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?



Connective spectra and infinite loop spaces
Ok, this discussion has grown beyond the level of comments so I'll collect the facts here. A bit of terminology: a $(-1)$-connected space is a space with a choice of basepoint and the category of $(-1)$-connected spaces $Top\_{>-1}$ is the category of pointed homotopy types with basepoint-preserving maps. The category of $n$-connected spaces $Top\_{>n}$ is the full subcategory of $Top\_{>-1}$ of spaces $X$ which have all homotopy groups $\pi\_k(X) = 0$ for $k \leqslant n$. For any space $X$ we can take its $(n-1)$-truncation $X\_{<n}$ which has all homotopy groups $\pi\_k = 0$ for $k\geqslant n$. There is a fiber sequence $X\_{\geqslant n} \to X \to X\_{<n}$ with the right arrow an isomorphism for $\pi\_{k<n}$ and the left arrow an isomorphism for $\pi\_{k\geqslant n}$.


Now, informally a *loop space* $X$ is just a space with a homotopy equivalence $X \simeq \Omega Y$ for some other space $Y$, however this definition is not precise enough since there can be many non-equivalent choices of $Y$. We would want some structure on $X$ that would guarantee it is a loop space without explicitly specifying such $Y$. A famous theorem of May tells us that that a space is a loop space if an only if it is grouplike and is equipped with the structure of an algebra over the $A\_\infty$ (aka $E\_1$) operad, in particular it is $(-1)$-connected. In this case there is a canonical choice of delooping denoted $\mathbb B X$ which is a 0-connected space such that $\Omega \mathbb B X \simeq X$ as homotopy types and as $A\_\infty$-spaces. Even more specifically, the functors $\Omega$ and $\mathbb B$ are well-defined on the categories of $(-1)$-connected spaces and $A\_\infty$-spaces respectively and establish an adjunction $\mathbb B \dashv \Omega$ between them, which restricts to an equivalence between the full subcategories of grouplike $A\_\infty$-spaces and $0$-connected spaces. In particular, $\mathbb B X$ is the *unique* 0-connected delooping of $X$, and for any other delooping $Y$ its 0-connected cover $Y\_{>0}$ is thus equivalent to $\mathbb B X$. Thus we don't need to consider non-connected deloopings when defining loop spaces.


So *formally* we can define a loop space as a grouplike $A\_\infty$-space, and a map of loop spaces as the map of spaces preserving the $A\_\infty$-structure (grouplikeness is a property rather than structure so doesn't impose extra conditions). These are exactly the maps induced on the loop spaces from the maps $Y \to Y^\prime$. Note that there can be many different $A\_\infty$ structures on the same topoolgical space, thus many nonequivalent deloopings (a classic example is $S^3$ which have different $A\_\infty$ structures different from $SU(2)$ group structure arising from mixing various structures at primes). For this reason it doesn't make sense to just say that a loop space is $X$ such that there *exists* $Y$ with $\Omega Y \simeq X$ --- the specific choice matters.


Generalizing to all $n$, an $n$-fold loop space can be delooped $n$ times and the category of $n$-fold loop spaces is equivalent to the category of $(n-1)$-connected homotopy types. An $\infty$ loop space is thus a space $X$ such that it can be delooped an arbitrary number of times. We can either define it as an algebra over the $E\_\infty$ operad (which is a colimit of all $E\_n$ with $E\_{n+1} = E\_n \otimes E\_1$) or as a sequence of spaces such that $n$-th space is $(n-1)$-connected and $Y\_n = \Omega Y\_{n+1}$. Such sequences are exactly connective spectra.


The definition via an *arbitrary* sequence $Y\_i$ requries a different notion of equivalence, since such sequences represent arbitrary spectra. We can say that a map of two spectra $Y\_i$ and $Y\_i^\prime$ is an equivalence if it is a homotopy equivalence on the 0-th component. Note that the sequence of $(i-1)$-connected covers $(Y\_i)\_{>i-1}$ is also a spectrum and its canonical map to $Y\_i$ is an equivalence of $\infty$-loop spaces. This construction realizes the subcategory of connective spectra as the localization of the category of all spectra. I understand this is Adams's approach in *Infinity loop spaces*.


