# Connective spectra and infinite loop spaces

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?

• Truncate $X_j$ to have no homotopy groups above degree $j$, and you get a connective spectrum with 0th space $Y_0$. The point is that there is a unique connective spectrum with $Y_0$ as it's zeroth space – Phil Tosteson Sep 12 '17 at 17:03
• You've got your definition of an $\infty$-loop space wrong. An $\infty$-loop space is a space admitting all deloopings (strictly speaking, equipped with a specific choice of them). An n-fold delooping of an n-fold loop space is by definition an $(n-1)$-connected basepointed space, so in your sequence each $Y_i$ must be $(i-1)$-connected, unlike the case of general spectra. – Anton Fetisov Sep 12 '17 at 17:07
• @Anton: Why would I require the delooping to be $(n-1)$-connected? I can surely take the $n$-fold loop space of a space that is not $(n-1)$-fold connected? Maybe this is the issue: If one makes this requirement, then one gets connective spectra, otherwise one gets any spectrum? – Matthias Ludewig Sep 12 '17 at 18:15

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.

• I've decided to delete my answer because I really do not want to defend a terminology that I find confusing and imprecise. I agree that in these days people saying "infinite loop space" mean group-like E_∞-space (a.l.a. a connectve spectrum). Just a small correction: the group-like condition is very important, and it is easy to give examples of E_∞-spaces that cannot be delooped ($\mathbb{N}$, for one). – Denis Nardin Sep 12 '17 at 21:21
• I have to strongly disagree with Fetisov's answer to this question. The definition of an infinite loop space should not involve operads, but should just involve what the word implies. The objects in the category of infinite loopspaces can be taken to be the same as the objects in spectra; but morphisms are different: if $X\langle 0 \rangle$ is the -1 connected cover of $X$, the map $X\langle 0 \rangle \rightarrow X$ becomes an isomorphism after applying $\Omega^{\infty}$. `user43326' (I hate anonymous postings) said it exactly right. – Nicholas Kuhn Sep 14 '17 at 14:12
• By the way, one can also consider the category of algebras over the monad $\Omega^{\infty} \Sigma^{\infty}$, but even in this case, it is a little theorem that this category is equivalent to the category of infinite loopspaces (a result of Beck in the 1960's). – Nicholas Kuhn Sep 14 '17 at 14:18
• Passing from $X$ to $X\langle 0\rangle$ is a co-localization. – Nicholas Kuhn Sep 14 '17 at 14:22

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 $\Omega ^{\infty}i$.