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8:08 AM
Notation question: how should I call the functor sending an E_∞-space to its corresponding connective spectrum? I don't want to call it group-completion because I have to introduce it in the class I'm teaching before I prove the recognition principle (so that E_∞-groups are actually equivalent to connective spectra).
Segal in cats & cohomologies calls it $\mathbf{B}$, so if there are no better proposals I'll go with that, but it feels a bit wrong (its underlying space is not the classifying space of the $E_\infty$-space)
 
8:20 AM
@DenisNardin what about (_) (x)S?
 
@S.carmeli It feels a bit clunky to write down. Ideally this would be a single letter symbol, since I use it all over the place in the proof of the recognition theorem
For context: this is for a class I'm teaching
 
 
8 hours later…
4:06 PM
I know the letter 'Q' comes up somewhere but I can't remember what that's for ...
 
4:20 PM
@Denis In your context, how would you define the corresponding connective spectrum? (Since it's before one knows the recognition principle?)
 
Since this is still being discussed, I would denote the functor Sigma^oo_tr, for what it's worth ^^
 
4:34 PM
I've always written B^∞ in analogy with Σ^∞.
 
@PiotrPstrągowski Well, as the spectrum {B^nM}_{n\ge 1} with the obvious bounding maps
@WilliamBalderrama Ooohh I like this suggestion, I might adopt it thanks!
@TomBachmann I know, but explaining my students why I chose this strange notation will take too much time :)
[I'm not introducing "spaces with transfers" at the same time, even if it was a very tempting idea...]
@DenisNardin For the record, I'm following Segal´s paper "Cats & cohomologies" for this section of the course
 
 
4 hours later…
8:58 PM
what are the dualizable objects in the symmetric monoidal $\infty$-category of $p$-complete $\mathbb{Z}_p$-module spectra with the completed tensor product?
 
 
2 hours later…
11:10 PM
@AaronMazel-Gee These are just the perfect complexes of $\mathbb{Z}_p$-modules (the dualizable objects with respect to the discrete tensor product).
 

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