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In "Rings, Modules, and Algebras in Stable Homotopy Theory" Elmendorf, Kriz, Mandell and May introduce a notation of spectra indexed over an universe $\mathcal{U}$ as a collection of pointed topological spaces index by finite subspaces of the universe. Has anyone seen a definition using pointed simplicial sets instead? What would be a simplicial model for the one-point compactification $S^V$ of a real finite dimensional vector space? Just its singular complex?

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3 Answers 3

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Yes to both interpretations of your question. It is not clear to me where you want to put pointed simplicial sets.

One interpretation of your question is that you want to replace pointed topological spaces with pointed simplicial giving the notion of a spectrum as a functor from supspaces of U to pointed sSet. This is a very common thing to do, and in some circles in homotopy theory is the standard definition of spectrum. Often "space" is interpreted as meaning simplicial set. This is because of the standard Quillen equivalence between Top and sSet.

The other possible interpretation of your question is that you are trying to replace vector spaces with simplicial sets. This is also (essentially) something which has been done. It gives a model of spectra known as W-spaces. This is one of the standard diagram category models of spectra. See the following paper for a comparison:

Model categories of diagram spectra, by M. A. Mandell, J. P. May, S. Schwede, and B. Shipley

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I think one issue the original poster referred to with "universe-indexed" spectra is that in, say, the EKMM definition one has structure maps $S^V \wedge E(W) \to E(V \oplus W)$ for any orthogonal pair of subspaces of $\mathcal U$. The lack of simplicial structure on $S^V$ obstructs writing down a direct analogue. –  Tyler Lawson Jan 22 '10 at 13:35
Thanks! This is already partially helpful, but Tyler is right. As far as I see EKMM work with topological spaces only and I wonder how to handle the $S^V$ in the definition of structure maps, if I try to use simplicial sets and no topological spaces. –  user2146 Jan 22 '10 at 14:29
Ahh I see. If you really must work with EKMM, then you need to do a two stage comparison. (1) Go from EKMM spectra to orthogonal spectra: sequences of spaces indexed on inner product spaces together with structure maps, $S^V \wedge E(W) \to E( V \oplus W)$. There is no (overt) operad in the game though. Then you want to pass to the simplicial version of orthogonal spectra: simplicial sets indexed by inner product spaces together with maps like the above, but where $S^V$ is the singular simp. set, as you guessed. I believe both of these transitions are explained in the paper I cited. –  Chris Schommer-Pries Jan 22 '10 at 19:33

Chris, that is not actually what we did. Personally, I find indexing simplicial sets by inner product spaces to be unnecessary and unhelpful, and I've not coauthored any paper with such a construction. One can easily compare symmetric spectra in simplicial sets with symmetric spectra in topological spaces, and one can easily compare symmetric spectra in topological spaces with orthogonal spectra. I see no point in a hybrid. As a matter of detail, in defining orthogonal spectra one can perfectly well work with all finite dimensional inner product spaces, without choosing a universe, whereas the universe is needed to define the linear isometries operad used in the EKMM construction. It is nice to keep the S^V as they are: that makes generalization to G-spectra effortless, where G is a compact Lie group, and that works for both orthogonal spectra of spaces and EKMM S-modules.

I prefer eclecticism: the different models have different advantages. Here is an eclectic correct definition: a map of symmetric spectra (of spaces) is a weak equivalence iff its pushforward map of orthogonal spectra induces an isomorphism of homotopy groups. (Proven in the paper MMSS Chris cites.)

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ps: I really don't like ``if you really must ...''. There are serious advantages to working in a model category in which every object is fibrant, and, related to that, both for theory and computations it is very helpful to have a clean zeroth space functor from spectra to highly structured spaces.

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