In Segal's paper on $\Gamma$-spaces, he gives a functor $Spectra \rightarrow \Gamma-Spaces$ defined by taking a functor $E$ and sending it to the $\Gamma$-space $AE$ with $AE(n) = Mor(S \times \cdots \times S, E)$, where $S$ is the sphere spectrum. Now, since this is supposed to define a $\Gamma$-space, in particular the sets $Mor(S \times \cdots \times S, E)$ should be topological spaces... but they don't seem to come with any obvious topology, at least not obvious to me.

On the other hand, it seems like there should be some sort of spectrum that acts like $Mor(S \times \cdots \times S, E)$; could he mean, possibly, the 0th space of this spectrum?

EDIT: The reference is

Segal, Graeme Categories and cohomology theories. Topology 13 (1974), 293--312

  • $\begingroup$ For some reason the tags don't seem to be automatically filling out when I try to post... $\endgroup$ – Dylan Wilson Aug 10 '11 at 5:59
  • $\begingroup$ @Dylan: I've had the same problem with tags, and found refreshing the page to be a solution. Also, please could you give the reference to Segal's paper? $\endgroup$ – Mark Grant Aug 10 '11 at 6:37
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    $\begingroup$ I'm a bit disturbed to see the cartesian product of spectra, as in $S\times S\times \cdots\times S$ (rather than say, the smash or wedge product). Is this a common construction in stable homotopy theory? $\endgroup$ – Mark Grant Aug 10 '11 at 9:05
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    $\begingroup$ Yeah... this was disturbing. I think that perhaps we don't see it very much because the category is additive, so we use wedge sums instead. $\endgroup$ – Dylan Wilson Aug 10 '11 at 17:53
  • $\begingroup$ @Mark: the inclusion of a finite wedge of spectra into the corresponding finite product is a weak equivalence. $\endgroup$ – John Klein Aug 19 '11 at 13:19

If $X = (X_n)$ and $Y = (Y_n)$ are spectra, one can define a morphism just to be a collection of maps $x_n \to Y_n$ commuting with the suspensions. Thus the set of morphisms between $X$ and $Y$ is a subset of $\prod_n Map(X_n,Y_n)$ - and we give it just the subspace topology.

An alternative way is the simplicial set approach: we define an $n$-simplex of the mapping space between $X$ and $Y$ to be a map $X\wedge \Delta[n]^+ \to Y$. If we want a topological space back, one can geometrically realize.

If you want mapping spectra, it is perhaps more reasonable to go to symmetric spectra. You can find an exposition of mapping spectra (and also of mapping spaces) in Stefan Schwede's book project on symmetric spectra: http://www.math.uni-bonn.de/people/schwede/SymSpec.pdf 2.24 & 2.25.

If you're interested in the relationship between Gamma-spaces and spectra from a homotopical view, you might also be interested in the Bousfield-Friedlander paper: http://club.pdmi.ras.ru/~topology/books/bousfield-friedlander.pdf

  • $\begingroup$ Ah, brilliant! And thanks for the reference as well- looks great! $\endgroup$ – Dylan Wilson Aug 10 '11 at 17:54

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