a implicit way. I take it defined from the homotopy pushout definition of the join of two spheres. Indeed we have then (for a CW product of spheres) $Z\_{n+m-1}=\mathbb S^{n}\vee\mathbb S^{m}$ and this implies that the wanted attaching map is of type $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$, where by definition $\mathbb S^{n+m-1} = \mathbb S^{n-1} \*\mathbb S^{m-1}$.
Now I want this bit to be generalizable to every product of finite CW-complex (it can have just a finite number of cells if nedded).


So instead of a simple map $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$ want some function $\mathbb S^{n+m-1}\to Z\_{p-1}$ where obviously $Z\_{p-1}$ can be more complex than just a wedge. The definition of this map would look like that of the map $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$, so we should look a the pushout span $\mathbb S^{n-1}\leftarrow \mathbb S^{n-1}\times\mathbb S^{m-1}\to \mathbb S^{m-1}$ but then the attaching maps of the cells are not trivial like they were for products of sphere. This looks very closely like the Whitehead product but I guess there is some non trivial things to consider at this point and I feel like I can't find the good way to tackle the problem.


I would like to know if some work about this had already be done (in a synthetic "up-to-homotopy" way), I'd be very glad that you have some reference about this problem.


Thank you very much



CW Product via Whitehead map
The attaching map of the product of cells is sometimes described as an *exterior join construction*.


Let $F:D^n\to Cf\subseteq X$ be the characteristic map of an $n$-cell of $X$ with attaching map $f:S^{n-1}\to X\_{n-1}$, and let $G:D^m\to Cg\subseteq Y$ be the characteristic map of an $m$-cell of $Y$ with attaching map $g:S^{m-1}\to Y\_{m-1}$.


To describe the attaching map of the product of these cells, note that there is a homeomorphism 
$$
S^{n+m-1} \approx D^n\times S^{m-1} \cup S^{n-1}\times D^m \subseteq D^n\times D^m.
$$
Under this homeomorphism, the attaching map of the product cell is
$$
F\times g \cup f\times G: D^n\times S^{m-1} \cup S^{n-1}\times D^m\to Cf\times Y\_{m-1} \cup X\_{n-1}\times Cg,
$$
the cofibre of which is $Cf\times Cg$. 


The exterior join construction is perhaps not very well documented in the literature, but here are some references:


*Baues, Hans Joachim*, [**Iterierte Join-Konstruktionen**](http://dx.doi.org/10.1007/BF01213826), Math. Z. 131, 77-84 (1973). [ZBL0244.55016](https://zbmath.org/?q=an:0244.55016).


*Marcum, Howard J.*, Fibrations over double mapping cylinders, Ill. J. Math. 24, 344-358 (1980). [ZBL0459.55009](https://zbmath.org/?q=an:0459.55009).


*Stanley, Donald*, [**On the Lusternik-Schnirelmann category of maps**](http://dx.doi.org/10.4153/CJM-2002-022-6), Can. J. Math. 54, No. 3, 608-633 (2002). [ZBL1002.55002](https://zbmath.org/?q=an:1002.55002). 


