nLab cylinder spectrum

1. Idea

For $S_\bullet$ a spectrum, its cylinder spectrum $Cyl(E)_\bullet$ has as component spaces the reduced cylinder spaces over the original component spaces:

$$Cyl(E)_n \coloneqq I_+ \wedge E_n \,,$$

i.e. the smash product (of pointed topological spaces) of $E_n$ with the unit inveral with a dijoint basepoint adjoined.

Cylinder spectra serve to define left homotopy of spectra as for other cylinder objects.

2. References

Early lecture notes include

• Frank Adams, part III, section 2 Stable homotopy and generalised homology, 1974