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edited Feb 19, 2013 at 3:30

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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).

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.

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).

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created Feb 19, 2013 at 3:06

30.4k
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96
140

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.