---
tag: Z38TW6347X
kind: definition
name: Spectrum
msc: [55P42]
defines: 1GHF4EKBXS
uses: [T9DG8DDE75, 2VD7GR64JJ, 2JFF5F3EGF, P976WYEFMB, 4Q3WY1RX91, AJHEYJE5XY]
sources:
  - url: https://en.wikipedia.org/wiki/Spectrum_(topology)
---

## Definition

A *spectrum* $E$ is a sequence $(E_n)_{n \ge 0}$ of [[pointed spaces|T9DG8DDE75]] together with pointed maps
$$\sigma_n \colon \Sigma E_n \to E_{n+1}, \qquad n \ge 0,$$
called structure maps, where $\Sigma$ is the reduced [[suspension|2VD7GR64JJ]].
A *map of spectra* $f \colon E \to F$ is a sequence of pointed maps $f_n \colon E_n \to F_n$ with $f_{n+1} \circ \sigma_n = \sigma_n \circ \Sigma f_n$ for all $n$.
For $k \in \Z$, the $k$-th *homotopy group* of $E$ is
$$\pi_k(E) = \operatorname{colim}_n \, \pi_{k+n}(E_n),$$
where the colimit runs over $n \ge \max(0, 2 - k)$ along the maps $\pi_{k+n}(E_n) \to \pi_{k+n+1}(\Sigma E_n) \to \pi_{k+n+1}(E_{n+1})$.

## Remarks

A spectrum is an *$\Omega$-spectrum* if each adjoint map $E_n \to \Omega E_{n+1}$ into the [[loop space|2JFF5F3EGF]] is a [[weak homotopy equivalence|4Q3WY1RX91]].
Then $\pi_k(E) = \pi_{k+n}(E_n)$ for all $n \ge 0$ with $k + n \ge 0$.
A map of spectra is a *stable weak equivalence* if it induces isomorphisms on all $\pi_k$.

Example: for a pointed space $X$, the *suspension spectrum* $\Sigma^\infty X$ has $(\Sigma^\infty X)_n = \Sigma^n X$ and identity structure maps.
Its homotopy groups are the stable homotopy groups $\pi_k^s(X)$ for $k \ge 0$, and they vanish for $k < 0$.

Example: for an abelian group $G$, choose [[Eilenberg–MacLane spaces|AJHEYJE5XY]] $K(G, n)$ and weak equivalences $K(G, n) \to \Omega K(G, n+1)$; their adjoints make $(K(G, n))_n$ an $\Omega$-spectrum $HG$ with $\pi_0(HG) = G$ and $\pi_k(HG) = 0$ for $k \neq 0$.

Non-example: a sequence of pointed spaces alone is not a spectrum, since the structure maps are part of the data.
For instance, $E_n = S^n$ with every $\sigma_n$ constant is a spectrum, but it is not the sphere spectrum.
All its homotopy groups are zero, because the maps in the colimit are zero.

An $\Omega$-spectrum $E$ defines a generalized cohomology theory by $E^n(X) = [X, E_n]_*$ for $n \ge 0$, extended to all $n \in \Z$ by suspension.
Other models, such as symmetric spectra and orthogonal spectra, add structure to get a well-behaved smash product.