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Is there a standard way to construct the shift map on an infinite product or coproduct of a direct or inverse system of spectra that induces the standard shift map of abelian groups in homology? Is it constructed differently for the product and coproduct? If there is some standard construction(s), what is it?

Can we just assume that we are (up to homotopy) dealing with $\Omega$-spectra which have some natural group structure?

Thanks.

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    I suppose up to homotopy we can always just add and subtract maps.2011-12-28
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    Additionally, does anyone know if the limit obtained by taking the (co)fiber of the shift map is "universal" at least up to homotopy? It is known not to be universal on the nose. I guess that would mean that somehow any map with the necessary properties factors through a space homotopy equivalent to the limit space. I imagine this follows from this object being a homotopy limit, i.e. the (co)limit in $hoSpectra$.2011-12-28
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    I am pretty sure you can do all of the above. For starters, yes I am sure you can just work with $\Omega$-spectra. I am pretty sure it will all work the way you expect. I will think a bit more and then maybe post something.2012-01-02
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    Thanks @SeanTilson. I suspect this might follow really easily from the fact that the space of (homotopy classes of?) maps between spectra form a group?2012-01-02
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    abelian group even since everything is the suspension at least 2 of something else.2012-01-02
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    It's obvious you can construct a map and it will do the right thing on homotopy groups, but I'm a little worried about generalized homology since smash products don't always commute with infinite products (at least it feels like they don't...)2012-01-02
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    (apologies for the word "obvious": what I mean is that the category is triangulated, and in particular additive, so you can just write down the "matrix" corresponding to the shift map and it will make sense.)2012-01-02
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    as for your question in the first comment: if you do the construction all in the homotopy category, then you won't get very much nice universality... But if you work upstairs in a model category (or the $\infty$-category, or whatever), then homotopy limits will be unique up to a contractible space of choices...2012-01-02
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    @Dylan Wilson, I think you're right in calling it obvious, sounds like kind of what I was thinking. Thanks.2012-01-02

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