# nLab free spectrum

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

## Idea

For sequential spectra and for highly structured spectra such as symmetric spectra and orthogonal spectra, the functor $(-)_n$ which picks their $n$th component space, for any $n \in \mathbb{N}$, has a left adjoint $F_n$.

A structured spectrum in the image of this free functor is called a free symmetric spectrum or free orthogonal spectrum, respectively (Hovey-Shipley-Smith 00, def. 2.2.5, Mandell-May-Schwede-Shipley 01, section 8, Schwede 12, example 3.20).

For a general abstract account see at Model categories of diagram spectra the section Free spectra.

###### Proposition

Explicitly, these free spectra look as follows:

For sequential spectra: $(F_n K)_q = K \wedge S^{q-n}$;

for orthogonal spectra: $(F_n K)_q = O(q)_+ \wedge_{O(q-n)} K \wedge S^{q-n}$;

for symmetric spectra: $(F_n K)_q = \Sigma(q)_+ \wedge_{\Sigma(q-n)} K \wedge S^{q-n}$.

## Properties

For $n = 0$ the free construction is isomorphic to the corresponding structured suspension spectrum construction: $F_0 \simeq \Sigma^\infty$. Generally, the stable homotopy type of $F_n K$ is that of $\Omega^n (\Sigma^\infty K)$ (Schwede 12, example 4.35).

## References

Last revised on April 21, 2016 at 12:34:47. See the history of this page for a list of all contributions to it.