@ClarkBarwick any thoughts on how to construct quotients of $E_n$-algebras?

@SaulGlasman @AaronMazel-Gee what about this... if $Y$ is an $E_n$-loop space there's a functor picking out $Y$, $f:E_n\to Top$, and then there's the inclusion $Top\hookrightarrow Cat_\infty$, so we have this map $E_n\to Cat_\infty$ picking out $Y$ as an $E_n$-monoidal quasicategory, yes? What category IS this?

I mean, okay stupid question... it's clear what category this is.

I guess this is different from the full sub-$\infty$-groupoid of $Top$ generated by $Y$.

yeah, i think you mean $\mathbb{E}_n \to \mathrm{Top}^\otimes$ (with respect to the cartesian monoidal structure), and the map $\mathrm{Top} \to \mathrm{Cat}_\infty$ is (strong) symmetric monoidal, so i don't think you're gaining anything new

(by passing to $\mathrm{Cat}_\infty$, i mean -- i agree that this isn't the full subcategory (or subgroupoid) of $\mathrm{Top}$ on $\{ Y \}$)

it's obviously true on objects/elements, i'm just not completely convinced that this is truly a pushout in categories

@Jon: I still don't know what you mean by "quotient of an n-fold loop space by another n-fold loop space." what universal property are you asking this quotient to satisfy?

(also what's the data? is it a morphism or an action or what?)

@jon I also don't know what you mean by "$BG/BH$".

I mean that if you think about $BG$ as being a point with a morphism for each element of G, I want to put in cells to make each such morphism coming from H equivalent to the identity morphism.

A map $H\to G$ induces an $H$-action on $G$, so for any other map $G\to X$ for $X$ an algebra object such that the induced map $H\to G\to X$ induces the trivial $H$-action on $X$, we have a factorization $G\to G/H\to X$.

@QiaochuYuan that's the "universal property" I'm asking for.

I don't know, I guess I'm not being clear, but really the picture in my head is just making all the morphisms in $BG$ coming from $BH$ be equivalent to the identity morphism.

@PeterNelson if $G$ and $H$ are both $n$-fold loop spaces, then $BG$ and $BH$ are $n-1$-fold loop spaces, and the induced map $BH\to BG$ means that $BH$ acts on $BG$ (through the multiplication on $BG$) and so we should be able to take the quotient $BG/BH$, once we determine what that quotient IS

of course, intuitively it's quite clear, an action of an $n$-fold loop space $X$ on another $n$-fold loop space $Y$ should induce a morphism (by some mysterious mechanism) $BX\to Top$ that picks out $Y$ and some collection of morphisms $Y\to Y$ (along with some more data, of course), and we can take the colimit of this diagram in Top to produce, I would hope, a sensible notion of $Y/X$.

Apologies for being dense and not particularly $\infty$-categorical, but if you're thinking of $BG$ as a 1-category with one object (or the quasicategory that is its nerve), what is $G$?

An actual topological space.

Like, with a bunch of elements, and a group structure.

Alright. Sorry for bugging everyone with all this business. Going to bed now. Cya.

@Jon: yes, but that notion of the quotient only sees the E_1-structure of X, and it seems like you were complaining about this earlier

there's a lot of detail hidden in how you specify the universal property; to specify a universal property of an object in an infinity-category you need to specify mapping spaces, and so you need to tell us what the "space of maps G -> X + trivializations of the induced H-action" is in order to tell us a universal property

I think I want to try to mimic ando blumberg and gepner and do something with Pic(GMod) and Pic(HMod), and notice that H-->G induces a functor Pic(HMod)-->Pic(GMod) and then notice that there's this map H-->Aut(H)-->Pic(HMod)-->Pic(GMod)-->Top, and take the colimit of that, or something along these lines...

spectral sequences look like good stuff. i'm currently picking this up from this exposition, hoping to study some of these from Hatcher's (other) book. can someone suggest some references which will be accessible to me, preferably containing some geometric applications of spectral sequences? (background : i have studied homology from Hatcher, but don't know cohomology yet)

I find Hatcher's spectral sequence book to be really clear and helpful.

ah, that's nice to know.

Although I think it's unfinished.

But just the first chapter makes it really clear what spectral sequences are all about.

on a different note, i was wondering what use can we put spectral sequences to. the theory developed is intellectually very satisfying, but i am not seeing any direct applications. i have heard that these are used to compute homology of fiber bundles though, which is nice

there are many many applications

one of which is, as you say, computing the homology of fiber bundles, using, e.g. the serre spectral sequence

there are also Adams type spectral sequences, these compute the homotopy groups of any finite cell complex

when I first learned spectral sequences I spent some time computing the cohomology of Eilenberg-MacLane spaces using the Serre spectral sequence

The entire point of spectral sequences is to compute things. Start with the Serre spectral sequence.

which is a lot of fun, and you can get quite a long way

one can also use homotopy fixed point spectral sequences to compute the fixed points of objects under group actions

interesting! i don't know cohomology though, as i have already mentioned above.

but i'll definitely read Serre spectral sequences from Hatcher.

it works just as well for homology, if you know homology

that's nice. yeah, i am familiar with homology.

though the multiplicative structure helps you a lot.

true

@BalarkaSen spectral sequences are cool because they literally contain ALL the information one could want. it just happens that as a result, they're often very hard to actually pull apart and GET that information out of.

they're like magical mountains full of gemstones, and only the most determined dwarves with their magical pickaxes can get inside without being crushed.

are E_n-monoidal categories of modules over E_{n+1}-ring spectra closed (i.e. do the tensor products have right adjoints)?

@JonBeardsley i'd expect so, though in general it should depend on what your underlying category / symmetric monoidal structure really is (i'm assuming you mean in Spaces here?). you should use the adjoint functor theorem for this

@AaronMazel-Gee ah sure. it should come down to whether or not smashing preserves colimits.

for E_1, smashing preserves colimits

E_n smash product is the same as E_1 + structure, isn't it?

so it looks like it preserves colimits

hm, so there's definitely a monoidal forgetful functor from modules over an E_n-ring to modules over the same ring thought of as an E_1-ring

what do you mean by "is the same"?

sorry, i've bugged this room enough recently. just figured someone would know the answer immediately. don't worry about it. =P

wow, hatcher indeed provides wonderful intuition for spectral sequences!

so I guess these are really "higher versions" of cellular homology (which is what the E^2-page is about).

i think of it as kind of writing down all possible cells something might have, and then weeding out the pieces that don't make any sense.

by sort of... checking them against the data you have about attaching maps.

yeah. that makes sense.

Of course, I don't do any computations, so always check anything I say with someone who actually regularly works with spectral sequences.

Also - if you ever get around to learning sheaf cohomology, you'll realize that most spectral sequences are just derived sheaf cohomology.

except for those ones that aren't

;)

Haha, examples please?

most of the homologically graded ones, e.g. the spectrals sequence for the homology of a filtered space

Ah I see.

TIL

i'm constrained a little because I have this feeling like saying "the Adams spectral sequence" is going to make you tell me that it's secretly a sheaf cohomology spectral sequence

Well...

=P

there's the EHP spectral sequence for computing unstable homotopy groups too

Seriously Tyler, I'm probably not going to try too hard to contradict you here. You've been at it a lot longer than I have.

no worries, the comment was meant to aim more towards "mild teasing" than "snark"

Haha, well thankyou. I guess I was just wondering though, do you not think of the ANSS that way?

I'm kind of single minded about these things.

not always, sometimes it's handy to think of it as completion or as obstruction theory or as "detecting maps by iterative comodule extensions"

i see what you mean.

yeah, i mean, in my mind sheaf cohomology IS obstruction theory

or at least, a TYPE of obstruction theory

and yeah, i guess i get the completion idea as well.

anyway...

spectral sequences are cool.

right. i hate those ones.

the bockstein sseq kind of makes me mad.

vitaly works with that one all the time.

sometimes the bockstein sseq is a generalized Adams spectral sequence, associated to e.g. the HZ-module HZ/p

sometimes the algebra is important too, i don't really know how to address serious convergence problems without knowing the "interlocking exact sequence" perspective

yes. i mean... the few times i've actually sat down and computed stuff with a sseq, i've basically just started by drawing those staircase diagrams

at that point i completely ignore any kind of broad theoretical perspective