Homotopy Theory

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Pieter

05:45

I recall a similar confusion back when the paper appeared online. It's probably still there in the logs.

7 hours later…

Cuntero

13:12

Is there a description of the fundamental groupoid of the classifying space of a category purely in terms of the category one has started with, without switching to the topological world?

Is it the groupoid completion of the category?

Tyler Lawson

13:50

yes

or, it's naturally equivalent to that (the object set is bigger because it contains more than just the vertices)

Cuntero

14:15

Thank you! I'm confused by the following example: Consider the category 0->1. Its classifying space is a unit interval, so its fundamental groupoid should be equivalent to the category with one object and one morphism. Isn't its group completion the category with objects 0 and 1 and two mutually inverse morphisms in between?

Zhen Lin

14:28

that is indeed a contractible groupoid

1 hour later…

Cuntero

15:31

Oh, I see. Thank you for clarifying.

1 hour later…

pro

16:45

@Pieter wait, what? you mean you already asked the exact same question here?

Tyler Lawson

Dec 10 '15 at 5:50, by Aaron Mazel-Gee

just skimming the arxiv, and came across this paper: http://arxiv.org/abs/1512.02740
i'm not sure what to make of it, it's a little hard to understand the motivation (maybe partly a language thing). but then, searching further, i came across this:
https://ldtopology.wordpress.com/2010/11/09/gauthier-update/
...and this:
http://bookstore.xlibris.com/Products/SKU-0054161049/On-The-Purpose-Of-An-Earthly-Life.aspx

oops. that was just supposed to be a link to the conversation itself.

Eric Peterson

17:23

is there a word that means 'greater than 1'? like there's 'positive codimension', but there are also things that start to hold in codimension >= 2. it'd be nice to have a reasonable & flexible, even if esoteric, word for that sort of condition

the closest i've heard is 'plural', like 'pluridimensional' for dimensions >= 2, but that's not very flexible & i feel i'd have a hard time selling a reader on that or any other use of it

Jon Beardsley

metalinear

Eric Peterson

'multi' is also insuitable b/c 'multiheight' sounds like something other than 'heights >= 1', which is also a use case i am interested in

& it doesn't sound like it excludes the case of 1. multivariable calculus subsumes calculus, eg

Jon Beardsley

Whoops, I mixed Latin and Greek there.

Eric Peterson

yeah, i've heard people get scolded for that too

^^ these are my research concerns these days

Jon Beardsley

superlinear

or... metagrammic?

postunital

Peter Nelson

17:31

"unital" has too much baggage

Jon Beardsley

commutative dimensions

epidual

Okay I'll stop.

surdeux

Ok really.

Pieter

@pro Not me, I think Adeel was involved.

Peter Nelson

I'm trying to think of what surface people use, and I think I've always just heard them say "genus at least 2" or "hyperbolic" or something

Tyler Lawson

17:47

unfortunately "hyper" also appears in "hyperplane" etc, which is the exact opposite

Peter Nelson

and in "hyperbolic" which has problems at height 2

Jon Beardsley

double-and-up

I'm writing the "acknowledgements" part of my thesis right now, because I'm too tired to do math.

After the first few sentences it's pretty much become just a really long list of names.

user105491

18:16

Superheight? (Like supersingular elliptic curves being of height 2)

Jon Beardsley

People would just think he's referring to himself.

Peter Nelson

Z/2-graded height?

Jon Beardsley

18:51

Given an equivalence of quasicategories $C\to D$, and an $O$-monoidal structure on $D$, $D^\otimes\to O^\otimes$, do we get an induced $O$-monoidal structure on $C$? It seems like the nature of $\infty$-operads makes this significantly more complicated than the usual way of doing this in a category.

For instance, does everything go through smoothly in replacing all the levels of $D^\otimes$ with $C^n$?

Jon Beardsley

19:23

The main thing is that I'd like to be sure that the straightening/unstraightening equivalence between "spaces over an n-fold loop space" and "functors from thaat n-fold loop space into spaces" is an equivalence of E_n-monoidal quasicategories. I know that the latter one, using Day convolution, has such a structure.

2 hours later…

Rune Haugseng

21:15

@JonBeardsley A perhaps cheeky answer is that the correct definition of "a symmetric monoidal structure on C" is a coCartesian fibration X -> Gamma^op satisfying the Segal condition together with an equivalence of infinity-categories between C and the fibre of X at <1> (and similarly for a unital infinity-operad, like E_n) - so it's essentially true by definition.

Jon Beardsley

21:59

@RuneHaugseng that's actually an excellent point. not cheeky at all.

Rune Haugseng

Maybe the cheeky part is that it doesn't tell you how an automorphism of C gives you a symmetric monoidal automorphism.

Jon Beardsley

Ah... I see. So, the point being that it doesn't tell you how induce the full structure on the morphism?

Rune Haugseng

Not quite - as long as you just want to "transfer a monoidal structure" along an equivalence there's nothing to do.

The issue is that if I have a monoidal structure on C, i.e. I have some X and a chosen equivalence X_{<1>} ~ C, then an automorphism of C should induce an automorphism of X that restricts to the automorphism of C via that chosen equivalence.

But that's not an issue as far as your question goes

Jon Beardsley

22:21

right. so is it the case that you can't build up the structure from the automorphism of C?

Rune Haugseng

Surely you can, it's just not immediately obvious to me right now

Jon Beardsley

Okay. Well, fair enough. Your basic insight above is more than enough for my purposes. :)

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