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Stream: community: general

Topic: classifying fibrations

sarahzrf (Apr 03 2020 at 02:27):

is there a nice moral reason why pseudofunctors into Cat classify just [op]fibrations over their domain rather than arbitrary functors into their domain?

Max New (Apr 03 2020 at 02:31):

lax functors into Prof classify arbitrary functors

Jonathan Beardsley (Apr 03 2020 at 03:07):

@sarahzrf basically, the rules of the pseudofunctor translate into being able to take pullbacks along morphisms in the base in a very concrete way

sarahzrf (Apr 03 2020 at 03:08):

actually they have to send identities to identities, max

sarahzrf (Apr 03 2020 at 03:08):

iirc

Jonathan Beardsley (Apr 03 2020 at 03:08):

The main thing that these fibrations are "for" is taking these kinds of pullbacks (imho)

sarahzrf (Apr 03 2020 at 03:09):

@Jonathan Beardsley hmmmm, i feel like that's a bit less "moral" than i was looking for

sarahzrf (Apr 03 2020 at 03:09):

like

Jonathan Beardsley (Apr 03 2020 at 03:09):

Yeah I thought it might be

sarahzrf (Apr 03 2020 at 03:09):

mm

sarahzrf (Apr 03 2020 at 03:09):

functions into Set classify functions into the domain

sarahzrf (Apr 03 2020 at 03:09):

what's different between sets and categories?

sarahzrf (Apr 03 2020 at 03:09):

or maybe the moral is that all functions of sets are fibrations?

Jonathan Beardsley (Apr 03 2020 at 03:10):

Well sets have no morphisms

sarahzrf (Apr 03 2020 at 03:10):

sure they do

sarahzrf (Apr 03 2020 at 03:10):

the morphisms are the equalities

Jonathan Beardsley (Apr 03 2020 at 03:10):

Somehow the "true" story is happening higher categorically, and you get these accidents when you truncateb

Dan Doel (Apr 03 2020 at 03:11):

What about groupoids?

sarahzrf (Apr 03 2020 at 03:11):

what about them?

Jonathan Beardsley (Apr 03 2020 at 03:12):

Yeah I know I'm not giving you a great intuitive answer tho.

Dan Doel (Apr 03 2020 at 03:13):

They're a better higher version of sets, for one. Do they have analogous properties?

Jonathan Beardsley (Apr 03 2020 at 03:13):

Also I think that from the POV of fibrations, my statement about sets not having morphisms stands

Jonathan Beardsley (Apr 03 2020 at 03:13):

The same story is true for groupoids

Dan Doel (Apr 03 2020 at 03:13):

Does the fact that all their morphisms are invertable make all the functors automatically fibrations?

Dan Doel (Apr 03 2020 at 03:14):

Or something like that?

Jonathan Beardsley (Apr 03 2020 at 03:14):

No. You still need liftings of the morphisms I believe.

Dan Doel (Apr 03 2020 at 03:14):

Ah.

Jonathan Beardsley (Apr 03 2020 at 03:15):

I can't see any reason that invertibility would give you that for free.

Jonathan Beardsley (Apr 03 2020 at 03:15):

I'm mostly making analogies from the (∞,1) case tho

Jonathan Beardsley (Apr 03 2020 at 03:16):

Recall that pseudofunctors into groupoids just give you "categories fibered in groupoids."

Jonathan Beardsley (Apr 03 2020 at 03:16):

Or "stacks."

sarahzrf (Apr 03 2020 at 03:17):

we should be talking about pseudofunctors from a groupoid into groupoids classifying a groupoid over the domain

Jonathan Beardsley (Apr 03 2020 at 03:17):

But there is still this equivalence between pseudofunctors and certain kinds of fibrations.

Dan Doel (Apr 03 2020 at 03:17):

Well, I haven't studied enough to recall that. :slight_smile:

Jonathan Beardsley (Apr 03 2020 at 03:17):

Sorry, different backgrounds.

Dan Doel (Apr 03 2020 at 03:17):

I can only throw out random ideas.

sarahzrf (Apr 03 2020 at 03:17):

if you just carry out the standard grothendieck construction on such a pseudofunctor, do you automatically get a groupoid as the domain of the fibration?

Jonathan Beardsley (Apr 03 2020 at 03:18):

My guess is yes.

Jonathan Beardsley (Apr 03 2020 at 03:19):

I believe the analogous thing is true for (∞,1)-groupoids... But I'd have to check.

Jonathan Beardsley (Apr 03 2020 at 03:20):

Yeah it's true. Just checked.

Dan Doel (Apr 03 2020 at 03:21):

I guess the opposite version of my earlier question is "what about posets?"

Jonathan Beardsley (Apr 03 2020 at 03:21):

Basically if you've got a fibration whose fibers are groupoids, and whose target is a groupoid, so is the total category.

David Michael Roberts (Apr 03 2020 at 03:22):

Jonathan Beardsley said:

Recall that pseudofunctors into groupoids just give you "categories fibered in groupoids."
Or "stacks"

You can have stacks that are not fibred in groupoids. Just saying. The usual category of vector bundles on spaces, over the category of spaces, is a stack, because descent is still effective etc.

sarahzrf (Apr 03 2020 at 03:22):

/me hears someone mention specializing a categorical notion to the (0, 1) case

sarahzrf (Apr 03 2020 at 03:22):

/me 's eyes start glowing

sarahzrf (Apr 03 2020 at 03:23):

functors from a poset to Pos classify fibrations, not arbitrary posets over the domain :)

Jonathan Beardsley (Apr 03 2020 at 03:23):

@David Michael Roberts indeed, in fact I'd generally consider a stack to be just fibered in categories, but I think the classical notion uses groupoids

Dan Doel (Apr 03 2020 at 03:23):

Okay, then it seems like sets are weird.

sarahzrf (Apr 03 2020 at 03:24):

well, i dunno tho?

sarahzrf (Apr 03 2020 at 03:24):

because my intuition for slice categories in general is like

sarahzrf (Apr 03 2020 at 03:24):

C/A is supposed to be the category of A-indexed families of C objects

sarahzrf (Apr 03 2020 at 03:24):

have i been wrong?

sarahzrf (Apr 03 2020 at 03:24):

i mean, that's the basis for the categorical semantics of dependent types, right?

Jonathan Beardsley (Apr 03 2020 at 03:24):

Any time you're outside of sets, yes

sarahzrf (Apr 03 2020 at 03:25):

:scream:

Jonathan Beardsley (Apr 03 2020 at 03:25):

Anything even slightly "higher" has to have some suitable notion of parallel transport

Jonathan Beardsley (Apr 03 2020 at 03:25):

Ok let me try saying this

sarahzrf (Apr 03 2020 at 03:25):

hmmmm

sarahzrf (Apr 03 2020 at 03:26):

ooh, hold on

sarahzrf (Apr 03 2020 at 03:26):

what characterizes something as being "higher" is "collection of them is a higher category", maybe?

Jonathan Beardsley (Apr 03 2020 at 03:27):

well, higher, for me, means "has some notion of n-morphisms for n>0"

sarahzrf (Apr 03 2020 at 03:27):

so what i said is right when C is a (1, 1)-category, just not for n > (1, 1)

Jonathan Beardsley (Apr 03 2020 at 03:27):

"not discrete"

Jonathan Beardsley (Apr 03 2020 at 03:27):

it's not though, right? for C an ordinary category, you get fibrations

Jonathan Beardsley (Apr 03 2020 at 03:28):

here's the thing

sarahzrf (Apr 03 2020 at 03:28):

i meant if you have a (1, 1)-category C, then taking slices over objects in it is the right thing

Jonathan Beardsley (Apr 03 2020 at 03:28):

oh like, you're INSIDE of a (1,1)-category?

sarahzrf (Apr 03 2020 at 03:28):

but if the objects are categories, then the category is Cat, and that's (2, 2)

sarahzrf (Apr 03 2020 at 03:28):

yeah!

Jonathan Beardsley (Apr 03 2020 at 03:28):

right right

Jonathan Beardsley (Apr 03 2020 at 03:28):

here gimme a second to say something maybe helpful

sarahzrf (Apr 03 2020 at 03:29):

:thumbs_up:

Jonathan Beardsley (Apr 03 2020 at 03:29):

given any kind of functor out of a category, you need to know what to take the morphisms to

Jonathan Beardsley (Apr 03 2020 at 03:29):

if you have an arbitrary functor into a category, say E-->B, you have no idea what to take the morphisms to

Jonathan Beardsley (Apr 03 2020 at 03:30):

in other words, clearly the functor P:B-->Cat should take b to the inverse image of b in E (as a subcategory)

Jonathan Beardsley (Apr 03 2020 at 03:30):

but given a morphism f:b-->c, there isn't really any clear way to canonically produce a functor from p^{-1}(c) to p^{-1}b

sarahzrf (Apr 03 2020 at 03:31):

ah yeah ive thought thru this bit before :)

Jonathan Beardsley (Apr 03 2020 at 03:31):

and, basically, a fibration is PRECISELY the sort of data you need to choose such functors, AND for them to compose correctly

David Michael Roberts (Apr 03 2020 at 03:31):

@Jonathan Beardsley "classical" as in Giraud's book _Cohomologie Nonabelienne_? Nope, he uses fibred in categories.

sarahzrf (Apr 03 2020 at 03:31):

you can produce a profunctor between those fibers tho

Jonathan Beardsley (Apr 03 2020 at 03:31):

@David Michael Roberts okay you're right and also smarter than me

sarahzrf (Apr 03 2020 at 03:31):

& then its a fibration if the profunctor is representable

David Michael Roberts (Apr 03 2020 at 03:32):

Also, there are algebraic stacks (so have a cover by a scheme) that are fibred in categories, too. I can't get people to take me seriously about these, I don't know why.

David Michael Roberts (Apr 03 2020 at 03:32):

And by "people" I mean number theorists/algebraic geometers.

Jonathan Beardsley (Apr 03 2020 at 03:33):

I literally only ever talk about stacks fibered in categories. But when I'm talking to people who I don't know, I try to use the terminology that every algebraic geometer I ever talk to uses.

David Michael Roberts (Apr 03 2020 at 03:34):

@Jonathan Beardsley not smarter, who knows what that means.

Jonathan Beardsley (Apr 03 2020 at 03:35):

All right, then I concede in whatever way will make you stop talking about it.

David Michael Roberts (Apr 03 2020 at 03:35):

I find myself betwixt several fields of mathematics, and so I have weird ideas and focus on stuff none of them particularly grok.

David Michael Roberts (Apr 03 2020 at 03:35):

OK, I'll pipe down :-)

sarahzrf (Apr 03 2020 at 03:37):

wait shit why is this topic in #general i meant to post it in #category theory :face_palm:

Jonathan Beardsley (Apr 03 2020 at 03:37):

i was confused about that

Joe Moeller (Apr 03 2020 at 03:45):

Not sure if someone already said this, but any functor between groupoids can be replaced by a fibration between groupoids equivalent to the original ones.

sarahzrf (Apr 03 2020 at 03:46):

oh!

sarahzrf (Apr 03 2020 at 03:46):

good to know!

Jonathan Beardsley (Apr 03 2020 at 03:46):

i didn't know that, but it makes me think of the usual thing in model categories

Jonathan Beardsley (Apr 03 2020 at 03:47):

e.g. how you can replace any map between spaces with a fibration without changing the homotopy type of anything

Joe Moeller (Apr 03 2020 at 03:53):

One way of thinking about the original question is that if you take a random functor, you still have fibres, but you aren't guaranteed in general that the stuff that crosses between fibres will arrange themselves into functors between the fibres.

Jonathan Beardsley (Apr 03 2020 at 03:53):

Yeah I think that's what I was maybe trying to say, but saying poorly.

Jonathan Beardsley (Apr 03 2020 at 03:54):

The morphisms are really crucial. And if you're just over a set, you don't have to keep track of any of that stuff.

Jonathan Beardsley (Apr 03 2020 at 03:54):

you just need fibers over objects, which you have.

sarahzrf (Apr 03 2020 at 03:54):

hmm

Jonathan Beardsley (Apr 03 2020 at 03:55):

like, the fact that you don't need "fibrations" for sets is, like, an accident

sarahzrf (Apr 03 2020 at 03:55):

(for the record, tho, a lot of things are clearer when u recognize that sets do have morphisms)

Jonathan Beardsley (Apr 03 2020 at 03:56):

but those morphisms have to go to something VERY specific under any kind of functor

Jonathan Beardsley (Apr 03 2020 at 03:56):

you don't have to choose anything for them

Joe Moeller (Apr 03 2020 at 03:56):

I think my answer is not maximally "moral" as opposed to technical.

Joe Moeller (Apr 03 2020 at 03:56):

but maybe halfwayish

Jonathan Beardsley (Apr 03 2020 at 03:57):

i mean, in my mind, the question should be totally reversed

Joe Moeller (Apr 03 2020 at 03:57):

how?

Jonathan Beardsley (Apr 03 2020 at 03:57):

i.e. "how come for reasonable notions of functors into nCat you always get fibrations over the base, but for 0Cat you just get all maps down to the base?"

sarahzrf (Apr 03 2020 at 03:58):

actually i have a take on this

sarahzrf (Apr 03 2020 at 03:58):

there's a pathology in the typical approach to thinking about 0Cat, i'd say

Dan Doel (Apr 03 2020 at 03:58):

If you do this in HoTT, does it not work anymore?

sarahzrf (Apr 03 2020 at 03:58):

if you're paying attention to strictness issues, then really, 0Cat should be the category of setoids, not sets

sarahzrf (Apr 03 2020 at 03:59):

i.e., sets equipped with equivalence relations

Jonathan Beardsley (Apr 03 2020 at 03:59):

I'm never really paying attention to strictness issues, unless I absolutely have to.

sarahzrf (Apr 03 2020 at 03:59):

same (:

sarahzrf (Apr 03 2020 at 03:59):

but we seem to be here!

Jonathan Beardsley (Apr 03 2020 at 03:59):

I just think about dots and paths in my head.

Alexander Campbell (Apr 03 2020 at 04:00):

But even with setoids, every functor is equivalent to a fibration. The difference here is really a groupoid vs category thing.

sarahzrf (Apr 03 2020 at 04:00):

but so: if we use setoids instead of sets, then i bet we get a similar phenomenon to groupoids or in homotopy theory

Joe Moeller (Apr 03 2020 at 04:00):

Could setoids also be called posetal groupoids :thinking:

sarahzrf (Apr 03 2020 at 04:00):

sure :)

sarahzrf (Apr 03 2020 at 04:00):

where every map is equivalent to a fibration, but not equal to one if you are willing to be evil

Jonathan Beardsley (Apr 03 2020 at 04:01):

okay alexander's comment is making me rethink my whole position

sarahzrf (Apr 03 2020 at 04:01):

it's also what i was gonna say next ^_^

Dan Doel (Apr 03 2020 at 04:01):

Or in HoTT does the same thing happen for groupoids as sets, because every functor being equivalent to a fibration means there's a path between the two?

Joe Moeller (Apr 03 2020 at 04:02):

Was it Voevodsky that emphasized that we should think of sets:groupoids as posets:categories?

sarahzrf (Apr 03 2020 at 04:02):

@Dan Doel the more category theory i learn the more the idea of doing category theory in hott terrifies me

Dan Doel (Apr 03 2020 at 04:02):

@Joe Moeller Yes, I think so.

sarahzrf (Apr 03 2020 at 04:02):

someone i talked to at popl told me that some of his work was on the open problem of defining the appropriate notion of a presheaf when working in hott

sarahzrf (Apr 03 2020 at 04:02):

iirc

Jonathan Beardsley (Apr 03 2020 at 04:03):

okay so are we saying that groupoids behave MORE like sets than categories, in this aspect?

sarahzrf (Apr 03 2020 at 04:03):

it sounds like it to me

Jonathan Beardsley (Apr 03 2020 at 04:03):

okay so here's a fact i know

Joe Moeller (Apr 03 2020 at 04:03):

That's what I was trying to get at.

Jonathan Beardsley (Apr 03 2020 at 04:03):

which is actually what Dan said a while back

sarahzrf (Apr 03 2020 at 04:03):

the only real difference between sets and groupoids i see is that you dont have to replace w/ an equivalent fibration in sets, but i bet that that's an artifact of using sets instead of setoids

Dan Doel (Apr 03 2020 at 04:04):

Is it 'evil' to distinguish between the functor on groupoids and the equivalent fibration?

Dan Doel (Apr 03 2020 at 04:04):

If so, then I think in HoTT you'd try to make them identical.

Dan Doel (Apr 03 2020 at 04:04):

Up to paths.

sarahzrf (Apr 03 2020 at 04:04):

i know that "grothendieck fibration" is an evil concept

sarahzrf (Apr 03 2020 at 04:05):

i think "street fibration" or something is the reformed version

Joe Moeller (Apr 03 2020 at 04:05):

That's right.

Jonathan Beardsley (Apr 03 2020 at 04:05):

hmmmmm okay... i'm trying to fit this into my backwards framework.... i guess iirc when you do the $\infty$-categorical Grothendieck construction stuff, you do indeed get that the $\infty$-category of "fibrations over an $\infty$-groupoid $X$" is equivalent to the $\infty$-slice category $(Gpd_{\infty})_{/X}$.

Dan Doel (Apr 03 2020 at 04:06):

So this might be the kind of situation that HoTT is intended to clear up. Groupoids act like sets, and categories act like posets. Or something.

sarahzrf (Apr 03 2020 at 04:07):

more like sets act like ∞-groupoids

Jonathan Beardsley (Apr 03 2020 at 04:07):

I suppose I would like to say that one of the underlying philosophical moves in HoTT is to replace sets with $\infty$-groupoids

Joe Moeller (Apr 03 2020 at 04:07):

@Jonathan Beardsley then does that equivalence tell you which fibration corresponds to your arbitrary map?

Dan Doel (Apr 03 2020 at 04:07):

Unless it also eliminates the fibration part for categories, too.

Jonathan Beardsley (Apr 03 2020 at 04:08):

I think it could if, instead of using the $\infty$-categories, you used the Quillen model structures.

John Baez (Apr 03 2020 at 04:08):

"Street fibrations" are the non-evil fibrations: any functor naturally isomorphic to a Grothendieck fibration is a Street fibration, and any functor naturally isomorphic to a Street fibration is again a Street fibration.

But almost nobody uses Street fibrations.

The most pushback I've ever gotten on using the word "evil" was from people who complained they weren't being :ogre: evil :ogre: for using Grothendieck fibrations.

Jonathan Beardsley (Apr 03 2020 at 04:08):

And replaced the map with a fibration in the correct model structure.

Jonathan Beardsley (Apr 03 2020 at 04:10):

John Baez said:

"Street fibrations" are the non-evil fibrations: any functor naturally isomorphic to a Grothendieck fibration is a Street fibration, and any functor naturally isomorphic to a Street fibration is again a Street fibration.

But almost nobody uses Street fibrations.

The most pushback I've ever gotten on using the word "evil" was from people who complained they weren't being :ogre: evil :ogre: for using Grothendieck fibrations.

Me and Liang Ze Wong used Street fibrations in the first draft of our paper on the enriched Grothendieck construction

Jonathan Beardsley (Apr 03 2020 at 04:10):

because it makes doing enriched fibrations easier in some ways

Jonathan Beardsley (Apr 03 2020 at 04:10):

b/c you can just work with adjoints between fibers

Joe Moeller (Apr 03 2020 at 04:10):

I think people have given either a negative or an idk to the question "are Grothendieck fibrations the fibrations of some model structure?"

Joe Moeller (Apr 03 2020 at 04:11):

@Jonathan Beardsley that sounds nice, why did you change it?

Jonathan Beardsley (Apr 03 2020 at 04:11):

but it turned out to be unnecessary

Jonathan Beardsley (Apr 03 2020 at 04:11):

and made the paper a lot longer

Jonathan Beardsley (Apr 03 2020 at 04:12):

basically we were anxious about picking cartesian lifts in enriched categories

Jonathan Beardsley (Apr 03 2020 at 04:12):

where you don't have honest sets of morphisms

Joe Moeller (Apr 03 2020 at 04:12):

Is there a theorem like "every Street fibration is equivalent to a Grothendieck fibration"?

Jonathan Beardsley (Apr 03 2020 at 04:12):

i can't remember how it all worked out in the end, but it turned out you didn't need to worry about this

Jonathan Beardsley (Apr 03 2020 at 04:13):

@Joe Moeller at least in $\infty$-cosmoi this type of thing is true

Jonathan Beardsley (Apr 03 2020 at 04:13):

Riehl and Verity have some theorem saying that things that should be called "Cartesian fibrations" are the same as things that should be called "Street fibrations."

Reid Barton (Apr 03 2020 at 04:14):

A Street fibration is a Grothendieck fibration iff it is an isofibration, and you can always replace a functor by an equivalent isofibration (and being a Street fibration is equivalence-invariant).

Joe Moeller (Apr 03 2020 at 04:14):

I think I wanted the other direction though, Reid. :thinking:

sarahzrf (Apr 03 2020 at 04:14):

∞-cosmoi 🤩

sarahzrf (Apr 03 2020 at 04:15):

what's an ∞-cosmos?

Jonathan Beardsley (Apr 03 2020 at 04:15):

"a place where one can do $\infty$-category theory"

sarahzrf (Apr 03 2020 at 04:15):

i don't know what i expected

Jonathan Beardsley (Apr 03 2020 at 04:15):

Do you know the classical notion of a cosmos?

sarahzrf (Apr 03 2020 at 04:15):

yeah

sarahzrf (Apr 03 2020 at 04:15):

it's a (1, 1)-quantale :upside_down:

Joe Moeller (Apr 03 2020 at 04:15):

Is it an ($\infty$,2)-category?

Jonathan Beardsley (Apr 03 2020 at 04:16):

@Joe Moeller well one would really need a good model for that type of thing I think, haha

Joe Moeller (Apr 03 2020 at 04:16):

I don't know about the classical notion of a cosmos :looking:

Jonathan Beardsley (Apr 03 2020 at 04:16):

https://ncatlab.org/nlab/show/cosmos

sarahzrf (Apr 03 2020 at 04:16):

something nice enough for enriching over that you don't have to sweat about it, basically

sarahzrf (Apr 03 2020 at 04:17):

enough of Set's properties that you use all the time as the enriching category, except maybe not cartesian

sarahzrf (Apr 03 2020 at 04:18):

let's regularize our naming conventions, a cosmos should be called a quantos :triumph:

Joe Moeller (Apr 03 2020 at 04:19):

{space, cosmos} is a nice pair of words.

Jonathan Beardsley (Apr 03 2020 at 04:19):

if we could replace $\infty$-topos with "space" we'd be in good shape

sarahzrf (Apr 03 2020 at 04:20):

@Jonathan Beardsley ok, i shouldve clarified ive only ever rly thought about benabou cosmoi

sarahzrf (Apr 03 2020 at 04:20):

are those the ones you have in mind

Jonathan Beardsley (Apr 03 2020 at 04:20):

nah, more like Street cosmoi

sarahzrf (Apr 03 2020 at 04:20):

ah, oof

Reid Barton (Apr 03 2020 at 04:20):

If you start with a Street fibration, you can replace it by an isofibration--the new functor is still a Street fibration so now you found an equivalent Grothendieck fibration.

sarahzrf (Apr 03 2020 at 04:20):

/me peeks

Joe Moeller (Apr 03 2020 at 04:20):

Reid, oh I see.

Jonathan Beardsley (Apr 03 2020 at 04:20):

i.e. instead of "a good thing to enrich over," we're talking about a "good place to do category thoery"

John Baez (Apr 03 2020 at 04:21):

A Benabou cosmos is a nice sort of 2-rig, @Joe Moeller.

sarahzrf (Apr 03 2020 at 04:21):

fibrational cosmos or the 2nd one, jon?

Jonathan Beardsley (Apr 03 2020 at 04:22):

oh i dunno, i guess @Emily Riehl might know, haha

Jonathan Beardsley (Apr 03 2020 at 04:23):

I would guess, from the language of the nlab article, that it's the "fibrational" definition that caused them to name these structures $\infty$-cosmoi

Jonathan Beardsley (Apr 03 2020 at 04:24):

ok good night

John Baez (Apr 03 2020 at 04:24):

Emily Riehl hints at a notion of $\infty$-cosmos here:

Riehl

She may have made this into a definition of $\infty$-cosmos somewhere.

sarahzrf (Apr 03 2020 at 04:25):

:o

Jonathan Beardsley (Apr 03 2020 at 04:25):

I mean she has written many papers on the theory of $\infty$-cosmoi

Jonathan Beardsley (Apr 03 2020 at 04:26):

She has definitely made it into a definition.

Jonathan Beardsley (Apr 03 2020 at 04:26):

https://ncatlab.org/nlab/show/infinity-cosmos

Jonathan Beardsley (Apr 03 2020 at 04:26):

crap i forgot i was going to bed

Reid Barton (Apr 03 2020 at 04:27):

There's even a book in progress now. Riehl-Verity http://www.math.jhu.edu/~eriehl/elements.pdf

sarahzrf (Apr 03 2020 at 04:37):

anyway returning to the title of the topic—

sarahzrf (Apr 03 2020 at 04:38):

• seems like groupoidy things generally have morphisms in equivalent to fibrations, which are classified by functors to the containing category

sarahzrf (Apr 03 2020 at 04:40):

• was there a consensus on whether i was correct with "in an arbitrary (1, 1)-category, you should think of objects in a slice as families parameterized by the base, but possibly not for n ≠ (1, 1)"?

Oscar Cunningham (Apr 03 2020 at 07:47):

@sarahzrf I'd say that for a category $A$ you can think of functors $B\to A$ as families of categories parameterised over $A$. You just have to think of the collection of all categories as being $\mathrm{Prof}$.

sarahzrf (Apr 03 2020 at 07:47):

ha

sarahzrf (Apr 03 2020 at 07:48):

oh i misread "the collection of all categories" as "all categories"

Max New (Apr 03 2020 at 13:19):

Yea sorry functors into a category are equivalent to normal lax functors into Prof, meaning identity is preserved up to equivalence which is the same as arbitrary lax functors into Span.

Oscar Cunningham (Apr 03 2020 at 13:29):

@Max New What's 'Span' here?

Max New (Apr 03 2020 at 13:31):

bicategory whose objects are Sets, morphisms are Spans and 2-cells are morphisms of spans. Categories are monoids in that bicategory, which explains the relationship to Prof which is Cats, profunctors and natural transformations

Max New (Apr 03 2020 at 13:31):

(Both Span and Prof can also be naturally extended to double categories)

Max New (Apr 03 2020 at 13:33):

Here is the nlab page about this stuff: https://ncatlab.org/nlab/show/displayed+category

Oscar Cunningham (Apr 03 2020 at 13:39):

Thanks! I didn't know that. It's nice to be able to drop the 'normal'.

Max New (Apr 03 2020 at 13:41):

It also has a simple explanation which is that Prof is the result of taking monoids in Span and that construction has a universal property that normal lax functors into Mod(C) are the same as lax functors into C.

Oscar Cunningham (Apr 03 2020 at 13:48):

Can we also drop the 'lax'? I know there's a 'laxification' functor $A\mapsto \hat A$ such that the lax functors $A\to B$ correspond to ordinary functors $\hat A\to B$. Might there also be a 'right laxification' such that lax functors $B\to A$ correspond to ordinary functors $B\to \check A$? In other words, is $\mathrm{Lax}(-,A)$ representable?

Max New (Apr 03 2020 at 14:29):

meaning the inclusion i : Pseudo -> Lax is a left adjoint? Maybe check if it preserves colimits

Oscar Cunningham (Apr 03 2020 at 14:51):

Ah, wait, of course that can't work. If there was a $2$-category $C$ such that $\mathrm{Cat}/A$ is the $2$-category of ordinary functors $A\to C$ then we would have $C=\mathrm{Hom}(1,C)=\mathrm{Cat}/1 = \mathrm{Cat}$, which we know doesn't work.

Mike Stay (Apr 05 2020 at 22:52):

John Baez said:

Emily Riehl hints at a notion of $\infty$-cosmos here:

Riehl

She may have made this into a definition of $\infty$-cosmos somewhere.

Why did she put gamma between 1 & 2 when γ ≈ 0.577?

John Baez (Apr 05 2020 at 22:54):

I'm afraid the answer is lost in the $\infty$-cosmos.

Emily Riehl (May 14 2020 at 16:47):

I thought γ was the Golden ratio.

John Baez (May 14 2020 at 18:25):

No, $\gamma$ is the Euler-Mascheroni constant. The golden ratio is called $\Phi$ or sometimes $\phi$ or sometimes even $\tau$. $\Phi$ is for the Greek sculptor Phidias.

I advocate $\Phi$ for the "big" golden ration 1.6180339... and $\phi$ for the "small" golden ratio 0.6180339....