12:15 AM
Yeah true, I just meant for G-symmetric spectra there
And similarly to what you say, commutative monoids in GL-motivic spectra are probably richer. Hornbostel has a skeleton draft about those, but somebody has to write that down at some point!

8 hours later…
7:55 AM
@Bogdan for what it's worth, you need to work with "symmetric G-spectra" to make this true (because G-symmetric spectra is Hausmann's name for a model, where commutative monoids are equivalent to commutative monoids in orthogonal G-spectra)

4 hours later…
12:06 PM
I am slightly confused now. Is the main point that being a commutative algebra in these other categories is not the same as being an $E_{\infty}$-algebra?

12:39 PM
In LMS and in symmetric G-spectra (as Tom says) commutative algebras are the same as "naive" E_oo algebras
Marc Hoyois told me once that it's only with the black magic of orthogonal specctra that commutative algebras are richer (and I thought that this sentence was pretty badass)

ok. interesting. I am either having deja vu or remembering that Marc Hoyois quote.

1 hour later…
2:01 PM
@SeanTilson Strictly commutative rings in orthogonal G-spectra are a model for "G-commutative" ring spectra. They are basically E_∞-rings that come equipped with additional "norm maps" N^GR→R. Essentially you can multiply not only a set of elements of R but also a G-set of them. Computationally this expresses itself in the homotopy groups forming a Tambara functor.

2:18 PM
Has anyone developed a coherent nerve for simplicially enriched bicategories? I'm fine assuming that all the $2$-morphisms are invertible.
I guess one way is to first strictify, then apply the coherent nerve to the morphism categories to obtain a simplicial category, then apply the coherent nerve again.