Timeline for Connective spectra and infinite loop spaces
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 14, 2017 at 20:01 | vote | accept | Matthias Ludewig | ||
| Sep 14, 2017 at 14:22 | comment | added | Nicholas Kuhn | Passing from $X$ to $X\langle 0\rangle$ is a co-localization. | |
| Sep 14, 2017 at 14:18 | comment | added | Nicholas Kuhn | By the way, one can also consider the category of algebras over the monad $\Omega^{\infty} \Sigma^{\infty}$, but even in this case, it is a little theorem that this category is equivalent to the category of infinite loopspaces (a result of Beck in the 1960's). | |
| Sep 14, 2017 at 14:12 | comment | added | Nicholas Kuhn | I have to strongly disagree with Fetisov's answer to this question. The definition of an infinite loop space should not involve operads, but should just involve what the word implies. The objects in the category of infinite loopspaces can be taken to be the same as the objects in spectra; but morphisms are different: if $X\langle 0 \rangle$ is the -1 connected cover of $X$, the map $X\langle 0 \rangle \rightarrow X$ becomes an isomorphism after applying $\Omega^{\infty}$. `user43326' (I hate anonymous postings) said it exactly right. | |
| Sep 12, 2017 at 21:21 | comment | added | Denis Nardin | I've decided to delete my answer because I really do not want to defend a terminology that I find confusing and imprecise. I agree that in these days people saying "infinite loop space" mean group-like E_∞-space (a.l.a. a connectve spectrum). Just a small correction: the group-like condition is very important, and it is easy to give examples of E_∞-spaces that cannot be delooped ($\mathbb{N}$, for one). | |
| Sep 12, 2017 at 20:46 | history | edited | Anton Fetisov | CC BY-SA 3.0 |
added 618 characters in body
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| Sep 12, 2017 at 20:22 | history | answered | Anton Fetisov | CC BY-SA 3.0 |