nLab
free spectrum

Contents

Contents

Idea

For sequential spectra and for highly structured spectra such as symmetric spectra and orthogonal spectra, the functor () n(-)_n which picks their nnth component space, for any nn \in \mathbb{N}, has a left adjoint F nF_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 nK) q=KS qn(F_n K)_q = K \wedge S^{q-n};

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

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

Properties

For n=0n = 0 the free construction is isomorphic to the corresponding structured suspension spectrum construction: F 0Σ F_0 \simeq \Sigma^\infty. Generally, the stable homotopy type of F nKF_n K is that of Ω n(Σ K)\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.