TITLE: Connective spectra and infinite loop spaces
QUESTION [11 upvotes]: It seems to be standard that connective spectra are "the same" as infinite loop space. However, I do not understand the reason why the associated spectrum is connective.
For me, an infinite loop space is a space $Y_0$ together with a collection of pointed spaces $Y_1, Y_2, \dots$ and homotopy equivalences (or homeomorphisms) $f_j: Y_{j-1} \rightarrow \Omega Y_j$, $j=1, 2, \dots$. Now there is an obvious way to associate an $\Omega$-spectrum $X$ to this data: Just set
$$ X_j = \begin{cases} Y_j & j \geq 0 \\ \Omega^j Y_0 & j < 0 \end{cases}$$
for the underlying spaces with the map $g_j: X_{j-1} \rightarrow \Omega X_j$ being given by $f_j$ for $j \geq 0$ and the identity for $j < 0$.
Clearly, there is no reason why $X$ should be connective: Just take $Y_0$ the zero space of a non-connective $\Omega$-spectrum.
So what is the precise meaning of this statement that infinite loop spaces are the same as connective spectra?

REPLY [4 votes]: A simple way to answer this question is:
denote by $Spec$ the homotopy category of spectra, $Spec _0$ the category of
(-1)-connected spectra, $InfL$ be the homotopy category of infinite loop spaces.  The assertion that  "connective spectra are the same as infinite loop space" simply means that the composition of functors
$$Spec _0 \stackrel{i}{\to} Spec\stackrel{\Omega ^{\infty}}{\to} InfL$$ is an equivalence, where $i$ denotes the inclusion.
You gave an example of infinite loop space in the image of $\Omega ^{\infty}$, which doesn't mean that it is not in the image of $\Omega ^{\infty}i$.