nLab free spectrum

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

• Mark Hovey, Brooke Shipley, Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149-208 (arXiv:math/9801077)

• Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, Model categories of diagram spectra, Proceedings of the London Mathematical Society, 82 (2001), 441-512 (pdf)

• Stefan Schwede, section I.3 of Symmetric spectra (2012)