Suppose I've got a symmetric monoidal quasicategory, is it naturally tensored over spaces, or pointed spaces?

Also, does anyone know if it's true in general, or under some conditions, whether or not tensoring commutes with totalization?

Or, rather, I suspect that it's not true in general, but am interested in knowing conditions under which this might occur, if any.

dang, another heat wave

Oh yeah? Where?

pacific northwest

What part?

@JonathanBeardsley if X_* is a cosimplicial object, let {Tot^k X_*} denote the tower of partial totalizations; then tensoring commutes with taking Tot if {Tot^k X_*} is pro-constant

if you're in spectra, then there's a condition --- i think due to bousfield --- which provides conditions on when a tower is pro-constant

this is what's used in the proof of the smash product thm

@skd Ah right, great. All right, I'll try to use that. I'm pretty sure I remember where the references for this stuff are.

Question on nomenclature: A set of invariants is complete if a certain map to the invariants is injective. What do we call it if the map is surjective?

that is every invariant is realized?

@ThomasRot I want to call it replete, but I just made that up.

Hey @JamesCameron! You in LA yet?

@JonathanBeardsley I am! I am liking LA, but it is very hot.

Summers are terrible. Make sure to get good AC.

@JamesCameron Ah jeez, I'm sure. I mean, it's unpleasantly hot in Seattle, so I can't even imagine southern California...

I would like to say that, given a Quillen equivalence of "nice" simplicial model categories, I get a pair of 1-simplices in $Cat_\infty$ and a couple of 2-simplices telling me that their compositions equivalent to the identity functors. Is this true? And if so, where is it written down?

I think that this is pretty close to being in a paper by @AaronMazel-Gee but I can only find it for adjunctions rather than equivalences.

More generally, I can't seem to find a good discussion of adjoint equivalences of quasicategories anywhere.

I feel like I just want to say that the unit transformation, which is required to exist to have an adjunction of quasicategories, is an equivalence.

It seems like maybe this could be finessed from knowing that the quasicategory functor underlying a Quillen equivalence is fully faithful.

Ah, and I guess the condition of being fully faithful and essentially surjective is just checked on homotopy categories, so the Quillen equivalence gives it to you.