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Stream: theory: category theory

Topic: finitary functors and "cofinitary" (?) functors

sarahzrf (May 07 2020 at 18:58):

1. is there a name for functors that preserve cofiltered limits?
2. what are good resources on functors that are finitary and/or "cofinitary" as above, that deal with them in more generality than just the ones on Set?

sarahzrf (May 07 2020 at 19:00):

i have 2 examples in a topic over in #practice: applied ct of what look suspiciously like terminal coalgebra constructions, but i'm not really sure if the endofunctors in question preserve the limit, and i'm not sure what a good source is for this kind of thing

Nathanael Arkor (May 07 2020 at 19:00):

I've seen cofinitary being used for functors preserving cofiltered limits (e.g. https://arxiv.org/pdf/1501.02834.pdf).

sarahzrf (May 07 2020 at 19:00):

one is in Grp, one is in a hom-category of Prof

sarahzrf (May 07 2020 at 19:00):

oh cool

sarahzrf (May 07 2020 at 19:01):

third question: is there literally anyone other than adámek who works on these things :sweat_smile:

Nathanael Arkor (May 07 2020 at 19:01):

For question 2, Adámek's works would be a good place to start.
Edit: as you've evidently noticed :big_smile:

sarahzrf (May 07 2020 at 19:02):

yeah i'm scrolling thru results for him on arxiv rn but i'm not sure which ones are good for my purposes

sarahzrf (May 07 2020 at 19:03):

e.g., for concreteness: one of my cases is that i have a fixed group G and my endofunctor is H ↦ G + H

sarahzrf (May 07 2020 at 19:03):

is that cofinitary?! which paper of his would be useful for this?!

sarahzrf (May 07 2020 at 21:49):

damn image.png

sarahzrf (May 08 2020 at 01:43):

okay, wait a second, i'm starting to think that in the group example the terminal coalgebra actually isn't the limit of the tower

sarahzrf (May 08 2020 at 01:44):

so the example comes from topology: https://wildtopology.wordpress.com/2013/11/23/the-hawaiian-earring/

sarahzrf (May 08 2020 at 01:45):

and:

1. the hawaiian earring group $\mathbb H$ is not the entire limit—but
2. i think it might be the final coalgebra of $G \mapsto G + \mathbb Z$

sarahzrf (May 08 2020 at 01:47):

...$\lim_{n}{F_n}$ is the same as the limit you'd take when trying to get the final coalgebra of that functor if it were cofinitary, right?

sarahzrf (May 08 2020 at 02:21):

okay let's identify $F_n$ with the n-fold coproduct of $\mathbb Z$. say $Z : \mathbf{Grp} \to \mathbf{Grp}$ is $Z(G) = G + \mathbb Z$. say we have a Z-coalgebra $f : G \to G + \mathbb Z$. define a homomorphism $h : G \to \mathbb H$ by $F_0 \ni h(g)_0 = 1$ and $F_n + \mathbb Z \ni h(g)_{n + 1} = (h_n + \operatorname{id}_{\mathbb Z})(f(g))$

sarahzrf (May 08 2020 at 02:21):

um slightly imprecise notation there in the successor case but

sarahzrf (May 08 2020 at 02:28):

actually wait

sarahzrf (May 08 2020 at 02:28):

i was about to try to argue that that really does have image in $\mathbb H$, but now i'm unsure...

sarahzrf (May 08 2020 at 02:41):

ok no i think it does

Morgan Rogers (he/him) (May 08 2020 at 10:09):

I enjoy how much of your thought process ends up verbalised on here :laughing: I look forward to hearing your conclusions

sarahzrf (May 08 2020 at 13:50):

i like to think out loud <.<

sarahzrf (May 08 2020 at 14:14):

anyway though this seems like a fairly interesting property to have but im having trouble finding anything about it on google o.O

sarahzrf (May 08 2020 at 14:14):

on the other hand i guess the hawaiian earring group isnt the subject of intense study and maybe it's not the most obvious property to notice if you aren't categorically inclined

sarahzrf (May 08 2020 at 14:14):

or maybe i just made a mistake :upside_down:

John Baez (May 08 2020 at 17:11):

Thanks! On MathOverflow I claimed that all finitely generated commutative monoids are finitely presented; then I forgot where I got this information and started to doubt it. Now I see it's called Somebody's Theorem. I'll add that information.

John Baez (May 08 2020 at 17:12):

But it should be pretty easy to cook up some groups with 2 generators where finitely many relations aren't enough.

John Baez (May 08 2020 at 17:12):

Hmm, I said it should be easy, so I should do it.

sarahzrf (May 09 2020 at 23:39):

@Samuel Hsu is starting to convince me that the terminal coalgebra is the entire limit rather than just $\mathbb H$

John Baez (May 10 2020 at 00:03):

Anyone wanting more on the Brauer 3-group can go here:

• Ross Street, Descent, Section 5.

John Baez (May 10 2020 at 00:06):

Unfortunately here he uses an approach to n-categories called "n-files" which was later shown to be flawed. (James Dolan came up with similar idea and discovered it was flawed; when he saw Street doing the same thing he told Street, and that was the end of "files".) But if you just skip the stuff about files and read "n-file" as "weak n-category", you'll be fine.

Alexander Campbell (May 10 2020 at 00:27):

John Baez said:

Unfortunately here he uses an approach to n-categories called "n-files" which was later shown to be flawed. (James Dolan came up with similar idea and discovered it was flawed; when he saw Street doing the same thing he told Street, and that was the end of "files".) But if you just skip the stuff about files and read "n-file" as "weak n-category", you'll be fine.

Here's an updated version which, among many other ameliorations, doesn't use n-files. http://web.science.mq.edu.au/~street/DescFlds.pdf

Alexander Campbell (May 10 2020 at 00:29):

Brauer groups now appear in §12.

John Baez (May 10 2020 at 00:30):

Thanks!

John Baez (May 10 2020 at 00:31):

This section is almost the same.

sarahzrf (May 10 2020 at 17:30):

i changed my mind again, i think the hawaiian earring group is the terminal coalgebra after all :sweat_smile:

sarahzrf (May 13 2020 at 01:01):

!
https://ncatlab.org/nlab/show/distributivity+of+limits+over+colimits#filtered_colimits

sarahzrf (May 13 2020 at 01:01):

this is very useful to know! and i somehow hadn't stumbled on it while browsing a lot of similar-looking material!

sarahzrf (May 13 2020 at 01:02):

apparently small limits commute with filtered colimits in any locally finitely presentable category

EDIT: oops, they only distribute :(

John Baez (May 13 2020 at 01:11):

Yes, this is the sort of stuff I've been learning lately. There's a nice treatment of some such facts in Borceux.

John Baez (May 13 2020 at 01:12):

Why do you care about this?

sarahzrf (May 13 2020 at 01:13):

obsession with filtered colimits lately :)

sarahzrf (May 13 2020 at 01:13):

but also i initially posted this topic in particular because of the application to construction of initial algebras

sarahzrf (May 13 2020 at 01:14):

although the construction that motivated that was actually a terminal coalgebra, which you might remember from the applied ct stream...

sarahzrf (May 13 2020 at 01:26):

so here's a question:

sarahzrf (May 13 2020 at 01:27):

an object A is "compact" or "finitely presentable" if Hom(A, -) preserves filtered colimits

sarahzrf (May 13 2020 at 01:27):

is there a notion of an object for which [A, -] preserves filtered colimits?

sarahzrf (May 13 2020 at 01:29):

akin to the distinction between a tiny object (Hom(A, -) preserves small colimits) and an infinitesimal object ([A, -] has a right adjoint—but this is equivalent to preserving small colimits, for nice enough categories)

Reid Barton (May 13 2020 at 01:30):

what setting are you in and what is [A, -]?

sarahzrf (May 13 2020 at 01:30):

ah, sorry, i mean for that to be the internal hom of a presumably cartesian closed category

Reid Barton (May 13 2020 at 01:31):

If it's a lfp category then [A, -] preserves filtered colimits iff A x - preserves finitely presentable objects (general statement about adjoints)

Reid Barton (May 13 2020 at 01:32):

In general, this might not happen even if A is finitely presentable

Reid Barton (May 13 2020 at 01:35):

But often it does happen that finitely presentable objects are closed under finite products (you can check whether it holds it on generators, e.g., representables in a presheaf category) and then it's equivalent to A being finitely presentable.

Reid Barton (May 13 2020 at 01:36):

Reid Barton said:

In general, this might not happen even if A is finitely presentable

It could also happen even when A is not finitely presentable (e.g., if the terminal object is not finitely presentable).

Reid Barton (May 13 2020 at 01:38):

There's also a paper by Kelly Structures defined by finite limits in the enriched context, I about an enriched analogue of locally finitely presentable categories (although I found it was easier to work out the theory I needed in my own way than to read it :upside_down:)

Reid Barton (May 13 2020 at 01:42):

If it's not a lfp category, then I don't know what happens.

Reid Barton (May 13 2020 at 01:47):

Some examples of apparently nice categories where these Hom(A, -) and [A, -] concepts differ that you may want to consider include $\mathrm{Set}^S$ for $S$ an infinite set and $\mathrm{Set}^{\mathrm{B}G}$ for an infinite group $G$ ($\mathrm{B}G$ being the one-object groupoid corresponding to $G$).

John Baez (May 13 2020 at 05:00):

This is interesting; actually @Joe Moeller and I were wondering about when the internal hom [A, -] preserves filtered colimits for A a compact object in a locally finitely presentable category!

John Baez (May 13 2020 at 05:02):

Unfortunately we care about situations where the terminal object is not compact.

John Baez (May 13 2020 at 05:02):

It may be that we're going against the grain here and need to do something different.

sarahzrf (May 13 2020 at 05:33):

i just found out that a groth. topos w/ a compact terminal object is called "strongly compact"

sarahzrf (May 13 2020 at 05:35):

ordinary compact topos is if global sections functor preserves filtered colimits of subterminals

sarahzrf (May 13 2020 at 05:35):

...well, nlab phrases it as "directed joins" but i figure that has to be equivalent

Morgan Rogers (he/him) (May 13 2020 at 09:21):

sarahzrf said:

apparently small limits commute with filtered colimits in any locally finitely presentable category

Where are you reading this on the page you linked? Should that read "small products distribute over flitered colimits"?

Morgan Rogers (he/him) (May 13 2020 at 09:21):

sarahzrf said:

...well, nlab phrases it as "directed joins" but i figure that has to be equivalent

Indeed it is.

Morgan Rogers (he/him) (May 13 2020 at 09:24):

sarahzrf said:

i just found out that a groth. topos w/ a compact terminal object is called "strongly compact"

The relativised equivalent concept is tidiness, which I mentioned on the finite object topic; unfortunately, this concept is frustratingly much harder to verify than it seems it should be.

Reid Barton (May 13 2020 at 10:38):

Morgan Rogers said:

sarahzrf said:

apparently small limits commute with filtered colimits in any locally finitely presentable category

Where are you reading this on the page you linked? Should that read "small products distribute over flitered colimits"?

No, it should read finite limits commute with filtered colimits.

Morgan Rogers (he/him) (May 13 2020 at 10:52):

That's a general fact (or in some cases the definition). I was quoting from the page that @sarahzrf linked.

...a category is precontinuous if and only if it has small limits and filtered colimits, filtered colimits commute with finite limits, and small products distribute over filtered colimits. In particular, any locally finitely presentable category, equivalently the category of algebras over some finitary essentially algebraic theory, is precontinuous.

Reid Barton (May 13 2020 at 11:07):

Oh I didn't actually look at the page itself

Reid Barton (May 13 2020 at 11:09):

and small limits distribute over filtered colimits, i.e., the functor colim:Ind(C)→C is continuous.

Reid Barton (May 13 2020 at 11:13):

I don't know what to make of this.

Reid Barton (May 13 2020 at 11:15):

I mean, I can accept the second statement as a definition of the first statement but then I don't have any intuition for when it holds.

Morgan Rogers (he/him) (May 13 2020 at 11:20):

Well... in any locally presentable category, apparently!

Morgan Rogers (he/him) (May 13 2020 at 11:20):

(ie me neither haha)

sarahzrf (May 13 2020 at 13:10):

oh damn, i confused distributivity for commutation :(

sarahzrf (May 13 2020 at 13:10):

shouldve known it seemed too good to be true!

sarahzrf (May 13 2020 at 13:14):

hmm... but wait, shouldn't the "the colimit-taking functor from Ind(C) is continuous" phrasing give actual commutation as long as limits in Ind(C) are computed pointwise? or...

Mike Shulman (May 13 2020 at 15:44):

Reid Barton said:

and small limits distribute over filtered colimits, i.e., the functor colim:Ind(C)→C is continuous.

Yeah, something seems wrong there: I would expect "colim:Ind(C)→C is continuous" to be the statement that small limits commute with filtered colimits.