Annoying question: Gepner and Haugseng show that a lax monoidal functor $V \to W$ of monoidal $\infty$-categories induces a change of base functor $Cat^V \to Cat^W$ from $V$-enriched categories to $W$-enriched categories, and also that there is a delooping functor $B: Alg_{E_1}(V) \to Cat^V$. Obviously these two constructions should commute, but is that written somewhere?

I'm particularly interested in the case of the monoidal functor $h: Cat_\infty \to Cat$ sending an $\infty$-category to its homotopy category, so that I know that delooping commutes with passing to the homotopy (2)-category.

@TimCampion i think this is essentially immediate from the construction. or are you worried about "completeness" issues?

@AaronMazel-Gee No, I was just being lazy and not taking the time to fully understand the construction. The construction of the change of base functor, for instance, doesn't seem to be spelled out in a theorem statement but rather given in the course of a proof at some point.

In the end it's a model-independent statement, but if checking this requires getting into the model then so be it.

I see. In their model, B is really an inclusion functor, so it should be straightforward.

oh, okay gotcha. i wouldn't say it's a model-dependent assertion; the entire theory is essentially model-independent. (it does use $\Delta^{op}$ "for real", but not e.g. quasicategories or anything like that.) it's just this: a V-category is a certain morphism $codisc(X) \to BV$, and the resulting W-category is the composite $codisc(X) \to BV \to BW$. i'm using my own notation here, but these are certain simplicial $\infty$-categories.

$codisc(X)$ is RKan from $[0] \in \Delta^{op}$ -- it's a Segal space presenting the $(-1)$-truncation of the space $X$. and $B$ is just "categorical deloop".

all totally analogous to the presentation of ordinary enriched categories as functors from "the codiscrete category on the set of objects" to the deloop of the enriching category

i see. that makes sense.

(part of the reason i was being slapdash is that this is just one in a string of model comparison results i need to chain together for something. Specifically, I want to apply Riehl and Verity's results on the cofibrancy of the free adjunction as a quasicategory-enriched category to talk about duals a monoidal infinity-category.)

sounds good! incidentally, do you have a "model-independent" way of thinking about that cofibrancy? i've also heard emily say the word "computad", but i don't really know what that means.

I don't have a deep understanding of the cofibrancy. It's another case of me reading just enough to get the results to quote.

A similar thing came up recently in the work of Bauer, Burke, and Ching on tangent infinity-categories. They model monoidal infinity-categories as strict monoids in quasicategories, and by a miracle of cofibrancy, they're able to import a corepresenting object for tangent structures directly from monoidal 1-categories.

That was a case where I tried to read about it a bit more, but their proof of cofibrancy was very short, essentially saying that it was obvious.

For quasicategory-enriched categories, cofibrancy is the same as Bergner-cofibrancy, if I remember right.

So it basically means that your simplicial category needs to be levelwise a free category.

It's a strong requirement.

So it seems like a miracle for it to hold for naturally-occurring objects.

But apparently it does!

to connect the dots, I believe "computad" is just the Australian word for "Bergner-cofibrant simplicial category"

But I may be missing some subtlety.

@TimCampion We define this as the inclusion of E_1-algebras into the "pre-completed" category (AKA precategories, or categorical algebras in the paper) followed by completion; the first part Alg_{E_1}(V) -> Alg_cat(V)_* should be functorial "by inspection", while I think we discuss functoriality of the localization somewhere

Right, that's 5.7.1 in the paper

In his paper "THH and base-change for Galois extensions of ring spectra", Mathew proves that if a faithful Galois extension A-> B satisfies descent for THH (so $THH(B)^{hG}\simeq THH(A)$), and if moreover B satisfies some mild finiteness condition, then A -> B has to be étale; in other words cases of Galois descent for THH are almost all accounted for by étale descent. Are there Galois extensions where we don't know whether they satisfy descent for THH ?

Such an extension would have to not be étale (otherwise we know it does), and not be of "almost finite presentation" (otherwise we know it doesn't)

(I have to say I don't have a great intuition for the notion of "almost finite presentation"; I know the definition and I know what finite presentation is intuitively, but I'm not sure how to intuitively understand the "almost". It's not even clear to me whether KO -> KU is of almost finite presentation)

Hey someone knows what is the definition of generically surjective linear map between two graded R-modules? I do not understand the generically part and I did not find a good definition online.

@RuneHaugseng, Thanks that helps