Filtration on Smash Product of Cofibers

Question

I have seen some similar questions to this one on here recently, so I hope this isn't redundant. Basically, suppose I have two cofiber sequences of spectra (or perhaps just work in some general homotopy category or something) $X\overset{f}\to Y\to Cf$ and $X'\overset{g}\to Y'\to Cg$. I'd like to look at an induced filtration on $Cf\wedge Cg$ along the lines of the bottom level being $X\wedge X'$, the middle level being something like $(X\wedge Y')\cup (Y\wedge X')$ and the top level being $Y\wedge Y'$, or something along those lines. Or I guess I should say I'd like to have filtration quotients that look like that. Honestly, I'd like to do this all in a cellular context. Intuitively this all seems pretty obvious, but does anyone have a good reference where such a filtration is discussed really rigorously, or a good way to think about it?

Thanks, as always.

Answer 1

This is an excellent question that I have thought a lot about. I'd rather answer it in a more general context that was motivated by what I knew to be true in the stable homotopy category. The reference is "The additivity of traces in triangulated categories", #99 on my web site. The essential point there is to formulate axioms relating triangulations to symmetric monoidal structures, and Axiom (TC3), with $Z=Cf$ and $Z'=Cg$, formulates exactly what happens when you smash two triangles (cofiber sequences) together. The motivation in terms of your question is given on page 49; I think the diagrams on pages 49 and 50 display the `filtration' you ask for, in appropriate generality, and they are what you see when you follow through on your concrete question. The equivalent form (TC3') of (TC3) may perhaps be easier to compare with your question, since it has $Z \wedge Z'$ conveniently displayed as the end target. The description of the proof of (TC3) on pages 60-61 follows the lines of your question. Edit: the page numbers above are those of the published paper; the page numbers of the file on my web site are 12-13 (for 49-50) and 21-22 (for 60-61).

Answer 2

So it seems to me that, using Peter May's paper above, for two distinguished triangles $X\to Y\to Z$ and $X'\to Y'\to Z'$, we have a 'filtration':

$$Z\wedge Z'\overset{f_1}\leftarrow Y\wedge Y'\overset{f_2}\leftarrow X\wedge X'$$

where the cofiber of $f_1$ is $Z\wedge Z'\to\Sigma V=\Sigma(X\wedge Y'\cup_{X'\wedge X}X'\wedge Y)$ and the cofiber of $f_2$ is just what it is, though Peter May's paper denotes it by $W$ and gives other distinguished triangles as well as several Cartesian squares that $W$ fits inside of. That way, we can attempt to understand what it is. Perhaps there is a better way to combine these things, but this is the best way I've determined thus far.