nLab connective spectrum

Idea

General

A connective spectrum is a connective object in the stable $\infty$-category of spectra, hence a spectrum $S$ whose homotopy groups in all negative degrees are trivial: $\pi_{\bullet \lt 0}(S) \,=\, 0$.

These are equivalently:

• infinite loop spaces;

• grouplike$\;$ symmetric monoidal ∞-groupoids;

• abelian ∞-groups.

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

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

There are objects in Spectra, 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.

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 $\tau_{\geq 0}$ from spectra to connective spectra is called the connective cover construction, part of the canonical t-structure on spectra (see there).

Related concepts

• connective object, connective cover

• t-structure

• model structure for connective spectra

Literature

• Jacob Lurie, pp. 150 in: Higher Algebra

	(∞,1)-operad	∞-algebra	grouplike version	in Top	generally
	A-∞ operad	A-∞ algebra	∞-group	A-∞ space, e.g. loop space	loop space object
	E-k operad	E-k algebra	k-monoidal ∞-group	iterated loop space	iterated loop space object
	E-∞ operad	E-∞ algebra	abelian ∞-group	E-∞ space, if grouplike: infinite loop space $\simeq$ ∞-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis