nLab
connective spectrum

Idea

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

• infinite loop spaces;

• grouplike symmetric monoidal ∞-groupoids.

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

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

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

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.

$$\begin{array}{ccc}{\mathrm{Ch}}_{+}& \stackrel{\mathrm{Dold}-\mathrm{Kan}\mathrm{nerve}}{\to }& \mathrm{ConnectSp}(\mathrm{\infty }\mathrm{Grp})\subset \mathrm{\infty }\mathrm{Grpd}\\ (\cdots A_2\stackrel{\partial }{\to }{A}_1\stackrel{\partial }{\to }{A}_0\to 0\to 0\to \cdots )& \stackrel{}{\mapsto }& N(A_{•})\end{array}$$\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 $\mathrm{Sp}(\mathrm{Top})$.

$$\begin{array}{ccc}\mathrm{Ch}& \stackrel{\mathrm{Dold}-\mathrm{Kan}\mathrm{nerve}}{\to }& \mathrm{Sp}(\mathrm{\infty }\mathrm{Grp})\\ (\cdots A_2\stackrel{\partial }{\to }{A}_1\stackrel{\partial }{\to }{A}_0\stackrel{\partial }{\to }{A}_{-1}\stackrel{\partial }{\to }{A}_{-2}\to \cdots )& \stackrel{}{\mapsto }& N(A_{•})\end{array}$$\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.

Related concepts

	(∞,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 $\simeq$ Γ-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis