# nLab spectrum object

### Context

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Idea

Every (∞,1)-category with finite limits has a stabilization to a stable (∞,1)-category $\mathrm{Stab}\left(C\right)$. This stabilization may be defined by abstract properties, but it may also be constructed explicitly as the category of spectrum objects in $C$.

In the special case that $C=$ Top, a spectrum object in $C$ is just an ordinary spectrum. There is an evident generalization of the notion of spectrum from Top to any $\left(\infty ,1\right)$-category $C$ with finite limits: a spectrum object ${X}_{•}$ is essentially a list of pointed objects ${X}_{i}$ together with equivalences ${X}_{i}\to \Omega {X}_{i+1}$, from every object in the list to the loop space object of its successor.

## Definition

For $C$ an (infinity,1)-category, a prespectrum object of $C$ is

• a $\left(\infty ,1\right)$-functor $X:ℤ×ℤ\to C$

• such that for all integers $i\ne j$ we have $X\left(i,j\right)=0$ a zero object of $C$

Notice that this definition is highly redundant. The point is that writing $X\left[n\right]≔X\left(n,n\right)$ a spectrum object is for all $n\in ℤ$ a (homotopy) commuting diagram

$\begin{array}{ccc}X\left[n+1\right]& \to & 0\\ ↓& & ↓\\ 0& \to & X\left[n\right]\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ X[n+1] &\to& 0 \\ \downarrow && \downarrow \\ 0 &\to& X[n] } \,.

Recalling that in an (infinity,1)-category with zero object

• $\Omega X\left[n+1\right]$ denotes the pullback of such a diagram;

• $\Sigma X\left[n\right]$ denotes the pushout of such a diagram

this induces maps

${\alpha }_{n}:\Sigma X\left[n\right]\to X\left[n+1\right]$\alpha_n : \Sigma X[n] \to X[n+1]
${\beta }_{n+1}:X\left[n\right]\to \Omega X\left[n+1\right]\phantom{\rule{thinmathspace}{0ex}}.$\beta_{n+1} : X[n] \to \Omega X[n+1] \,.

A prespectrum object is

• a spectrum object if ${\beta }_{m}$ is an equivalence for all for all $m\in ℤ$ (a spectrum below $n$, if ${\beta }_{m}$ is an equivalence for all $m\le n$);

• a suspension spectrum if ${\alpha }_{m}$ is an equivalence for all $m\in ℤ$ (a suspension spectrum above $n$, if ${\alpha }_{m}$ is an equivalence for all $m\ge n$).

One writes

• $\mathrm{Sp}\left(C\right)$ for the full sub-$\left(\infty ,1\right)$-category of $\mathrm{Fun}\left(ℤ×ℤ,C\right)$ on spectrum objects in $C$;

• $\mathrm{Stab}\left(C\right)≔\mathrm{Sp}\left({C}_{*}\right)$ – the stabilization of $C$ for the $\left(\infty ,1\right)$-category of spectrum objects in the $\left(\infty ,1\right)$-category ${C}_{*}$ of pointed objects of $C$.

## Properties

• If $C$ is a pointed $\left(\infty ,1\right)$-category with finite limits, then $\mathrm{Sp}\left(C\right)$ is a stable (infinity,1)-category.

## Examples

For $C=\mathrm{Top}$, $\mathrm{Stab}\left(C\right)$ is the $\left(\infty ,1\right)$-category version of the classical stable homotopy category of spaces: the stable (infinity,1)-category of spectra.

E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ Γ-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object