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I have a quick question which I think should go like this, but I am not really sure and that is why I would like someone more knowledgeable than me to weigh in and say if I am correct.

Let us say that $X$ is an infinite loop space - it is well-known ( to the initiated) that this is equivalent to $X$ being a connective spectrum. I can see that from a connective spectrum $\{Y_n\}_{n \in \mathbb{Z}}$ it is easy to get an infinite loop space - take $Y_0$ to be your space. In the other way around, if X is an infinite loop space, do we get the spectrum $\{Y_n\}$ from X by setting $Y_i = 0 $ if $i <0$ (and the obvious maps) and $Y_0 = X = \Omega X_1 $, $Y_1= X_1$ , where the map $SY_0 \rightarrow Y_1$ is taken from the map $\Omega Y_0 \rightarrow \Omega Y_1$ (using that X is an infinite loop space) and so on?

Tedar
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  • Very often, people define an infinite loop space to be the zeroth space of an $\Omega$ spectrum. – Baby Dragon Sep 25 '13 at 16:27
  • @BabyDragon With infinte loop space here, I mean a space that is a loop space "infinitely many times" so to say. – Tedar Sep 25 '13 at 16:46

1 Answers1

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For a connected space $X$ it is reasonable well known that $B \Omega X \simeq X$.

In fact more in true; there is a natural transformation $B^n \Omega^n X \to X$ that is a weak equivalence if $X$ is $(n-1)$-connected. It shouldn't be a surprise then that for a spectrum $X$ there is a '$B^\infty$' functor such that $B^\infty \Omega^\infty X \simeq X$ if $X$ is connective. $B^\infty$ is the functor from your $E_\infty$ spaces to connective spectra.

This is covered in Adams' "Infinite Loop Spaces" book

Drew
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  • There is more said here: http://math.stackexchange.com/questions/81472/can-spectra-be-described-as-abelian-group-objects-in-the-category-of-spaces-in – Drew Sep 25 '13 at 23:47