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Stream: learning: questions

Topic: computing a 2-limit

sarahzrf (Nov 05 2020 at 06:32):

I have a 2-limit i want to compute—but id also be interested in resources that explain how to compute things of this sort in general:

suppose we have a functor F : BG → C for a group G and category C, so G is acting on the object F(*) ∈ C. then we can compose with C's self-indexing C^op → Cat to get F' : BG^op → Cat (even if C doesnt have pullbacks in general, it will at least have these ones since we are pulling back along isomorphisms). we can make this a strict functor even, by defining pullback along F(g) as composition with F(g⁻¹).

now i want to know: what's the 2-limit of F' like? (i think the ordinary 1-limit is gonna be something like the slice over the limit of F)

sarahzrf (Nov 05 2020 at 08:19):

i want to guess the objects are gonna be something like families of objects over F(*) coherently connected by isomorphisms between pullbacks—some kind of cocycley thing

sarahzrf (Nov 05 2020 at 08:33):

ooh i just realized this has something to do with galois shit, found some relevant-looking stuff googling

Reid Barton (Nov 05 2020 at 14:11):

Did you work it out? My guess is the category of "equivariant $G$-objects over the $G$-object $F(*)$", i.e., the slice category $[BG, C]_{/F}$.

sarahzrf (Nov 05 2020 at 15:03):

did you mean something like $[BG, C/F(*)]$? and no i havent worked it out yet exactly

Reid Barton (Nov 05 2020 at 15:09):

No, I mean an object $E$ of $C$ with a $G$-action and a map $E \to F(*)$ which is equivariant for the $G$-actions.

Reid Barton (Nov 05 2020 at 15:11):

But I didn't work it out exactly either

Reid Barton (Nov 05 2020 at 15:19):

The construction of the 2-limit of a pseudofunctor F : I -> Cat is pretty simple: an object consists of

• for each object i, an object X(i) of the category F(i),
• for each morphism of I, an isomorphism between whatever you can build out of X(i), X(j) and F,
• for each composable pair of morphisms of I, an equation that relates the above isomorphisms.

Reid Barton (Nov 05 2020 at 15:21):

You can read this off from the definition of a limit by considering what it means to give a cone with vertex the "walking object" category $*$.

sarahzrf (Nov 05 2020 at 15:35):

okay, ive glanced at this but im gonna need to think it thru properly later

sarahzrf (Nov 05 2020 at 15:35):

ty!

sarahzrf (Nov 05 2020 at 15:35):

wait, this is the weak 2-limit and not the strict 2-limit, right?

Reid Barton (Nov 05 2020 at 15:38):

Yes, otherwise you would have equalities instead of isomorphisms (and then no equations at the next stage)

sarahzrf (Nov 05 2020 at 16:19):

Reid Barton said:

Did you work it out? My guess is the category of "equivariant $G$-objects over the $G$-object $F(*)$", i.e., the slice category $[BG, C]_{/F}$.

wait omg, i dont know how i misparsed this :face_palm:

sarahzrf (Nov 05 2020 at 16:19):

of course F is an object of [BG, C]

sarahzrf (Nov 05 2020 at 16:20):

i need to stop trying to do category theory on an empty stomach

sarahzrf (Nov 05 2020 at 17:53):

riiiight okay

sarahzrf (Nov 05 2020 at 17:53):

so ive chewed thru the fact that a 2-limit in Cat is gonna be given by the category of [weighted] 2-cones

sarahzrf (Nov 05 2020 at 17:53):

and that this is like a weak version of image.png

sarahzrf (Nov 05 2020 at 17:54):

and ive fiddled with what that looks like in this particular case & more or less confirmed that it should be what you described

sarahzrf (Nov 05 2020 at 17:54):

thanks :)

sarahzrf (Nov 05 2020 at 18:40):

neat, if C is a grothendieck topos then this 2-limit should be as well then, right?

John Baez (Nov 05 2020 at 19:00):

Some related simpler stuff:

Let $BG$ be the category with one object $\ast$ and the group $G$ as endomorphisms.

A functor $F: BG \to \mathsf{Set}$ is called a $G$-set since it's a set $X$ with an action of $G$ as permutations of this set.

The limit of $F$ is the set of fixed points, $\{x \in X: gx = x\}$. A more fancy name for it is $H^0(G, X)$, the zeroth cohomology of the group $G$ with the $G$-set $X$ as coefficients. In group cohomology we can define $H^n(G,A)$ for any integer $n \ge 0$ and any abelian group $A$ on which $G$ acts, but for $n = 1$ we don't need $A$ to be an abelian group and for $n = 0$ it can just be a set.

John Baez (Nov 05 2020 at 19:03):

Next we can boost this up a bit and consider a pseudofunctor $F: BG \to \mathsf{Cat}$, and look at the pseudolimit of $F$.

sarahzrf (Nov 05 2020 at 19:03):

galois descent time

sarahzrf (Nov 05 2020 at 19:05):

actually, what source do i read if i want the coherence law to become obvious

sarahzrf (Nov 05 2020 at 19:06):

i bet i could work out what it would have to be just ad-hoc, but how do i learn what the pattern is

John Baez (Nov 05 2020 at 19:06):

You just draw some diagrams and you see what you need - that's much better.

sarahzrf (Nov 05 2020 at 19:06):

hmm.

John Baez (Nov 05 2020 at 19:06):

Anyway, let me just keep talking. Maybe I should write a huge block of text and post it all at once.

John Baez (Nov 05 2020 at 19:21):

Now let's consider a pseudofunctor $F: BG \to \mathsf{Cat}$. This is a category $X$ on which $G$ has a 'weak' action: we don't demand $F(g) F(h) = F(gh)$, all we need is natural isomorphisms $\phi_{g,h} : F(g) F(h) \to F(gh)$ obeying the obvious coherence law that makes the two isomorphisms from $F(g) F(h) F(k)$ to $F(ghk)$ equal.

Also, we don't demand $F(1) = 1_X$; all we need is a natural isomorphism $\phi : 1_X \to F(1)$ obeying some obvious coherence law. If $\phi$ is the identity we say $F$ is normalized, and I think it's pretty innocuous to impose this restriction.

Now we want to look at the pseudolimit or 'weak limit' of $F : BG \to \mathsf{Cat}$. This could be called the category of weak fixed points of $F$, or homotopy fixed points.

Instead of objects $x \in X$ such that $F(g) x$ equals $x$, this category of weak fixed points will have, as objects, $x \in X$ equipped with isomorphisms $\alpha_g : F(g) x \to x$, obeying the obvious coherence law. This coherence law makes the two isomorphisms from $F(g) F(h) x$ to $x$ agree.

There will also be morphisms in the category of weak fixed points. One can again guess what these must be like, but the good news is that the pseudolimit of $F$ automatically spits out the answer.

To make contact with group cohomology, we can suppose $X$ is a one-object category. Since only isomorphisms matter in this game, there's no further loss of generality in assuming $X$ is a one-object groupoid, say $B A$ for some group $A$... not necessarily abelian: I'm just using the letter $A$ since it's standard in group cohomology.

In this case $F: BG \to X = BA$ will be just an action of $G$ on the group $A$, as automorphisms.

Then we can look at the category of weak fixed points. The objects in here have a name: they are called 1-cocycles on $G$, valued in $A$. It's good to work out what they are explicitly. Two objects will be isomorphic if they are cohomologous: again one can work out what this means explicitly.

So, the set of isomorphism classes of weak fixed points, i.e. the decategorification of the pseudolimit of $F$, is called the set of cohomology classes of 1-cocycles on $G$, valued in $A$, or $H^1(G,A)$ for short.

John Baez (Nov 05 2020 at 19:28):

It may seem sort of pathetic to limit our category $X$ to being $B A$ for some group $A$, but it's actually not, if all we care about is understanding the groupoid of weak fixed points of a pseudofunctor $F: BG \to \mathsf{Cat}$. We can reduce the case of a general groupoid $X$ to the case of (a bunch of) groupoids of the form $B A$. So, the traditional approach to this stuff, using group cohomology, is not as pathetic as it may seem... especially since people have gotten really good at actually computing things like $H^1(G,A)$.

But, the more modern approach in terms of pseudolimits is more conceptual: it explains what you're 'really doing'.

John Baez (Nov 05 2020 at 20:24):

I'm done with my rant, @sarahzrf. There's nothing mystical about the coherence laws in this context: you just start drawing diagrams using your isomorphisms (like the $\alpha_g$ or the $\phi_{g,h}$ above) and decree that you want them all to commute. You could just say they all commute! But if you look at a bunch, you'll see that a few simple diagrams imply all the rest.

sarahzrf (Nov 05 2020 at 20:26):

hmmm, i dunno

sarahzrf (Nov 05 2020 at 20:26):

if you take that naïve of an approach, you'd assume that all braided monoidal categories should be symmetric, wouldn't you?

sarahzrf (Nov 05 2020 at 20:26):

but entirely aside from that, i guess like—i suppose, to draw an analogy...

sarahzrf (Nov 05 2020 at 20:30):

A: where does associativity and unitality come from, in monoids? it's exactly what you need to make any word be unambiguous to parenthesize, right? what's up with that?
B: well, there's nothing mystical about them—try writing down some equations in a monoid and seeing which ones ought to be true, and you should be able to work out that you just need associativity and unitality.

...and indeed, A would be well-served by getting hands on like that—but also, eventually, the real answer is going to have to be about how monoids are algebras of the list monad.

sarahzrf (Nov 05 2020 at 20:32):

it's true that i should probably spend some time drawing a ton of diagrams, but i would very much also like to hear something about why the diagrams i'm drawing are the ones they are

Dan Doel (Nov 05 2020 at 20:44):

Yes, in that sense the coherences 'come from' presenting arbitrary length lists in terms of an empty list, singleton inclusion, and binary concatenation.

John Baez (Nov 06 2020 at 00:12):

sarahzrf said:

if you take that naïve of an approach, you'd assume that all braided monoidal categories should be symmetric, wouldn't you?

Yes, you'd even assume they were better than symmetric: you'd assume the braiding of any object and itself,

$B_{x,x} : x \otimes x \to x \otimes x,$

is the identity!

John Baez (Nov 06 2020 at 00:14):

So these coherence laws - the coherence laws for $k$-tuply monoidal $n$-categories - cannot be guessed simply by saying "all diagrams should commute".

John Baez (Nov 06 2020 at 00:15):

But there are a certain bunch of coherence laws which secretly say some simplicial set is contractible, and when applied to 1-categories they amount to saying "every diagram commutes".

sarahzrf (Nov 10 2020 at 03:13):

this might deserve to be its own thread, but i'm asking it here since it involves the same constructions as in this thread:

if F : BG → C has a colimit, is that colimit generally van Kampen?

sarahzrf (Nov 10 2020 at 03:14):

im considering just working this out explicitly, cuz im having trouble finding much reference to this notion that i could check

sarahzrf (Nov 10 2020 at 03:14):

im just procrastinating on it since it seems like a pain <.<

Morgan Rogers (he/him) (Nov 10 2020 at 11:19):

Based on the examples on that nLab page, colimits being van Kampen seems to be more a property of the category in which they're being taken than a property of the colimit's shape?

sarahzrf (Nov 10 2020 at 21:22):

well, i think it's both

sarahzrf (Nov 10 2020 at 21:22):

but true—i should give more context

sarahzrf (Nov 10 2020 at 21:23):

first: im particularly interested in the case of topoi, or even just grothendieck topoi

sarahzrf (Nov 10 2020 at 21:23):

second: im particularly interested in the case where the colimit "injection", the quotient map to the orbit space, is a split epi

sarahzrf (Nov 10 2020 at 21:23):

...actually, does that situation have nice properties? i dont suppose thats an absolute colimit or anything?

sarahzrf (Nov 10 2020 at 21:23):

like how split coequalizers are

sarahzrf (Nov 10 2020 at 21:28):

ohhh hold on.... i was thinking about this presentation of the orbit space (EDIT: oh wait that's actually not quite the orbit space >_< but it's something quite closely related) as a reflexive coequalizer: image.png

sarahzrf (Nov 10 2020 at 21:28):

and it now occurs to me that umm

sarahzrf (Nov 10 2020 at 21:30):

maybe you can make an argument like... if the category is lextensive up to the cardinality of the group, then the coproduct there gets turned into a product of slices, and then if the coequalizer is split, you can use that to go the rest of the way on the same grounds as why split coequalizers are absolute

sarahzrf (Nov 10 2020 at 21:31):

nice