# nLab connective spectrum

### Context

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Idea

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

$\mathrm{Top}\stackrel{\supset }{←}\mathrm{ConnectSp}\left(\mathrm{Top}\right)↪\mathrm{Sp}\left(\mathrm{Top}\right)\phantom{\rule{thinmathspace}{0ex}}.$Top \stackrel{\supset}{\leftarrow} ConnectSp(Top) \hookrightarrow Sp(Top) \,.

There are objects in $\mathrm{Sp}\left(\mathrm{Top}\right)$, 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

Non-connective spectra are well familiar in as far as they are in the image of the nerve operation of 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.

$\begin{array}{ccc}{\mathrm{Ch}}_{+}& \stackrel{\mathrm{Dold}-\mathrm{Kan}\phantom{\rule{thickmathspace}{0ex}}\mathrm{nerve}}{\to }& \mathrm{ConnectSp}\left(\infty \mathrm{Grp}\right)\subset \infty \mathrm{Grpd}\\ \left(\cdots {A}_{2}\stackrel{\partial }{\to }{A}_{1}\stackrel{\partial }{\to }{A}_{0}\to 0\to 0\to \cdots \right)& \stackrel{}{↦}& N\left({A}_{•}\right)\end{array}$\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 $\mathrm{Sp}\left(\mathrm{Top}\right)$.

$\begin{array}{ccc}\mathrm{Ch}& \stackrel{\mathrm{Dold}-\mathrm{Kan}\phantom{\rule{thickmathspace}{0ex}}\mathrm{nerve}}{\to }& \mathrm{Sp}\left(\infty \mathrm{Grp}\right)\\ \left(\cdots {A}_{2}\stackrel{\partial }{\to }{A}_{1}\stackrel{\partial }{\to }{A}_{0}\stackrel{\partial }{\to }{A}_{-1}\stackrel{\partial }{\to }{A}_{-2}\to \cdots \right)& \stackrel{}{↦}& N\left({A}_{•}\right)\end{array}$\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.

(∞,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 November 1, 2012 17:37:29 by Urs Schreiber (131.174.41.102)