# nLab connective spectrum

### Context

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Idea

### General

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.

### Strict

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.

## Properties

### Inclusion into all spectra

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.

### Connective cover

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.

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ Γ-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object
stabilizationspectrumspectrum object

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