User Bruno Stonek | chat.stackexchange.com

Bruno Stonek

Mar 16, 2021 17:06

it's great that you're writing that, @AaronMazel-Gee , thanks!

Bruno Stonek

Mar 16, 2021 11:17

That question is 5+ years old now. I guess several people now have experience teaching higher algebra graduate courses. I think it'd be great to hear about that (and to share resources, perhaps)

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Bruno Stonek

Mar 16, 2021 11:16

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Q: teaching higher algebra

Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)? I'm asking out of curiosity (and also hoping for more resources). The kind of course I had in mind would be a class for graduate students, with background in standard top...

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Bruno Stonek

Mar 10, 2021 12:19

thanks for the suggestions!

Bruno Stonek

Mar 10, 2021 10:58

do you guys have some easy or intuitive way to explain to people not used to infinity-categories why, if X is a space, then the colimit of {*}:X\to Spaces is equivalent to X?

Bruno Stonek

Dec 1, 2020 11:02

oh, it seems this is in 4.8.4 of Higher Algebra, I hadn't seen it cause "Eilenberg-Watts" is not mentioned. anyway, any other references are still appreciated

Bruno Stonek

Dec 1, 2020 10:57

I read here mathoverflow.net/questions/159735/… that the Eilenberg-Watts theorem in $\infty$-categories has been discussed by some people. Have their conclusions appeared in print? Or perhaps one of you has these WCATSS13 notes David mentions there?

Bruno Stonek

Nov 23, 2020 12:39

the proof I know of this statement doesn't immediately generalize

Bruno Stonek

Nov 23, 2020 12:38

has anybody seen an infty-categorical adaptation of the following 1-categorical fact? Let C be a monoidal category with a free-forgetful adjunction (F,U), where U:C\to Set is faithful, and moreover F(*) =1, the monoidal unit of C. Then End_C(1) \cong Nat(id_C,id_C)

Bruno Stonek

Nov 10, 2020 13:06

thanks, yeah, that reference was brought up by Cisinski himself in the comments. I just wanted to make sure I wasn't missing any subtleties or something.

Bruno Stonek

Nov 10, 2020 13:00

sanity check: the colimit formula for Kan extensions (in quasicategories) always holds, right? not only when you're extending along a subcategory. this was brought up here mathoverflow.net/a/337428/6249 but there's a comment by @AdrianClough that went unanswered.

Bruno Stonek

Nov 3, 2020 11:43

hmm, a reference to the literature would still be nice

Bruno Stonek

Nov 3, 2020 11:22

thanks Rune!

Bruno Stonek

Nov 3, 2020 10:08

reference request: where can I find a quasicategorical version of: "to compute a colimit in pointed spaces, you compute it in spaces and collapse the basepoints to a point"?

Bruno Stonek

Oct 29, 2020 10:40

is there anything that might sensibly be called a homotopy-theoretical analog of an algebraically closed field?

Bruno Stonek

Oct 13, 2020 13:23

However, one doesn't need the full structure of a model category to present an infinity-category, it suffices to have a relative category, and (cdgas, quasi-isos) surely is one. So what can we say about the infty-category it presents? What's the precise relation to E_infty-algebras?

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Bruno Stonek

Oct 13, 2020 13:23

It's often said that cdgas are "bad" in characteristic p. One way I've heard this justified is that they don't even have a reasonable model structure. More precisely, for example, here mathoverflow.net/a/23885 Tyler showed that they don't have a model structure with weak equivalences=quasi-isos and fibrations=levelwise surjections.

Bruno Stonek

Oct 5, 2020 11:21

Thank you all for the personal reflections that have been posted here in the past months. I've read them (with some time delay) and they have been food for thought. I thought of many things to say, but the non-anonymity mixed with the fact that the chat keeps a log here forever, prevent me from taking the step and sharing them. This is just a statement of fact, luckily I don't currently feel the necessity to speak out, so all's well, in a sense.

At any rate, I'll take the opportunity to thank @JonathanBeardsley for opening up this chat years ago (and congratulations on that position!). It …

Bruno Stonek

Sep 14, 2020 09:44

oh, hmm. I wasn't expecting that. thanks for the reference!

Bruno Stonek

Sep 14, 2020 09:27

does somebody have an $\infty$-categorical reference for the fact that the projection of an undercategory $C_{c/}\to C$ creates connected colimits?

Bruno Stonek

Apr 28, 2020 13:19

(the reason for my feeling uneasy is because I feel I get too close to foundational, model-dependent stuff, which always makes me uncomfortable!)

Bruno Stonek

Apr 28, 2020 13:19

what's the deeper reason, quasicategorically? in complete segal spaces, I get it that they're a localization of simplicial spaces, so that explains why limits are easy

Bruno Stonek

Apr 28, 2020 13:17

limits of quasicategories make me nervous

Bruno Stonek

Apr 28, 2020 13:17

thanks. that's what I was thinking of but was afraid I might be ignoring some subtlety...

Bruno Stonek

Apr 28, 2020 13:04

let C be a quasicategory given as a limit of a diagram of quasicategories (I care about the sequential case). what's the easiest way of seeing that the mapping spaces in C are limits of the mapping spaces in the components of the diagram?

Bruno Stonek

Apr 14, 2020 10:55

@S.carmeli perhaps I should have said: how can one do so much homotopically flavored stuff with SCRs when additively they're homotopically so easy

Bruno Stonek

Apr 14, 2020 09:30

wrt to the original question, I'm not sure E_n-ring spaces (in the GGN sense) are really what fits in the analogy: shouldn't it rather be spaces which have a strict addition and an E_n multiplication?

Bruno Stonek

Apr 14, 2020 09:24

here's a vague question. how can one do so much with simplicial commutative rings when the underlying simplicial abelian groups are just generalized Eilenberg-Mac Lane spaces? seems awfully restrictive

Bruno Stonek

Apr 14, 2020 09:20

that one's about different degrees of associativity vs. different degrees of commutativity, though

Bruno Stonek

Apr 14, 2020 09:18

I mean the one in page 41 here arxiv.org/pdf/1302.5756.pdf

Bruno Stonek

Apr 14, 2020 09:17

I idly wondered about that as well. it's a similar question to what's going on in that big diagram in Barwick's operator categories paper, if I remember correctly

Bruno Stonek

Apr 14, 2020 09:09

that's reassuring

Bruno Stonek

Apr 14, 2020 09:03

more classical definitions can be found in this paper by May arxiv.org/abs/0903.2813

Bruno Stonek

Apr 14, 2020 09:02

with the monoidal structure which corresponds to the smash product of connective spectra, if I'm not reading it wrong

Bruno Stonek

Apr 14, 2020 09:01

one definition (the one in GGN, the reference above) is that it is an E_\infty-algebra in E_\infty-groups

Bruno Stonek

Apr 14, 2020 08:17

Also, this is perhaps naive, but... since E_infty-groups are connective spectra, why aren't E_n-ring spaces (in the sense of GGN) just connective E_n-ring spectra?

Bruno Stonek

Apr 14, 2020 08:17

@TylerLawson In Gepner-Groth-Nikolaus' Universality of multiplicative infinite loop spaces machines, Example 7.2.(ii), they give a definition of E_n-ring space. Would that be it? What are the tricky things about E_\infty-ring spaces you're alluding to? I just don't really know that theory.

Bruno Stonek

Apr 13, 2020 16:22

in the E_infty case, what I'm talking about would be: E_infty-ring spectra / cdgas / simplicial commutative rings. if I replace E_infty-ring spectra by E_n-ring spectra, I know what fits in the dg world, but not really what fits in the place of SCRs

Bruno Stonek

Apr 13, 2020 16:14

hmm. I think I rather meant something like E_n ring space

Bruno Stonek

Apr 13, 2020 16:11

phrasing it like that already answered it I guess :) thanks

Bruno Stonek

Apr 13, 2020 16:10

oh

Bruno Stonek

Apr 13, 2020 16:09

right

Bruno Stonek

Apr 13, 2020 15:06

there's E_n-algebras in the dg and in the spectral worlds. Out of curiosity, has somebody worked with the simplicial analog?

Bruno Stonek

Mar 30, 2020 17:46

@WilliamBalderrama: that was very useful. thanks

Bruno Stonek

Mar 30, 2020 15:20

in a form as simple as $Uf\simeq Ug$ implies $f\simeq g$ where U denotes the forgetful functor

Bruno Stonek

Mar 30, 2020 15:15

not conservativity

Bruno Stonek

Mar 30, 2020 15:15

yeah... I actually want something rather like faithfulness

Bruno Stonek

Mar 30, 2020 15:08

The forgetful functor for algebras over a monad is faithful. Is there an analogous statement in the world of quasicategories?

Bruno Stonek

Mar 22, 2020 12:08

@EdoardoLanari an unstraightened version is corollary 2.3 in Gepner-Kock arxiv.org/pdf/1208.1749.pdf

Bruno Stonek

Mar 19, 2020 16:12

that you get the full adjunction data / coherences

Bruno Stonek

Homotopy Theory