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edited May 18, 2017 at 20:58

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Edit: As explained in the comments, this is not an answer to the question. I leave it as it might at least serve as an intuition for why the direct sum axiom plus excision should be enough to imply the direct limit axiom.

Working $\infty$-categorically, we may think of a homology theory as a functor from spaces to spectra. The axioms about the suspension isomorphism and cofiber sequences are equivalent to preservation of pushouts. Now, a functor which preserves pushouts and all (small) coproducts, preserves all (small) colimits. This is proposition 4.4.2.7 in HTT by Lurie.

Remark: Actually, we can reverse the logic by saying that we just want our the functor to preserve all (small) colimits. One can break this into (1) preservation of finite colimits and (2) preservation of filtered colimits. One can also replace (2) by (2') preservation of coproducts. Additionally, one can replace (1) by (1') preservation of pushouts. Finally, one can test that a map of spectra is an equivalence by checking the induced map on the homotopy groups and for a pushout these can be described by a long exact sequence. This allows us to reformulate everything in classical terms (i.e. the Eilenberg-Steenrod axioms).

Working $\infty$-categorically, we may think of a homology theory as a functor from spaces to spectra. The axioms about the suspension isomorphism and cofiber sequences are equivalent to preservation of pushouts. Now, a functor which preserves pushouts and all (small) coproducts, preserves all (small) colimits. This is proposition 4.4.2.7 in HTT by Lurie.

Remark: Actually, we can reverse the logic by saying that we just want our the functor to preserve all (small) colimits. One can break this into (1) preservation of finite colimits and (2) preservation of filtered colimits. One can also replace (2) by (2') preservation of coproducts. Additionally, one can replace (1) by (1') preservation of pushouts. Finally, one can test that a map of spectra is an equivalence by checking the induced map on the homotopy groups and for a pushout these can be described by a long exact sequence. This allows us to reformulate everything in classical terms (i.e. the Eilenberg-Steenrod axioms).

Edit: As explained in the comments, this is not an answer to the question. I leave it as it might at least serve as an intuition for why the direct sum axiom plus excision should be enough to imply the direct limit axiom.

Working $\infty$-categorically, we may think of a homology theory as a functor from spaces to spectra. The axioms about the suspension isomorphism and cofiber sequences are equivalent to preservation of pushouts. Now, a functor which preserves pushouts and all (small) coproducts, preserves all (small) colimits. This is proposition 4.4.2.7 in HTT by Lurie.

Remark: Actually, we can reverse the logic by saying that we just want our the functor to preserve all (small) colimits. One can break this into (1) preservation of finite colimits and (2) preservation of filtered colimits. One can also replace (2) by (2') preservation of coproducts. Additionally, one can replace (1) by (1') preservation of pushouts. Finally, one can test that a map of spectra is an equivalence by checking the induced map on the homotopy groups and for a pushout these can be described by a long exact sequence. This allows us to reformulate everything in classical terms (i.e. the Eilenberg-Steenrod axioms).

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created May 18, 2017 at 13:11

2.3k
13
28

Working $\infty$-categorically, we may think of a homology theory as a functor from spaces to spectra. The axioms about the suspension isomorphism and cofiber sequences are equivalent to preservation of pushouts. Now, a functor which preserves pushouts and all (small) coproducts, preserves all (small) colimits. This is proposition 4.4.2.7 in HTT by Lurie.

Remark: Actually, we can reverse the logic by saying that we just want our the functor to preserve all (small) colimits. One can break this into (1) preservation of finite colimits and (2) preservation of filtered colimits. One can also replace (2) by (2') preservation of coproducts. Additionally, one can replace (1) by (1') preservation of pushouts. Finally, one can test that a map of spectra is an equivalence by checking the induced map on the homotopy groups and for a pushout these can be described by a long exact sequence. This allows us to reformulate everything in classical terms (i.e. the Eilenberg-Steenrod axioms).