# nLab Spectrum

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

## Idea

By $Spectrum$ or $Spectra$ one may denote the stable (∞,1)-category of spectra. It is also denoted $Sp$, or sometimes $Spec$ although that can be confusing.

Its homotopy category is the classical stable homotopy category.

For more see the entry stable (∞,1)-category of spectra.

## Properties

### Finite homotopy (co)limits of spectra

###### Proposition

A sequence of morphisms of spectra $E \longrightarrow F \longrightarrow G$ is a homotopy fiber sequence if and only if it is a homotopy cofiber sequence:

A proof is spelled out at Introduction to Stable homotopy theory (this Prop., following Lewis-May-Steinberger 86, chapter III, theorem 2.4 )

In fact:

###### Proposition

A homotopy-commuting square in Spectra is a homotopy pullback if and only it is a homotopy pushout.

This follows from Prop. by the fact that Spectra is additive (this Prop.).

###### Remark

This property of Spectra (Prop. , Prop. ) reflects one of the standard defining axioms on stable (∞,1)-categories (see there) and on stable derivators (see there).

### Other abstract characterizations

$Spectra$ is equivalently

category: category

Last revised on January 16, 2021 at 14:58:27. See the history of this page for a list of all contributions to it.