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

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

There are objects in Sp(Top)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.

Ch + DoldKannerve ConnectSp(Grp)Grpd (A 2A 1A 000) N(A ) \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)Sp(Top).

Ch DoldKannerve Sp(Grp) (A 2A 1A 0A 1A 2) N(A ) \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.


Inclusion into all spectra

The inclusion

Spectra 0Spectra 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 0Spectra_{\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)

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 (