hey, isn't this supposed to be a big deal? arxiv.org/abs/1705.07442

a big deal for HoTTies, at least

whoa

oh, neat

so a simple but technical question: are the pointwise formulae for calculating values of a left Kan extension and its homotopy variant "compatible"? I have a situation where the usual left Kan extension is given by a colimit that is already a homotopy colimit for each object. I want to deduce that the usual left Kan extension is naturally equivalent to any choice of homotopy Kan extension

@JoeBerner What do you mean by "compatible"? They are the same formulas (only with a homotopy colimit instead of a colimit), aren't they?

@JonathanBeardsley Well, sure the composition agrees, but if your regard it as being a monoidal structure on End(FinSet_bij), then you basically have no non-identity operads since the structure map O o O --> O is forced to be a natural isomorphism. That's what I meant about the difference between combinatorialists and topologists.

@DenisNardin that is only to compute particular values of the functors on objects, so my question is essentially if this is compatible for morphisms.

How much abstract algebra is in homotopy theory?

@Zee lots

damn...I guess I gotta examine my relationship with rings

@JoeBerner this isn't quite what you asked for, but if we wanted to be really honest we'd check that the strict fiber product $A \times_B B_{b/}$ of relative categories models the homotopy fiber product (in the barwick--kan model structure). i'm sure this is false in general; if they're model categories and $A \to B$ is a quillen adjoint (not sure which handedness offhand) and $b \in B$ is cofibrant, then this might have a shot of being true

i think lurie only proves consistency theorems in HTT for simplicial model categories, though i could easily be wrong

i guess i'd want $A \to B$ to be right quillen, so that its values on fibrant objects are fibrant, so they receive the maps from $b$ that they're supposed to

Anyone free to answer a dumb enriched category theory question?

Almost without fail, when I want to ask a question in here and start writing it down, I figure it out myself.

:-\

If E is an E_oo ring spectrum, does the tensor up functor Spt --> E-mod preserve E_oo algebras ?

@JonathanBeardsley yeah shoot! I probably won't be able to answer though

@Bogdan yeah, tensoring with an E_\infty ring spectrum is symmetric monoidal

iirc

So E-mod gets the relative smash product, right ? I see that the underlying spectra of X ^ Y ^ E is the same as X ^ E ^_E Y ^ E but wasn't sure about the E-module structure.

Okay thanks, do you know of a reference that I can cite ? Or is that just well-known and I don't need to ?

@Bogdan see Higher Algebra Theorem 4.5.3.1 and Remark 4.5.3.2 which follows it

Thanks!

I imagine there are other references, depending on what setting you're working in. Probably in, e.g., EKMM.

It's for motivic spectra, so I think your ref of Lurie works pretty good

Ah okay. Yeah, so long as you're working in, say, the quasicategory of motivic spectra.

Although operadic stuff in motives stresses me out, haha.,

yeah here I'm just talking about algebras over a simplicial E-oo operad, so it's just using the structure of oo-cat so I think it's okay and not too scary

if you use Jardine's motivic symmetric spectra, these are the same as strictly commutative monoids (like it's the case in equivariant stuff)

I would guess it's the same for Po Hu's motivic S-modules

@Bogdan Wait that is false for equivariant orthogonal G-spectra