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

I guess augmented means admitting an operad morphism to the unit operad... the identity operad?

I actually don't know what a non-unital $\infty$-operad is

the phrase "non-unital" doesn't occur in Higher Algebra

Me neither.

Although, "unital operad" appears.

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

Right.

there might be something to be said, though

gtg

cya

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 $Cat_\infty$?

Sorry I'm essentially away from keyboard right now.

No, watching football.

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

Actually I'm not sure about that, haha.

Yeah. Both games have enormous rulebooks.

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

Right.

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

@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"

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.

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.

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

yeah, i'm confused now too.

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.

Right. I agree.

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?

i'm confused too, is that really a thing you can build?

@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!

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?

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

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 $i$

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

Right

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

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...

Haha.

Well, luckily they won't be mandatory.

just replying to the poll

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.

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

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!

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.

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.

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?

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.

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

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.

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.

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.

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.

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

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.

@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

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.

surely the space of minimal Kan complexes mapping to a given Kan complex is contractible

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

Unless I misunderstood your comment.

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.

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.