1:08 AM
Is there a correspondence between non-unital $\infty$-operads and augmented $\infty$-operads? What should augmented mean in this context?

I actually don't know what a non-unital $\infty$-operad is
the phrase "non-unital" doesn't occur in Higher Algebra

Me neither.

ah, I see
yeah, that makes sense

Just none of the $\infty$-operads we're used to are non-unital. I'm not sure though how one might produce the "augmentation ideal."

so naïvely the answer to your question is no, because it seems like it's sensible to define the collection of "non-unital \infty-operads" as the collection of all $\infty$-operads

1:22 AM
Right.

there might be something to be said, though
gtg

user105491
1:34 AM
Pretty sure you know about what I'm talking about already, but there's a "relation between the ∞-categories of nonunital E∞-rings and that of E∞-rings augmented over the sphere spectrum" (Urs's wording); see my answer here.

user105491
But I don't know what a nonunital \infty-operad is

Yeah. That's a standard construction for rings. So, there should be something along those lines for $\infty$-operads inasmuch as they are algebras in a certain category.
Well, so... in the sense that $\infty$-operads are $Fin_\ast$-monoids in $Cat_\infty$... are they essentially the $E_\infty$-rings of $Cat_\infty$?

user105491
It seems so; because $\mathrm{N}(\mathcal{F}\mathrm{in}_{\ast})=\mathrm{Comm}^\otimes$, that should be right.

user105491
We should be able to use Proposition 5.2.3.15 in HA when $\mathcal{O}^\otimes=\mathrm{Comm}^\otimes$ and $\mathcal{C}$ is the symmetric monoidal $\infty$-bicategory of $\infty$-categories, so that "nonunital symmetric monoidal $\infty$-categories" are "commutative monoids" of $\mathcal{C}$.

user105491
And then somehow remove "symmetric monoidal" and replace it with "operad" by removing "coCartesian fibrations".

user105491
1:49 AM
@Jon I think I know the solution. Or something close to it. In Warning 2.3.1.2 of HA, Lurie says that the $\infty$-categorical generalization of "an operad $\{O_n\}_{n≥0}$ which has a distinguished unary operation $id ∈ O_1$, which is a left and right unit with respect to composition" is "s built-in to his definition of an ∞-operad".

user105491
So we need something more general than his definition.

user105491
2:04 AM
Lurie uses nonunital $\infty$-operad in a different sense, see the second paragraph on page 181 of HA, for example

user105491
He uses it to mean $\infty$-operads that are not unital, but as I pointed out above, I don't think this is what you want.

user105491
Is it?

Sorry I'm essentially away from keyboard right now.

user105491

No, watching football.

user105491
2:12 AM
Ah, OK

user105491
I don't even understand that game (except for the touchdowns). What are the fouls and stuff? :-)

Oh, ummm, well it's less complicated than rugby. ;-)

user105491
Suffice it to say that I don't understand rugby either. :-)

Actually I'm not sure about that, haha.
Yeah. Both games have enormous rulebooks.

user105491
Anyway, here are my thoughts - if the answer to my question is no, then I think the problem lies in the fact that since $\mathrm{Map}(\{X\},X)$ is the union of those components of $\mathrm{Map}_{\mathcal{O}^\otimes}(X,X)$ which lie over $\beta:\langle 1\rangle \to\langle 1\rangle$ such that $\beta^{-1}(\ast)=\ast$, but this is trivially always true,

user105491
2:16 AM
so that $\mathrm{Map}(\{X\},X)\simeq\mathrm{Map}_{\mathcal{O}^\otimes}(X,X)$, and there's obviously an element in $\mathrm{Map}_{\mathcal{O}^\otimes}(X,X)$ that acts as the unary operation id

user105491
But this is what we'd like to eliminate

Right. As you say, this notion is sort of embedded in Lurie's notion of $\infty$-operads.

user105491
Boiling it down to the bare essentials, we need to tweak these mapping spaces, which cause this problem

user105491
Basepoint-less mapping spaces or something

user105491
We should generalize his definition of $\infty$-operads by letting the mapping spaces to be not necessarily pointed

user105491
2:21 AM
In other words, replace $\mathcal{F}\mathrm{in}_{*}$ with the category of all finite sets that are not necessarily equipped with a distinguished point ∗.

user105491
This generalizes his definition to something that could be suitable for defining nonunital $\infty$-operads.

Right.

user105491
Does that look close to what you were expecting?

user105491
That was just off the top of my head, so this very likely contains some gaps

I didn't really have any expectations. Just thinking out loud.

user105491
2:26 AM
So a nonunital $\infty$-operad is map $\mathcal{C}^\otimes\to\mathrm{N}(\mathcal{F}\mathrm{in})$ satisfying some conditions

user105491
As for augmented $\infty$-operads - they should be $\infty$-operads with a map of $\infty$-operads to ... $E_0$, maybe?

user105491
I think $E_0$ because $\mathrm{Alg}_{/E_0}(\mathcal{C}^\otimes)\simeq \mathcal{C}^\otimes$, so in some sense because rings may be regarded as $\mathbf{Z}$-algebras, $\infty$-operads can be regarded as "$E_0$-algebras".

user105491
2:40 AM
As for the correspondence, I honestly have no clue.

@RingSpectra @JonBeardsley i have to run, but the basepoints in $Fin_*$ are about giving ourselves the ability to forget objects in a tuple, they don't have anything to do with units. rather, units are requested by maps $<I> \to <J>$ which are not surjective onto $J$ (notably the unique map $<0> \to <1>$)

@AaronMazel-Gee perhaps some other time, but i'm not sure what you mean by the phrase "units are requested"

user105491
OK, I see what you're saying, but what I was saying was that these chosen basepoints can be interpreted as the "unit", in some sense. I may be wrong.

I think, @RingSpectra that the warning you're referring to is actually telling us that we don't have to fiddle too much.
In other words, classically a unital operad has a nullary operation, called the unit.
and that lives in C(0), if C is the operad.
and that's exactly the circumstance Lurie says he's approaching, in that warning.
But... then again, I'm not sure what it would mean to have the space Map(\emptyset,X) not be contractible. So, it's not clear to me.

user105491
Ah, but that's the second point in the warning. The first point says that the operads, the way he's defined them, are implicitly unital.

user105491
2:46 AM
Unital in the sense of the first bullet.

No I know, but I'm saying that if we use the classical case as guide, then we don't need to change the definition.
Because what we would mean by unital is having a unique nullary operation.
And that is NOT built in to Lurie's definition.
In some sense the relevant fact is whether or not $O^\otimes$ is pointed, as a category, I think.

it is, the unit comes from the map {x} -> {1,x}
where x is the basepoint
this is <0> -> <1> that Aaron mentioned

@TylerLawson what does "it is" refer to?

but if you don't want units you might try working over finite pointed sets and surjective maps instead
"it is" means that the unit is built into Lurie's definition

Oh I see.
Hm.

user105491
2:50 AM
@Tyler Ah, OK. I was taking it differently and trying to remove the pointed assumption, but I forgot about the morphisms, which were the things that actually needed changing.

In other words, the nullary operation requires an additional definition, and the unary operation is built in.

user105491
I thought you were concerned about the existence of the unary operation; if not, you can just say that if this nullary operation does not exist, i.e., if the operad is not pointed as an infinity-category, then this operad is nonunital.

user105491
Or can be called nonunital.

yeah, i'm confused now too.

user105491
I guess I'm the source of this confusion; I thought Jon was trying to define nonunital operads by removing this unary operation $id$. But I now think that the real thing behind it is this nullary operation $e$. So this can be remedied by suggesting that this $\infty$-operad is not pointed as an $\infty$-category.

2:56 AM
Well, again, I don't really have some vision here.
I suppose, if one wants to parameterize, say, non-unital $E_\infty$-rings, one should indeed remove this unary operation, as you both suggest.
It's just that the indexing seems to be a little different than some of the indexing other sources use. I'm basically thinking about the non-unital operads of, say, Harper and Hess' paper on homotopy completion.

user105491
So nonunital rings are rings without the assumption that a multiplicative identity exists. As you were asking for some equivalence with augmented algebra, I assumed you were doing something without this unary operation, and this is what I was trying to do.

Right. I agree.

user105491
I guess @Tyler was closer to what you wanted; see for example Definition 5.4.4.1 of HA

user105491
He considers maps <n> -> <m> such that these maps are surjective.

user105491
In this subcategory of Fin_*

user105491
3:04 AM
So these surjectivity assumptions should tell us that a nonunital \infty-operad is a map C -> N(Surj) satisfying those conditions.

user105491
There is probably something better than what I said, though.

user105491
Gtg

5:41 PM
i'm confused: if i have an E_infty ring A, then i can build equivariant versions of A-cohomology from a commutative group scheme G over A. what is G when A is an elliptic cohomology theory? it's supposed to just be the elliptic curve or something, but that's a group scheme over \pi_0(A); how do i refine this to a derived group scheme over A itself?

5:52 PM
i'm confused too, is that really a thing you can build?

user105491
Are there any examples of this refinement in other cases (not when A is an elliptic cohomlogy thoery) that you can imitate in this case?

@ZhenLin what's the latest version of your notes? the 2014-12-16 version was 1121 pages, and i just noticed that any moment now you'll probably break Higher Algebra's record (currently 1178) for "longest document in all of homotopy theory"... if you haven't already!

user105491
@Aaron Where can I find these notes? I'm curious.

maybe it should come as the restriction of the universal one which lives over the derived moduli stack?
that was a dumb statement
maybe it wasn't, gah i'm confused

@Arpon do you have a reference for the construction you claim here?

6:07 PM
oh yeah this is in Lurie's survey on elliptic cohomology
ok i at least decided that statement was indeed dumb, but i'm still confused about my original question.... maybe i just assumed this could be done because the whole discussion is supposed to be working towards building equivariant elliptic cohomology, but now i'm not sure

@AaronMazel-Gee does the nLab count?
@RingSpectra i hope he doesn't mind me sharing the link here: zll22.user.srcf.net/writing/homotopical-algebra

user105491
6:24 PM

7:05 PM
Some more questions on enrichments: If a category C is tensored and cotensored over graded sets, I get for each object X two objects S^i X, and O^i X (think of it as suspension and loops), representing the co and contravariant nth degree homs. Are these objects neccesarily isomorphic?

I'm a little confused - suspension and loops are rarely isomorphic
maybe you want an isomorphism between $*_i \otimes X$ and $Fun(*_{-i}, X)$? where $*_i$ is the point in degree $i$

user105491
7:27 PM
Me and Jon were talking about this in the htpy chat room: I was thinking about something Sean Tilson was saying in the early day(s) of this chat room - "I think the the way to get people who know things in here is to talk about math and start linking to discussions in here on the main site".

user105491
Do you think that we can pursue this?

@SaulGlasman you're right, yes , I want an iso between those two you mention. To call it loops and suspensions is very wrong

user105491
@Dedalus So something like the tensor-hom adjunction

user105491
Right?

Right

7:33 PM
@RingSpectra I guess I understood more what you were saying was the possibility of having Seminar type activities in here, that are planned?

user105491
@Jon Something like that, yes

The other possibility is having this in separate chat rooms which can be created to have only a limited number of people with speaking privileges. They would then be saved in the MO chat database forever.

user105491
Actually, that's an amazing idea!

user105491
This can be like the relaxation room. :-)

user105491
How do you create a room and associate it to MO?

user105491
7:37 PM
I mean, before anyone does that, if no one's interested, it'll be sorta useless.

user105491
Thus the poll question above

I mean, I think if you go click on "MO Site Rooms" above, it lets you create a room.

more seminars is not really what my mathematical life needs right now...

user105491
@Peter It'll be an informal "seminar"; more like a discussion focused on something specific

Haha.
Well, luckily they won't be mandatory.

7:39 PM

I was thinking it would be nice to have an IRC channel, where things are not logged forever...

I don't know how to use IRC.
0

This was something that came up recently in the homotopy theory chat room. Even if it wasn't to homotopy theory, has MO ever considered asking active users with interesting research to host lectures or seminars in chat rooms specifically for this purpose? They could of course be in rooms where on...

It's not too complicated. Unfortunately people wouldn't find out about it through MO though

user105491

user105491
@Jon Awesome! That opens it up to a wider audience.

7:43 PM
Yeah, there are a lot of AG people on MO, so I think everyone could benefit from it. It would also foster interdisciplinary communication.
Well, right, that's what I'm saying!

user105491
@Jon My internet's really slow :-P

I know that #math on efnet had some lectures a long time ago. I can't say whether it worked out fine or not.

So did ##math in freenode

the freenode ones were more tedious than just reading a set of lecture notes
there's not really much opportunity to interrupt and ask a question, and there are no theatrical elements that you find in in-person lectures
so instead it's primarily choppy text
maybe a different crowd could do a better job, idk

I suspect the same holds for the ones I think of. I am not sure the seminar-format fits in a chat room. But yeah, maybe one could do it differently.

7:52 PM
You would probably need a better solution than texxing expressions to avoid constant pauses while the lecturer tweaks her mathjax code.
Perhaps it could work with a free drawing feature.

Yeah, in the early days of this chat room people were really interested in using a tablet to quickly draw diagrams.
It's sort of complicated. One way to do it is to draw something and have it be automatically saved to the internet, since you can "upload" images from the internet.

Probably possible using dropbox or something similar.

Right.

Using a public dropbox, it would be sufficient to save diagrams there, and refer to them by filename.

Basically if you're going to "give a seminar" you'd probably have to do some preparation beforehand.

8:00 PM
the freenode people did more than prepare diagrams, most of them prepared text and pasted it in, a line at a time, at a reasonable pace

Sure. Yeah.

(which is a lot like reading a set of notes)

Yeah, for it to have any value it'd definitely have to have a level of interactivity and be somewhat conversational.
I think the trick would be finding the happy medium (if any exists)
Between super informal conversation and one person pasting lecture notes line by line for an hour.

Perhaps a live "Khan academy" style feed with audio would be good?

user105491
I think @Espen's idea is great.

8:06 PM
Is that like, tablet and stuff?

I imagine either with a tablet, or a camera broadcasting a piece or paper.

I've thought about that before. Every summer I teach calculus over the internet with a live audio feed and a whiteboard that i draw on with a tablet.

The only downside is the hardware requirements, even though tablets have been getting increasingly popular lately.

Yeah. All the stuff I use I get from the school.

I gave one of the freenode lectures, and I can't say it was a good format. There were some questions but I agree that something much more interactive would be better.

8:10 PM
Haha, I just realized that at least 3 of my projects as a graduate student have been inspired by things John Rognes has written.

@Jon are you a computation oriented guy? :)

@EspenNielsen Nope, not at all! =P
I just realized that two projects my advisor gave me come straight out of Rognes' redshift paper and my own thesis project on hopf algebras in spectra also comes from Rognes.

Personally I get scared of algebraic K-theory. Is there any reason studying it if I'm not a (derived) algebraic geometer?

Um, I'm not sure. I'm certainly not the right person to ask though.
I suppose the fact that it's hard to compute, like the homotopy groups of spheres?

certainly there are people who study it that I doubt consider themselves algebraic geometers in any capacity

8:21 PM
Yeah, I get that. I mean, these days you can study abstract homotopy theory without considering yourself a topologist.
(No offense to any abstract homotopy theorists out there)

there are manifold geometry questions tied into the algebraic K-theory of spaces
pun-free

@EricPeterson Are you referring to E_n algebra stuff?

i mean A-theory and upside-down-A-theory and surgery theory and other things i don't understand but have sat through talks on

I dunno. It's kind of bizarre that THH approximates K-theory. What's that about?

@EricPeterson I found an article of Wald (found = first Google search result) which looks interesting. I might have a look at that.

8:26 PM
Waldhausen was originally a 3-manifolds guy IIRC

I'm so used to dealing with based spaces that I frequently forget that the empty set is a thing...

@JonBeardsley Recently I've been toying with the idea of looking at Kan complexes equipped with a trivial cofibration from a minimal Kan complex, as a strengthening of the basepoint-notion. I have yet to figure out what limits and colimits of such things should look like.

@Espn
haha oops. too much sports this weekend.
@EspenNielsen for any particular reason? or just cuz?

My motivation comes from a certain approach to classify homotopy types

Ah okay. Cool.

8:31 PM
For which it might be useful to have minimality of stuff
But the details are still fuzzy

I see.

@Espen why would that be offensive? Abstract homotopy theory has little to do with topology..

Well, I'll take issue with the statement that it has little to do with topology, haha.
Perhaps one could make the statement that it doesn't necessarily have to be strongly connected to topology.
haha @SaulGlasman you may not have found anything "nonunital" in higher algebra if you did a text search since there's a typo and it's spelled "nounital"
And then there's all this stuff about quasi-units (which are, I think, basically units only up to E_1 coherence)
This is just the stuff that @TylerLawson and @RingSpectra were trying to explain to me last night that I wasn't understanding.

8:52 PM
oh and also I was only searching for the hyphenated version
shoulda known

Yeah, somehow Tyler and Sanath both knew that precise part in chapter 5 of Higher Algebra.
Of course, Lurie seems way more interested in showing that you don't need all higher coherences to have a good unit, i.e. that quasi-unital algebras are equivalent to unital algebras. But he doesn't seem to go into great detail about nonunital algebras.
Oh actually, maybe that's not true. I'll keep reading, haha.

@JonBeardsley well, it has as much to do with topology as category theory has to do with sets, I would say.

9:10 PM
So then... what category are $\infty$-operads the $E_\infty$-algebras of?
I guess just $Cat_\infty$?

they're not
symmetric monoidal categories are the E_\infty algebras of Cat_\infty
\infty-operads are just their own thing

9:25 PM
Yeah.... Hrm.

@Adeel no, the nlab isn't really a single document. maybe nor should the stacks project count, that one for content reasons? i dunno, i've never really looked at it..
@EspenNielsen i feel like i should warn you: ive spent more hours of my life than i shouldve trying to brute-force the notion of a minimal kan complex to be somehow functorial. the inclusion {minimal-Kan} --> {Kan} of sset- (even Kan-)enriched categories induces an equivalence of quasicategories, and so there is a backwards functor on underlying quasicategories (which can be extracted directly as the homotopy-coherent nerves, no Bergner-fibrant-replacement necessary).
...but everything i tried to do at the sset-enriched level ended up breaking sset-enriched functoriality in some way or another; surely i was just running up against the same fact about the \infty-category of Spaces over and over
i think i was trying to construct a section of the "target" map $Fun([1],Spaces) \to Spaces$ which, for every space $X$, selected an effective epimorphism $S \to X$ from a set $S$ (i.e. an essentially discrete space)
i don't know if this has any bearing on what you're trying to do, but just f.y.i.

9:58 PM
@AaronMazel-Gee Thanks for the warning. That's interesting. I also thought that there was no way to have functorial minimal-Kan replacement, for the same reason that you cannot have a strictly functorial functor from spaces to based spaces preserving connectedness. I had resigned to assuming from the outset that every Kan complex comes equipped with a chosen minimal Kan replacement, and looking at maps preserving this replacement.
@Adeel Some mathematicians take pride in knowing that their practice is "grounded", in a sense. For example, I know a professor who views abstract algebra as worthless except as a tool to study geometry (he probably exaggerated for comedic effect). I imagine there are homotopy theorists to feel the same way about model (or higher) categories.

@EspenNielsen do you have some minimal counterexample for that (either 1- or \infty-categorical), the attempt to functorially choose basepoints? it's intuitively clear enough, probably even the circle should do it. but also, why not just study the minimal kan complexes themselves?

@AaronMazel-Gee I think the following example is sufficient. If you look at the 1-point space and the 2-point space, you can send the one point to either of the two points of the other. If you choose one of the two points as the base point, you cannot define the image under the choose-basepoint functor for the embedding into the other point.

oh haha sure, yeah i guess even that'll do it

@AaronMazel-Gee Do minimal Kan complexes have homotopy colimits?

@EspenNielsen hadn't thought about it, but i'd doubt it -- the "fat" that you're adding in is exactly what minimal kan complexes are built to avoid

10:11 PM
Unless I end up needing any homotopy colimits, I guess I can just work in minimal Kan complexes. :P The thought hadn't occurred to me.

10:22 PM
surely the space of minimal Kan complexes mapping to a given Kan complex is contractible

10:37 PM
@SaulGlasman I think this is true in general only if you restrict to the trivial cofibrations.

yes, sorry

Or even then, I think you need the condition that the Kan complex in question has no two homotopy equivalent path components.
Etc in higher dimensions

the way I'm thinking of it, I don't think automorphisms of the target matter
I'm thinking of a simplicial set where 0-simplices are trivial cofibs M -> X where M is minimal, 1-simplices are equivalences M -> M' over X, ...

If you quotient out by the homotopy automorphisms and require that the maps M -> X are homotopy equivalences, then I you are right, since given any two such M, M', you can inject one into X and deformation retract X onto the other to obtain a morphism between them, so the nerve is contractible.

10:59 PM
right, but you don't need to explicitly quotient
M has trivial homotopy automorphism group over X, no matter how many automorphisms X has

Yes, you are of course right about that.