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I will use the notation of this question. So, if $X$ is a (nice) topological space and $G$ is an abelian group, we can form its $G$-linearization $G[X]$. In McCord's article, this was denoted $B(G,X)$.

In that question it is mentioned that $\pi_*(G[X])=\tilde{H}_*(X;G)$. But more is true: the functor from Spaces to Graded Abelian Groups that maps a space $X$ to $\pi_*(G[X])$ is a homology theory, isomorphic to singular homology with coefficients in $G$. This says that the isomorphism commutes with the boundary maps in long exact sequences. And we have a good grip on what the boundary for $\pi_*(G[-])$ is, actually: if $A\to X$ is a cofibration, then McCord proves that

$$G[A]\to G[X]\to G[X/A]$$

is a fibration (actually, a principal bundle). The boundary map in homology for $A\to X$ corresponds to the boundary map in homotopy for this fibration.

With this set up, we have that $H_*(-;G)$ is naturally isomorphic to the composition $\pi_* \circ G[-]$.

If I understand correctly what I've heard, it is actually true that for any spectrum $E$ we have that $E_*$ (the associated homology functor from Spaces to Graded Abelian Groups) decomposes as a composition $\pi_* \circ E[-]$, where $E[-]$ is some functor from spaces to spaces which maps cofibrations to fibrations.

Question 1: what further hypotheses do we need on these functors to make the correspondence with the category of homology theories be a 1-1 correspondence? We certainly need $E[-]$ to map a point to a point (or to something contractible). Is this enough?

So, for $E=HG$, the Eilenberg-Mac Lane spectrum of $G$, we have that $E[-]=G[-]$: for $H\mathbb Z$, for example, $E$ is the infinite symmetric product functor.

For $E$ the sphere spectrum, we have $E[-]=Q$.

Question 2: what can be said about other spectra? Is there a clean description on how to get $E[-]$ from $E$ in general? Or maybe in other well-known cases, $KO, KU, MO, MU, K(n)$, etc...

Subsidiary question for the comments: any further references to this point of view are welcome.

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Bruno Stonek
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  • I now realize I haven't looked at Mostovoy's paper in the answer in the linked question, will do now... – Bruno Stonek Jan 09 '16 at 11:08
  • @SebastianGoette How so? $E\wedge -$ produces a spectrum, as I see it. – Bruno Stonek Jan 09 '16 at 11:16
  • I see, so you would have to take what people call $\Omega^\infty(E\wedge\mathord{-})$ - the space in degree $0$ of the associated $\omega$-spectrum. In the sphere example, this is what you wrote. – Sebastian Goette Jan 09 '16 at 11:18
  • I believe Question 2 is answered in the paper of Mostovoy, in sections 2.2 and 2.3, though it seems to be only for connective spectra... – Bruno Stonek Jan 09 '16 at 11:38
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    I am not sure what are you asking. If $E$ is a spectrum, then the homology theory $E_$ is defined* to be $E_(X) = \pi_ (E \wedge \Sigma^\infty X_+)$, where $X_+$ is $X$ with a disjointly added marked point, $\Sigma^\infty$ is stabilization and $\wedge$ is the smash product of spectra. Note that homology theories are ill-behaved in general, we can\t guarantee their representability in above form and even if representable the representation is generally highly non-unique. cont... – Anton Fetisov Jan 09 '16 at 13:50
  • A good-behaved object is a cohomology theory, where Brown's representability gives us a spectrum $E$ s.t. $E^(X) = \pi_{-}Hom(\Sigma^\infty X_+, E)$. Ordinary cohomology ($E^0 (pt) = G$, $E^n(pt) = 0$ if $n\ne 0$) are represented by Eilenberg-Maclane spectra $HG$, while for all other spectra $E^*(pt)$ is very nontrivial. – Anton Fetisov Jan 09 '16 at 13:53
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    What you are looking for is the notion of reduced 1-excisive functors (Goodwillie calculus is another good keyword). Note that this is going to give you the notion of spectrum, which is much better behaved than the notion of (co)homology theory. – Denis Nardin Jan 09 '16 at 17:13
  • I now realize this question is really quite badly formulated, because I was quite confused. There were two questions. 1) In @DenisNardin's words, where can I find a reference to the statement "there is an equivalence between the stable homotopy category and the category of reduced 1-excisive functors" ?, and 2) Is there a nice description of the functor $\Omega^\infty(E\wedge -)$ from Spaces to Spaces? The answer to the second qustion is Mostovoy's paper in the answer to the linked question. I do not yet have an answer for the second one, but it seems to be hidden somewhere in the relevant... – Bruno Stonek Jan 28 '16 at 17:46
  • ...Lydakis papers from 1998. This is related to $\Gamma$-spaces paraphernalia but I'm not comfortable with them so as to try to extract the quoted statement above from said papers. – Bruno Stonek Jan 28 '16 at 17:48
  • Of course, in the first comment of the last two, I meant "for general spectra $E$ other than Eilenberg Mac-Lane spectra", for which we have the nice description outlined in the post. – Bruno Stonek Jan 28 '16 at 17:58

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As a starting point you might well move backwards in time to 1960:

MR0115163 Whitehead, George W. Homology theories and duality. Proc. Nat. Acad. Sci. U.S.A. 46 1960 554–556.

The details of the results announced there were written up in

Whitehead, George W. Generalized homology theories. Trans. Amer. Math. Soc. 102 1962 227–283.

The representability of homology theories in terms of spectra dates from then. I encourage you to read the Math Review of the first by Peter Hilton. The representing spectrum is not unique, but I think the lim^1 error term measuring non-uniqueness was understood when I was a graduate student, or very soon thereafter. (I got my PhD in 1964). A clean exposition is in Section 1 of McClure's Chapter VII of $H_{\infty} ring spectra and their applications, published in 1986.

Peter May
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  • Dear Professor May, thank you for your reply. I now realize that this answer is addressing my confusion in the original post, wanting an equivalence between those functors Spaces --> Spaces with the category of homology theories, while what I really wanted (as Denis Nardin guessed) is the equivalence between those functors with the homotopy category of spectra. The issues of phantomness, which you had already addressed here, are not what I wanted to get at (but I didn't realize that when I wrote the question). – Bruno Stonek Jan 28 '16 at 17:52