Timeline for Multiplicativity of the homology Atiyah-Hirzebruch spectral sequence for a ring spectrum
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Jun 6, 2019 at 7:11 | comment | added | Tyler Lawson | And while letting fly with some $\infty$-categorical statement is perhaps irritating or irresponsible within the scope of the question, maybe that's what I should been more honest about and done instead; for the Whitehead tower, by its nature, is something that only wants to be defined up to contractible choice, and its natural pairing follows suit. | |
| Jun 6, 2019 at 7:08 | comment | added | Tyler Lawson | And so I think that, while it's bad to pretend that things are simpler than what they are, there is added value in understanding that there is some degree of canonicality here: e.g. that $Map(\tau_{\geq n} E \wedge \tau_{\geq m} E, \tau_{\geq n+m} E) \to Map(\tau_{\geq n} E \wedge \tau_{\geq m} E, E)$ is a weak equivalence, letting us say that the endomorphism operad of $E$ maps, perhaps after some zigzag of equivalences, to the endomorphism operad of the filtered object $\tau_{\geq n} E$. | |
| Jun 6, 2019 at 6:59 | comment | added | Tyler Lawson | The 2-categorical approach that you suggest works well to build the pairing. I worry about the gradual accumulation of complexity. Once you have constructed this filtered pairing, you probably want to know that the pairing on the spectral sequence is associative, commutative, unital; hence more homotopies and more 2-homotopies. And perhaps there is a clever way to do this; but the standard ways I can imagine proceeding look a lot like what one finds in Adams' blue-book proofs of associativity and commutativity of the smash product. | |
| Jun 6, 2019 at 6:53 | comment | added | Tyler Lawson | Dear John, you are completely correct, and to try and build this with a strict model is very difficult, and doing so functorially is even worse (requiring us to first contend with a model for a functorial Postnikov tower). I should have been more responsible and not pushed this issue under the rug. | |
| Jun 5, 2019 at 22:04 | history | answered | John Rognes | CC BY-SA 4.0 |