For sequential spectra and for highly structured spectra such as symmetric spectra and orthogonal spectra, the functor which picks their th component space, for any , has a left adjoint .
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.
Explicitly, these free spectra look as follows:
For sequential spectra: ;
for orthogonal spectra: ;
for symmetric spectra: .
For the free construction is isomorphic to the corresponding structured suspension spectrum construction: . Generally, the stable homotopy type of is that of (Schwede 12, example 4.35).
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)
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