Homotopy Theory

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Denis Nardin

08:08

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)

S. carmeli

08:20

@DenisNardin what about (_) (x)S?

Denis Nardin

@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…

Ian Coley

16:06

I know the letter 'Q' comes up somewhere but I can't remember what that's for ...

Piotr Pstrągowski

16:20

@Denis In your context, how would you define the corresponding connective spectrum? (Since it's before one knows the recognition principle?)

Tom Bachmann

Since this is still being discussed, I would denote the functor Sigma^oo_tr, for what it's worth ^^

William Balderrama

16:34

I've always written B^∞ in analogy with Σ^∞.

Denis Nardin

@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…

Aaron Mazel-Gee

20:58

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…