@AaronMazel-Gee Using a Hopf algebra structure such as iopscience.iop.org/article/10.1088/0305-4470/23/22/001/pdf (with some specific choices in the parameters) and comparing with a braiding structure from this docnum.univ-lorraine.fr/public/SCD_T_1997_0080_FRANZ.pdf it seems to work out in this specific context...

Nice the antipode structure seems to hold-up too! I had doubts but realized I had an error in my notes...

@AaronMazel-Gee I don't even know what y'all are talking about, but this only holds for so-called "quasitriangular" Hopf algebras

@AaronMazel-Gee I think what Peter and I object to is the cheapening of the term "triggered," but it seems like this was stated pretty clearly by Joe. I don't know about being "offended," but I think people for whom triggering events can be really painful and destructive, the idea that someone is "triggered" by, say, something that they find irritating, can feel sort of dehumanizing

@JonathanBeardsley could you please remove the "cheapening" comments in the interest of this not spiralling out of control?

I don't think he should. But we should move on.

having reminders on the starboard is, in my opinion, not conductive of "moving on."

Here are some facts: (1) if $\hat{p}\colon K^\rhd\to \mathrm{Fun}(S,C)$ is such that each composite $K^\rhd\to \mathrm{Fun}(S,C)\to \mathrm{Fun}(\{s\},C)=C$ is a colimit cone in $C$, then $\hat{p}$ is a colimit cone in $\mathrm{Fun}(S,C)$. (2) If $p\colon K\to \mathrm{Fun}(S,C)$ is such that each $K\to \mathrm{Fun}(S,C)\to \mathrm{Fun}(\{s\},C)=C$ admits a colimit, then $p$ admits a colimit.

Where does this get proved in HTT?

The discussion of functoriality of colimits in 4.2.2 gets close, but I don't see anything that addresses whether $p$ itself has a colimit.

Also, what is Remark 4.2.2.6 talking about?

In any case, I kinda want a more elementary proof of this.

@CharlesRezk 5.1.2.2 with X = C x S?

(Except it uses the "other" join K ◇ Δ^0)

Ah thanks. That's not where I expected to find it.

Yeah, 4.2.2 uses the other join too.

I still would like a more elementary proof. I.e., without cartesian fibrations (but probably I'd still need to use the alternate join/slice).

The proof that $C_{/p}\to C$ creates colimits is pretty easy.

@skull which comment? something starred?

@JonathanBeardsley Thinking about it, this might just be the answer I was looking for as it is the co-multiplication that enables the easy construction of tensor products of representations. In a quasi-triangular Hopf algebra, the coproduct implicitly defines a braiding through the universal R-matrix such that its representations form a braided monoidal category. Indeed, the Hopf algebras I work with are all quasi-triangular.

So, the answer to my initial question seems to be: ''Yes if the Hopf algebra is quasitriangular''.

@G.Bergeron yeah the slogan is that modules over Hopf-algebras are always monoidal, but they're braided monoidal if the Hopf-algebras are quasi-triangular.

@JonathanBeardsley in general, any comments where gentlemen are fighting in the war room! :-)

This is a war room?

i've noticed that the stars from those comments have been removed, which is a step in the right direction, imho

nah, @G.Bergeron that's just the metaphor that this ordeal started with ;-)

no. it wasn't

Honestly, I think we should let the matter rest, and not mention it again. Opinions were exchanged, everyone is richer for it and let's leave it at that. Please stop bringing it up

but, as denis says, we're done with that conversation

So Lurie replaces chain complexes in a stable infinity category with objects in this category that he calls "Gap," are people are familiar with this?

Like, for instance, if I hand you an honest chain complex in an Abelian category, how do I produce an object of Gap(Z,C)?

Is there some classical way of taking a chain complex and naturally producing from it a filtered object? Lurie shows that Gap(Z,C), which is sort of complicated, is equivalent to Fun(N(Z), C). So showing that an honest chain complex always resulted in a filtered object would do it I guess...

Where does he talk about Gap?

@SaulGlasman Definition 1.2.2.2

Of which work?

Oh sorry, haha, HA.

Ah, right

No, I could have figured that out

I guess, maybe somewhat orthogonally, does anyone know if there's a general notion of chain complexes in an arbitrary stable model category?

I think you want Remark 1.2.2.3

That shows how an object of Gap gives you something recognizable as a chain complex

If you start with a complex in an abelian category, though, I think your Gap object will take values in the derived category

@SaulGlasman So... there is an infinity category of chain complexes of abelian groups. Given an object of that, it feels like we should be able to produce something in Gap(Z, N(Ab))

I mean, maybe there's something to be said about like... chain complexes just don't really make sense in an arbitrary stable infinity category, and these Gap objects do.

It seems from 1.2.2.3 that if your complex is ...->C_n -> C_{n - 1} -> ..., then if F is the corresponding object of Gap, F(n - 1 , n) = C_n[n]

which is in D(Ab), not (Ab)

Yeah, I think that's exactly right

But that sort of seems strange to me, because of the classical Dold-Kan, i.e. given a strict chain complex you can produce a simplicial object, Fun(\Delta,C) and then just take nerves to produce a simplicial object with all higher coherences

So it's like, there's no way to produce a "chain complex up to all higher coherences" from a strict chain complex

Yeah.

That much makes sense in any pointed oo-category, I guess

But if you're in the nerve of a 1-category, then you don't need the coherences

Right. Hm.

But if you pursue those coherences, you must end up writing down something like Gap

Well, okay, but so there should be a sort of "nerve" functor that goes from strict chain complexes to some sort of "trivial" type objects of Gap

And I mean, of course there is, but it's like, going the long way around, haha, by going to simplicial objects, taking the nerve, getting a filtered object, and then taking the associated object in Gap

Yeah, I'm sure something like that is possible

Yeah, I mean, chain complex is in particular some kind of sequence of objects with conditions on the maps between them. So you could, for instance, start with a strict "chain complex" in spectra, and then take the nerve... And you'd want to see that this gives you some kind of very low-level object of Gap.

i.e. with a bunch of trivialized higher coherences...

A chain complex in a stable category gives you, in particular, a filtered object, and as Lurie discusses, the objects F(i, j) for general i and j are related to the higher pages of the resulting spectral sequence

If you started with a chain complex in an abelian category, then the spectral sequence you get out has to degenerate at E_1, and I'm guessing that means that the objects F(i, j) are zero whenever j - i > 2