Timeline for What are cospectra, and why have they received so little attention?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 16, 2021 at 8:05 | comment | added | Niall Taggart | Just to add a reference to the pile. The type of cospectra with maps $\Sigma X_{n+1} \to X_n$ also appear in "Stable frames in model categories" by Lenhardt. If my memory recalls correctly, Lenhardt uses cospectra to show that the homotopy category of any stable model category is a module over the stable homotopy category, and that (symmetric) spectra (of simplicial sets) is initial among stable model categories. (There's a nice account of this story in "Foundations of Stable Homotopy Theory" by Barnes and Rotizheim.) | |
| Feb 10, 2021 at 1:42 | comment | added | Tim Campion | I had completely forgotten about the very surprising point in Nicholas Kuhn's comment, but it turns out he told me this before! The point being that the Bousfield-Kuhn functor has a left adjoint (as we know now, it's in fact monadic, given by the "Lie algebra" monad), and the left adjoint carries in the "spectrum" structure from the (stable) category of $K(n)$-local spectra. | |
| Feb 9, 2021 at 17:33 | comment | added | Nicholas Kuhn | Ah, I bet Gray's cospectra have maps $\Sigma X_{n+1} \rightarrow X_n$. And in appropriate $v_n$-local categories, there are objects that one can think of as infinite suspensions (analogous to infinite loopspaces). | |
| Feb 9, 2021 at 16:57 | comment | added | Neil Strickland | I think that Brayton Gray's cospectra are different; they are related to the colimit of $[\Omega^kX,\Omega^kY]$ as $k\to\infty$, and I do not think that that interacts with the construction in this question. (But I also think that Brayton's version is very interesting and that it deserves to be investigated further.) | |
| Feb 9, 2021 at 14:53 | history | answered | Nicholas Kuhn | CC BY-SA 4.0 |