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12:37 AM
Suppose I've got a symmetric monoidal quasicategory, is it naturally tensored over spaces, or pointed spaces?
 
1:14 AM
Also, does anyone know if it's true in general, or under some conditions, whether or not tensoring commutes with totalization?
 
1:36 AM
Or, rather, I suspect that it's not true in general, but am interested in knowing conditions under which this might occur, if any.
 
1:59 AM
dang, another heat wave
 
2:29 AM
Oh yeah? Where?
 
pacific northwest
 
3:31 AM
What part?
 
skd
@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
 
3:56 AM
@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.
 
 
9 hours later…
1:04 PM
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?
 
 
7 hours later…
8:20 PM
@ThomasRot I want to call it replete, but I just made that up.
 
 
2 hours later…
9:56 PM
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...
 
10:45 PM
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.
 
11:14 PM
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.
 

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