Consider the homotopy category $\mathrm{hoSp}$ of spectra. It has a full subcategory $\mathrm{hoSp}\_{\geq 0}$ of connective spectra, equivalently of infinite loop spaces, equivalently $E\_\infty$-group spaces.


The inclusion $\mathrm{hoSp}\_{\geq 0} \hookrightarrow \mathrm{hoSp}$ has a right adjoint, sending a spectrum $T$ to its connective cover $T\langle0\rangle$. In other words, for any connective spectrum $A$, we have
$$ [A, T] = [A, T\langle 0\rangle].$$
I'm wondering about intermediate categories $\mathrm{hoSp}\_{\geq 0} \hookrightarrow \mathcal{C} \hookrightarrow \mathrm{hoSp}$ for which the inclusion $\mathrm{hoSp}\_{\geq 0} \hookrightarrow \mathcal{C}$ also has a left adjoint. Explicitly: for which spectra $T$ is there a connective spectrum $T'$ such that
$$[T,A] = [T', A]$$
for all connective $A$? This $T'$ would be some sort of universal "quotient" of $T$ that quotients it down to being connective.


Dually, I can consider the full subcategory $\mathrm{hoSp}\_{\leq 0} \hookrightarrow \mathrm{hoSp}$ of coconnective spectra. This inclusion has a left adjoint given by the truncation $T \mapsto \tau\_{\leq 0}T$. On which intermediate category does this inclusion also have a right adjoint?


I am also interested in variations of these questions, e.g. working with the $(\infty,1)$-categories rather than their homotopy categories, or looking at the inclusion of spectra with homotopy in some bounded range $[0,n]$ (perhaps $n=0$, say) among spectra with homotopy bounded on only one side, or looking at spectra with some finiteness built in.




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**Codicil:** The accepted answer below resolves the $\infty$-categorical variation of this question. Since the homotopy-categorical version is different, I have posted it as [new question](https://mathoverflow.net/questions/444845/).



Which spectra have a universal connective quotient?
This answer is about the $\infty$-categorical variant. This is a fancy way to say: on *spaces* of maps, the natural map
$$
Map(T',A) \to Map(T,A)
$$
is an equivalence for any connective $A$. Note that for any bounded-below spectrum $B$, we can use $Map(-,B) \simeq \Omega^n Map(-,\Sigma^n B)$ to conclude that the natural map above is an equivalence for $B$ as well, and hence by taking homotopy classes of maps we get that
$$
[T',B] \to [T,B]
$$
is an equivalence for any bounded-below $B$.




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I first claim: a map $T \to T'$ induces an equivalence $[T', B] \to [T,B]$ for all bounded-below $B$ if and only if it is an integral homology isomorphism.


To see that this is necessary:


Consider $B$ of the form $K(Q,n)$ for $Q$ an injective abelian group -- these are bounded-below. Then the universal coefficient theorem implies that the natural map
$$
Hom(H\_n(T'), Q) \to Hom(H\_n(T), Q)
$$
is isomorphic to the natural map
$$
H^n(T', Q) \cong [T', K(Q,n)] \to H^n(T,Q) \cong [T, K(Q,n)],
$$
and hence is an isomorphism. If this is true for all injective abelian groups $Q$, for a generic abelian group $M$ we can apply it to an injective resolution of $M$ and find that the natural map
$$
Hom(H\_n(T'), M) \to Hom(H\_n(T), M)
$$
is an isomorphism.


To see that this is sufficient:


Suppose we have a map $T \to T'$ which is an integral homology isomorphism, and consider the class of spectra $B$ such that $[T',B] \to [T,B]$ is an isomorphism.


* This class contains Eilenberg-Mac Lane spectra, by the universal coefficient theorem.
* This class is closed under fibers. If $B' \to B \to B''$ is a fiber sequence of spectra, then we get long exact sequences upon applying $[T,-]$ or $[T',-]$, and if the maps $[T',B] \to [T,B]$ and $[T',B''] \to [T',B'']$ are isomorphisms then so is $[T',B'] \to [T,B']$ by the five-lemma.
* This class is closed under spectra that have finitely many nonzero homotopy groups, by inductively applying this to the fiber sequences $K(\pi\_n B, n) \to P\_n B \to P\_{n-1} B$ in their Postnikov tower.
* This class is closed under products.
* This class is closed under homotopy limits of towers. If we have a tower $\dots \to B\_2 \to B\_1 \to B\_0$ of objects in this class, then the homotopy limit is part of a fiber sequence
$$
holim (B\_i) \to \prod B\_i \xrightarrow{1-\text{shift}} \prod B\_i
$$
and so the homotopy limit is in the class.
* This class contains all bounded-below spectra, because any bounded-below $B$ is the homotopy limit of its Postnikov tower $\dots \to P\_2 B \to P\_1 B \to P\_0 B$, and the individual $P\_n B$ all have only finitely many nonzero homotopy groups.




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Now I claim: the collection of $T$ for which such a connective $T'$ exists consists of the spectra $T$ such that the integral homology $H\_\*(T)$ is concentrated in nonnegative degrees. This is necessary: because $T \to T'$ is an integral homology isomorphism, the isomorphism $H\_n(T) \to H\_n(T')$ forces $H\_n(T) \cong H\_n(T') \cong 0$ for $n < 0$.


Suppose that $H\_\* T = 0$ in negative degrees. An equivalent formulation is that the smash product $H\Bbb Z \wedge T$ is connective. If this is true, then we can form the $H\Bbb Z$-Adams tower. Briefly, construct a fiber sequence $J \to \Bbb S \to H\Bbb Z$ (hence $J$ is 0-connected) and form the tower
$$
\Bbb S/J \wedge T \leftarrow \Bbb S/J^2 \wedge T \leftarrow \Bbb S/J^3 \wedge T \leftarrow \dots
$$
The homotopy limit $T'$ is called the $H\Bbb Z$-nilpotent completion of $T$, and it has a natural map $T \to T'$.


Something special happens in this case, however, that I'll just briefly describe. The associated graded, in degree $p$, is $\Bbb S/J \wedge J^{p} \wedge T \simeq (H\Bbb Z \wedge T) \wedge J^p$. This is $p$-connected because $J^p$ is $p$-connected and $H\Bbb Z \wedge T$ is connective. As a result, by induction this nilpotent completion is connective. Moreover, smashing with $H\Bbb Z$ can be moved inside the homotopy limit because of this connectivity growth, and the map $H\Bbb Z \wedge T \to H\Bbb Z \wedge T'$ turns out to be a weak equivalence (true for Adams towers whenever smashing can be moved inside the limit); so $T \to T'$ is a homology isomorphism.




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As a note, for this to happen we need for the fiber of $T \to T'$ to have trivial homology; this is impossible unless it is contractible or it is unbounded below. This is certainly interesting but carries it outside the usual library of basic examples.


