connective spectrum

A **connective spectrum** is a spectrum whose homotopy groups in negative degree vanish. These are equivalently

Connective spectra form a sub-(∞,1)-category of spectra

$Top \stackrel{\supset}{\leftarrow} ConnectSp(Top) \hookrightarrow Sp(Top)
\,.$

There are objects in $Sp(Top)$, though, that do not come from “naively” delooping a topological space infinitely many times. These are the **non-connective spectra**. For general spectra there is a notion of homotopy groups of negative degree. The connective ones are precisely those for which all negative degree homotopy groups vanish.

There is a subclass of connective spectra that are equivalent to possibly-more-familiar objects, namely nonnegatively graded chain complexes, via the Dold-Kan correspondence. This identifies ∞-groupoids that are not only connective spectra but even have a *strict* symmetric monoidal group structure with non-negatively graded chain complexes of abelian groups.

$\array{
Ch_+ &\stackrel{Dold-Kan \; nerve}{\to}&
ConnectSp(\infty Grp) \subset \infty Grpd
\\
(\cdots A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0 \to 0 \to 0 \to \cdots)
&\stackrel{}{\mapsto}&
N(A_\bullet)
}$

The homology groups of the chain complex correspond precisely to the homotopy groups of the corresponding topological space or ∞-groupoid.

The free stabilization of the (∞,1)-category of non-negatively graded chain complexes is simpy the stable (∞,1)-category of arbitrary chain complexes. There is a stable Dold-Kan correspondence that identifies these with special objects in $Sp(Top)$.

$\array{
Ch &\stackrel{Dold-Kan \; nerve}{\to}&
Sp(\infty Grp)
\\
(\cdots A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0 \stackrel{\partial}{\to} A_{-1}
\stackrel{\partial}{\to} A_{-2} \to \cdots)
&\stackrel{}{\mapsto}&
N(A_\bullet)
}$

So it is the homologically nontrivial parts of the chain complexes in negative degree that corresponds to the non-connectiveness of a spectrum.

The inclusion

$Spectra_{\geq 0} \hookrightarrow Spectra$

of the full sub-(∞,1)-category of connective spectra into the (∞,1)-category of spectra preserves small (∞,1)-colimits. Moreover, $Spectra_{\geq 0}$ is generated under small (∞,1)-colimits by the sphere spectrum.

These statements prolong to sheaves of spectra.

e.g. (Lurie, "Spectral Schemes", example 1.23)

By the above, connective spectra form a coreflective sub-(∞,1)-category of the (∞,1)-category of spectra. The right adjoint (∞,1)-functor from spectra to connective spectra is called the *connective cover* construction.

Revised on March 17, 2015 15:29:24
by Mike Shulman
(108.247.146.128)