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Stream: theory: category theory

Topic: distributivity of limits and colimits

Mike Shulman (Apr 28 2021 at 21:20):

This thread is for a continuation of this discussion on MO chat which was a continuation of the comments on this answer. Because Zulip is a better platform for conversations than either MO comments or MO chat.

Tim Campion (Apr 28 2021 at 21:23):

Oh cool!

Tim Campion (Apr 28 2021 at 21:26):

Something I've been meaning to mention in this area is that Charles Rezk has a draft up on his website investigating oo-categorical limit doctrines

Tim Campion (Apr 28 2021 at 21:26):

https://faculty.math.illinois.edu/~rezk/accessible-cat-thoughts.pdf

Tim Campion (Apr 28 2021 at 21:27):

The most exciting thing to me is that Charles discovered that oo-categorically, the doctrine of pullbacks is _sound_ (in the evident sense mimicking ABLR in oo-categories)

Tim Campion (Apr 28 2021 at 21:27):

Notoriously, this doctrine is not sound 1-categorically.

Mike Shulman (Apr 28 2021 at 21:31):

That's neat that there's another way in which limits/colimits are better $\infty$-categorically. I don't work enough with sound doctrines that it sounds exciting to me, though. Does it have useful applications?

Tim Campion (Apr 28 2021 at 21:34):

I'm not sure! One thing it tells you is that Cat_oo (which is sketchable by a pullback sketch of course) has certain limit/colimit commmutativity properties.

Tim Campion (Apr 28 2021 at 21:34):

PB-filteredness is equivalent, Charles shows, to being a space-indexed colimit of filtered categories

Tim Campion (Apr 28 2021 at 21:35):

So colimits in Cat_oo indexed by such categories commute with pullbacks.

Tim Campion (Apr 28 2021 at 21:35):

I think Charles gives some interesting unexpected examples like mapping tori

Tim Campion (Apr 28 2021 at 21:37):

But I guess mostly I just find it suggestive -- in 1-categories, it seems to me the main limitation of sound doctrines is that there just don't seem to be that many of them, and you can really generally get by treating them on an ad hoc individual basis.

Tim Campion (Apr 28 2021 at 21:37):

Maybe the true potential of the theory will be more fully realized in oo-categories.

Tim Campion (Apr 28 2021 at 21:38):

$\infty$-categories

Nathanael Arkor (Apr 28 2021 at 21:38):

@Tim Campion:

I've had occasion to wonder recently if maybe the whole apparatus of limit doctrines could be redone with distributivity rather than commutation of limits and colimits.

To my understanding, the paper Accessibility and presentability in 2-categories by @Ivan Di Liberti and @fosco is very much inspired by this perspective.

Tim Campion (Apr 28 2021 at 21:39):

@Nathanael Arkor Oh that's really cool! Could you explain more? I don't see the word "distribut-" appearing in the paper.

Tim Campion (Apr 28 2021 at 21:41):

Perhaps the version on my computer is old though

Nathanael Arkor (Apr 28 2021 at 21:42):

The notion of "envelope" (Definition 3.1) can be rephrased in terms of a distributive law of KZ doctrines (for S over D), so that P is the composite SD.

Tim Campion (Apr 28 2021 at 21:48):

@Dmitri Pavlov Thanks for the reference. Taking a look, I _think_ the idea generalizes as follows: any KZ-doctrine distributes over the doctrine Lim

Tim Campion (Apr 28 2021 at 21:49):

And a free algebra for the KZ doctrine is automatically an algebra for the composite doctrine

Tim Campion (Apr 28 2021 at 21:49):

I think it just boils down to the fact that the structure map for the algebra fits into an appropriate string of adjoints (by virtue of KZ ness), and in particular preserves limits.

Nathanael Arkor (Apr 28 2021 at 21:53):

Tim Campion said:

Dmitri Pavlov Thanks for the reference. Taking a look, I _think_ the idea generalizes as follows: any KZ-doctrine distributes over the doctrine Lim

This is certainly true for $\mathbb D\text{-}\mathrm{Ind}$ for sound $\mathbb D$.

Nathanael Arkor (Apr 28 2021 at 21:54):

(By Corollary 6.4 of A classification of accessible categories.)

Tim Campion (Apr 28 2021 at 21:54):

Right... my claim does seem fishy, given that ABLR have to do _some_ work to prove there's a distributive law.

Tim Campion (Apr 28 2021 at 21:54):

I suppose what I should really claim is that _if_ there's a distributive law, then the free algebras for the KZ monad are automatically algebras for the composite monad

Tim Campion (Apr 28 2021 at 21:55):

In the particular case of composing the one's KZ doctrine with the doctrine Lim

Mike Shulman (Apr 28 2021 at 21:55):

Surely that can't be true unless your algebra is free on a Lim-algebra?

Mike Shulman (Apr 28 2021 at 21:56):

(in which case it should follow formally from the existence of a distributive law, right?)

Tim Campion (Apr 28 2021 at 21:56):

I agree that case follows formally from the existence of a distributive law

Tim Campion (Apr 28 2021 at 21:57):

Okay -- there's an additional requirement.

Tim Campion (Apr 28 2021 at 21:58):

Let T be a KZ doctrine and suppose that T distributes over Lim. Let C be a category, and suppose that T(C) in fact has limits. Then T(C) is an algebra for $T \circ Lim$

Tim Campion (Apr 28 2021 at 21:58):

Proof: the only thing to check is that the structure map $T^2(C) \to T(C)$ preserves limits. But by KZness it has a left adjoint

Tim Campion (Apr 28 2021 at 21:59):

The left adjoint is $T(y)$ where $y$ is the unit for the KZ doctrine

Tim Campion (Apr 28 2021 at 22:00):

I would have to look up the definitions to verify that that particular adjunction is guaranteed to exist by the KZ property, but at least it seems to be the case when $T = Ind$ (and this adjunction is the core of the Johnstone-Joyal proof).

Tim Campion (Apr 28 2021 at 22:00):

So I feel pretty comfortable asserting that this is the correct generalization.

Mike Shulman (Apr 28 2021 at 22:03):

My memory of lax-idempotence is that $y_T$ is right adjoint to the structure map. I don't remember anything about $T(y)$ being left adjoint to it.

Tim Campion (Apr 28 2021 at 22:04):

I think they both hold

Tim Campion (Apr 28 2021 at 22:04):

You only get one adjunction for a generic T-algebra

Tim Campion (Apr 28 2021 at 22:04):

But we have a free T-algebra

Tim Campion (Apr 28 2021 at 22:04):

So both adjunctions potentially make sense

Tim Campion (Apr 28 2021 at 22:04):

And I believe they both hold

Mike Shulman (Apr 28 2021 at 22:05):

Ah, that could be.

Tim Campion (Apr 28 2021 at 22:06):

I think Marmolejo and Wood talk about this adjoint triple

Nathanael Arkor (Apr 28 2021 at 22:06):

Yes, it's on the first page of Kan extensions and lax idempotent pseudomonads.

Tim Campion (Apr 28 2021 at 22:24):

Let's see. Does this work in great generality?

Tim Campion (Apr 28 2021 at 22:25):

Let $\mathcal K$ be any class of small categories

Tim Campion (Apr 28 2021 at 22:25):

To be used for indexing colimits

Tim Campion (Apr 28 2021 at 22:26):

The free $\mathcal K$-cocompletion can always be computed as the closure in the presheaf category under $\mathcal K$-indexed colimits.

Tim Campion (Apr 28 2021 at 22:27):

Call this functor $P_{\mathcal K}$

Tim Campion (Apr 28 2021 at 22:27):

Suppose that $P_{\mathcal K}(C)$ is naturally reflective in $P(C)$ for any small $C$.

Tim Campion (Apr 28 2021 at 22:28):

Where $P(C)$ is the presheaf category.

Tim Campion (Apr 28 2021 at 22:28):

Then for every small $C$, $P_{\mathcal K}(C)$ is complete.

Tim Campion (Apr 28 2021 at 22:29):

And by the previous argment, since $P_{\mathcal K}$ is lax-idempotent, the structure map $P_{\mathcal K}^2(C) \to P_{\mathcal K}(C)$ has a left adjoint and so is continuous.

Tim Campion (Apr 28 2021 at 22:30):

So in order to get a distributive law with $Lim$, and have that $P_{\mathcal K}(C)$ is always an algebra for the composite monad, well, we just need the distributive law.

Tim Campion (Apr 28 2021 at 22:32):

(I am still perpetually confused about which monad is said to distribute over which other one)

Tim Campion (Apr 28 2021 at 22:34):

The thing to check is that $P_{\mathcal K}$ preserves completeness of categories and continuity of functors between complete categories.

John Baez (Apr 28 2021 at 22:35):

(It's good to remember that multiplication distributes over addition, so timesy things like the monad for cartesian categories tend to distribute over plussy things like the monad for categories with some class of colimits.)

John Baez (Apr 28 2021 at 22:35):

(But you probably know that.)

Tim Campion (Apr 28 2021 at 22:35):

@John Baez Ah, thanks. Yeah, that makes sense.

Tim Campion (Apr 28 2021 at 22:36):

I think the part which confuses me is that to verify the distributive law of $Lim$ over $P$ so that $P \circ Lim$ is a monad, you need to consider $Lim \circ P$, which feels backwards

Tim Campion (Apr 28 2021 at 22:36):

Of course, you just consider $Lim \circ P$ in order to get a functor from it to $P \circ Lim$, but it still feels strange.

Tim Campion (Apr 28 2021 at 22:37):

Er -- I guess I'm still a bit turned around :)

Tim Campion (Apr 28 2021 at 22:39):

What I want to say is that $P_{\mathcal K}$ will preserve right adjoints and that that will be enough to preserve continuity. Of course that doesn't quite work because we end up leaving adjoint-functor-theorem land.

John Baez (Apr 28 2021 at 22:39):

I guess this is how I remember this stuff:

In a ring you can rewrite any product of sums as a sum of products (but not vice versa), so you define the monad for rings using a distributive law PS $\to$ SP, where P is for products (the monad for monoids) and S is for sums (the monad for abelian groups).

John Baez (Apr 28 2021 at 22:40):

But then, yeah, the monad for rings is SP, with multiplication SPSP $\to$ SSPP $\to$ SP.

Tim Campion (Apr 28 2021 at 22:43):

Thanks. I think I got that well enough to at least rewrite my confusion in an accurate way!

Tim Campion (Apr 28 2021 at 22:44):

I managed not to be too confused by the fact that my $P$ corresponds to your $S$ :)

Tim Campion (Apr 28 2021 at 22:44):

I think

Tim Campion (Apr 28 2021 at 22:48):

However, $Lim$ is co-lax-idempotent.

Tim Campion (Apr 28 2021 at 22:49):

So $C$ is complete iff there is a right adjoint to the unit $C \to Lim(C)$

Tim Campion (Apr 28 2021 at 22:49):

So completeness is witnessed by an adjunction. Unfortunately maybe that doesn't help.

John Baez (Apr 28 2021 at 22:50):

Tim Campion said:

I managed not to be too confused by the fact that my $P$ corresponds to your $S$ :)

Darn, I was trying to confuse you.

Tim Campion (Apr 28 2021 at 22:57):

It really feels like all that needs to be juggled is size. It doesn't feel like soundness should actually be necessary for anything here.

Tim Campion (Apr 28 2021 at 23:01):

Let $L_\kappa$ be the doctrine for $\kappa$-small limits. If $C$ is small, then $L_\kappa C$ is small.

Tim Campion (Apr 28 2021 at 23:04):

Let $L = Lim$ be the dotrine for all small limits.

Tim Campion (Apr 28 2021 at 23:05):

In general, the unit $C \to L_\kappa C$ induces $P_{\mathcal K} C \to P_{\mathcal K} L_\kappa C$

Tim Campion (Apr 28 2021 at 23:07):

If $C$ is small, then $L_\kappa C$ is small. By hypothesis, this implies that $P_{\mathcal K} L_\kappa C$ is complete.

Tim Campion (Apr 28 2021 at 23:07):

and in particular $\kappa$-complete.

Tim Campion (Apr 28 2021 at 23:08):

By the universal property of $L_\kappa$, we get an induced functor $L_\kappa P_{\mathcal K} C \to P_{\mathcal K} L_\kappa C$.

Tim Campion (Apr 28 2021 at 23:08):

which hopefully gives us a distributive law.

Tim Campion (Apr 28 2021 at 23:09):

Ah, but the hypothesis was a bit problematic.

Tim Campion (Apr 28 2021 at 23:10):

Generally we expect that $P_{\mathcal K} C$ will only be complete, or even $\kappa$-complete, if $C$ is appropriately cocomplete.

Tim Campion (Apr 28 2021 at 23:10):

And that's where we might actually need some theory of limit doctrines to understand things.

Tim Campion (Apr 28 2021 at 23:46):

Perhaps an easier approach would be: prove once and for all that there is a distributive law $L \circ P \to P \circ L$.

Tim Campion (Apr 28 2021 at 23:46):

This is done in Day and Lack's "Limits of small functors"

Tim Campion (Apr 28 2021 at 23:47):

in the enriched setting. So it seems to be reasonably formal.

Tim Campion (Apr 28 2021 at 23:47):

Then all these other distributive laws are "sub-distributive laws" of that one.

Tim Campion (Apr 28 2021 at 23:49):

I think it follows that there is a distributive law $L \circ P_{\mathcal K} \to P_{\mathcal K} \circ L$ [which is a sub-distributive law of $L \circ P \to P \circ L$] if $P_{\mathcal K} L C$ is complete for every $C$.

Tim Campion (Apr 28 2021 at 23:51):

And showing this does indeed seem to be the substantive part of any proof in the literature that there is such a distributive law for a particular $\mathcal K$.

Nathanael Arkor (Apr 28 2021 at 23:56):

Tim Campion said:

I think it follows that there is a distributive law $L \circ P_{\mathcal K} \to P_{\mathcal K} \circ L$ [which is a sub-distributive law of $L \circ P \to P \circ L$] if $P_{\mathcal K} L C$ is complete for every $C$.

I think this is Theorem 5.4 of Karazeris–Velebil's Representability relative to a doctrine.

Tim Campion (Apr 29 2021 at 00:14):

@Nathanael Arkor Thanks, that is very nice! And they are treating sub-monads of $L$ at the same time!

Tim Campion (Apr 29 2021 at 00:15):

I bet one can get away with this holding for $C$ small.

Tim Campion (Apr 29 2021 at 00:17):

Just because these monads should commute with colimits of $Ord$-indexed chains.

Tim Campion (Apr 29 2021 at 00:37):

Here's a cute criterion. Suppose that the canonical functor $P_{\mathcal K}(C^I) \to P_{\mathcal K}(C)^I$ is an equivalence (or even just has a right adjoint) for every $C$ and every $I \in \mathcal L$ (where $\mathcal L$ is a collection of small categories used for indexing limits). Then if $C$ is $\mathcal L$-complete, the diagonal functor $C \to C^I$ has a right adjoint. By 2-functoriality, $P_{\mathcal K}(C) \to P_{\mathcal K}(C^I)$ has a right adjoint. Composing, we get that $P_{\mathcal K}(C) \to P_{\mathcal K}(C)^I$ has a right adjoint. Thus, under this hypothesis, we get that $P_{\mathcal K}$ preserves $\mathcal L$-completeness.

Tim Campion (Apr 29 2021 at 00:38):

So -- if $P_{\mathcal K}$ preserves exponentials in the above sense, then $L_{\mathcal L}$ distributes over $P_{\mathcal K}$ for every class of limits $\mathcal L$.

Tim Campion (Apr 29 2021 at 00:39):

I think this at least recovers the case where $\mathcal K$ is the filtered categories.

Tim Campion (Apr 29 2021 at 00:41):

Actually, I think the weaker form of the criterion -- just saying that $P_{\mathcal K}(C^I) \to P_{\mathcal K}(C)^I$ has a right adjoint -- holds... always?

Tim Campion (Apr 29 2021 at 00:41):

That sounds a bit strong.

Tim Campion (Apr 29 2021 at 00:42):

Maybe not always, but maybe if one simply assumes that $P_{\mathcal K}(C)$ may be "computed in one step"" -- i.e. assume that the closure of the representables under $\mathcal K$-colimits comprises precisely the $\mathcal K$-colimits of representables.

Tim Campion (Apr 29 2021 at 00:46):

That is... just assume that $\mathcal K$ is saturated in the sense of Albert and Kelly's "closed".

Tim Campion (Apr 29 2021 at 00:48):

No, there is an additional condition.

Morgan Rogers (he/him) (Apr 29 2021 at 09:27):

Tim Campion said:

But I guess mostly I just find it suggestive -- in 1-categories, it seems to me the main limitation of sound doctrines is that there just don't seem to be that many of them, and you can really generally get by treating them on an ad hoc individual basis.

This seems unlikely - there are lots of classes of commuting limits and colimits, for example, it's just that few of them are generic enough to have been studied in any depth.

Morgan Rogers (he/him) (Apr 29 2021 at 09:42):

A few tangents to this conversation:
First is the notion of codistributivity; the argument for constructing the canonical morphism $\mathrm{colim}_{K^I}\mathrm{lim}_I D \to \mathrm{lim}_I \mathrm{colim}_{K^I} D$ dualizes just fine. It's interesting to see just how far codistributivity can fail for simple cases!
Second, related to my previous reply, I have partially translated from French (and, in places, corrected) the work of F. Foltz giving precise but involved criteria for commutativity of limits and colimits in Set. An interesting one, quoted as Lemma 2.2 in the paper linked above, is that in order for a colimit diagram to commute with any non-trivial connected limits at all, the diagram has to have cones under all spans. I wonder if this result has an analogue for distributivity.

Tim Campion (Apr 29 2021 at 14:22):

@Morgan Rogers (he/him) This is fantastic! I would be very interested to read what you have on Foltz.

Tim Campion (Apr 29 2021 at 14:23):

I once made an MO post where my wording suggested I didn't fully trust Foltz's results.

Tim Campion (Apr 29 2021 at 14:23):

After years, I changed the wording because I thought it was unfair to Foltz.

Tim Campion (Apr 29 2021 at 14:24):

But I did have this nagging sense that because of the involved combinatorial nature of the results, it seemed not unreasonable to suppose there might be errors.

Tim Campion (Apr 29 2021 at 14:24):

@Morgan Rogers (he/him) Your point that there are many classes of commuting limits and colimits in Set is good -- but on the other hand, I know of only a handful of them which are sound.

Tim Campion (Apr 29 2021 at 14:25):

And I don't know anything about getting interesting results about limit doctrines without soundness.

Morgan Rogers (he/him) (Apr 29 2021 at 14:25):

Tim Campion said:

I once made an MO post where my wording suggested I didn't fully trust Foltz's results.

To some extent you were right not to trust them, a lot of the diagrams have morphisms incorrectly labelled or pointing in the wrong direction due to them being hastily added by hand. There is also a sticking point that I haven't yet resolved.

Morgan Rogers (he/him) (Apr 29 2021 at 14:26):

Tim Campion said:

Morgan Rogers (he/him) Your point that there are many classes of commuting limits and colimits in Set is good -- but on the other hand, I know of only a handful of them which are sound.

Would you mind elaborating? The nLab definition is not very enlightening.

Tim Campion (Apr 29 2021 at 14:27):

Soundness was introduced by Adamek, Borceux, Lack and Rosicky.

Morgan Rogers (he/him) (Apr 29 2021 at 14:28):

Tim Campion said:

Morgan Rogers (he/him) This is fantastic! I would be very interested to read what you have on Foltz.

I wasn't planning to finish drafting it any time soon (I have a thesis to write..!) but if you're willing to wait until, say, December, I can share it!

Tim Campion (Apr 29 2021 at 14:28):

I once wrote a blog post about their paper https://golem.ph.utexas.edu/category/2014/05/classifying_by_generalizing_th.html

Tim Campion (Apr 29 2021 at 14:28):

Morgan Rogers (he/him) said:

Tim Campion said:

Morgan Rogers (he/him) This is fantastic! I would be very interested to read what you have on Foltz.

I wasn't planning to finish drafting it any time soon (I have a thesis to write..!) but if you're willing to wait until, say, December, I can share it!

Oh sure -- no rush! I am not thinking about this stuff actively.

Tim Campion (Apr 29 2021 at 14:29):

One way of looking at it is the following: if you've got some class of limits D (a "limit doctrine")

Tim Campion (Apr 29 2021 at 14:29):

Say that a category I is "D-filtered" if I-colimits commute with D-limits in Set.

Tim Campion (Apr 29 2021 at 14:30):

In the case where D = finite categories for example, there's this rich theory of finitely accessible categories

Tim Campion (Apr 29 2021 at 14:30):

And free cocompletion under filtered colimits.

Tim Campion (Apr 29 2021 at 14:30):

There are several related constructions one can do for general D which end up being equivalent in this case, and the equivalence is kind of necessary to get accessiblity theory to tick.

Tim Campion (Apr 29 2021 at 14:31):

Soundness is a condition which guarantees that one has these equivalences.

Tim Campion (Apr 29 2021 at 14:31):

I'm alluding basically to the fact that there are several different ways to characterize Ind(C)

Tim Campion (Apr 29 2021 at 14:31):

for a category C

Tim Campion (Apr 29 2021 at 14:35):

In particular, one general result says that if D is sound, then the "free completion functor" Lim distributes over the "free D-filtered cocompletion functor" D-Ind.

Tim Campion (Apr 29 2021 at 14:35):

I don't know if there are examples where this holds when D is not sound.

Morgan Rogers (he/him) (Apr 29 2021 at 14:41):

Oh hey I've encountered this paper before! From those examples, it seems like the problem is one of saturation; their example of the doctrine of pullbacks and terminal objects not being sound in particular. I wonder if soundness could be reduced to "every diagram in the saturation of $\mathbb{D}$ admits an initial functor from a diagram in $\mathbb{D}$" or some condition to that effect.

Morgan Rogers (he/him) (Apr 29 2021 at 14:42):

Where by "saturation" I mean the maximal class of limits commuting with $\mathbb{D}$-filtered colimits

Tim Campion (Apr 29 2021 at 14:43):

Soundness and saturation are related in some weird way I don't fully understand.

Nathanael Arkor (Apr 29 2021 at 14:43):

There are examples of doctrines which are saturated but not sound. There's an example somewhere…

Tim Campion (Apr 29 2021 at 14:43):

The doctrine of L-finite simply-connected limits (the saturation of Pullbacks) is saturated but not sound.

Tim Campion (Apr 29 2021 at 14:44):

(However the oo-categorical analog is sound according to Charles Rezk!)

Tim Campion (Apr 29 2021 at 14:44):

For example, Dostal and Velebil make saturation a blanket assumption on their doctrines here https://arxiv.org/abs/1405.3090

Tim Campion (Apr 29 2021 at 14:44):

But soundness is still an important question in their setting.

Tim Campion (Apr 29 2021 at 14:45):

ABLR also give the weird example of the doctrine of pullbacks + terminal objects, which is not sound.

Tim Campion (Apr 29 2021 at 14:45):

even though its saturation is sound

Morgan Rogers (he/him) (Apr 29 2021 at 14:46):

Back in undergrad when I was doing my first CT research project (incidentally, also when I started translating Foltz :joy:) I started investigating the relationship between this sort of saturation and the sort that @Tim Campion just referenced, but didn't have enough background to see deeply into it.

Tim Campion (Apr 29 2021 at 14:46):

Note also that finite limits is not a saturated doctrine!

Tim Campion (Apr 29 2021 at 14:46):

Morgan Rogers (he/him) said:

Back in undergrad when I was doing my first CT research project (incidentally, also when I started translating Foltz :joy:) I started investigating the relationship between this sort of saturation and the sort that Tim Campion just referenced, but didn't have enough background to see deeply into it.

Oh sorry -- are we talking about different sorts of "saturation"?

Morgan Rogers (he/him) (Apr 29 2021 at 14:47):

One is "if $\mathcal{C}$ has limits in $\mathbb{D}$, what is the maximal class of limits it automatically has", whose answer for pullbacks is what you said;
The other is "which limits commute with the colimits commuting with the limits in $\mathbb{D}$"

Nathanael Arkor (Apr 29 2021 at 14:47):

Tim Campion said:

Note also that finite limits is not a saturated doctrine!

Oh, I didn't realise this. What is the saturation?

Tim Campion (Apr 29 2021 at 14:47):

Nathanael Arkor said:

Tim Campion said:

Note also that finite limits is not a saturated doctrine!

Oh, I didn't realise this. What is the saturation?

The L-finite limits!

Morgan Rogers (he/him) (Apr 29 2021 at 14:48):

Tim Campion said:

The doctrine of L-finite simply-connected limits (the saturation of Pullbacks) is saturated but not sound.

It's not clear to me that this is the saturation of pullbacks in the latter sense

Tim Campion (Apr 29 2021 at 14:48):

Oh I see.

Nathanael Arkor (Apr 29 2021 at 14:49):

Tim Campion said:

Nathanael Arkor said:

Tim Campion said:

Note also that finite limits is not a saturated doctrine!

Oh, I didn't realise this. What is the saturation?

The L-finite limits!

Is there a nice example of an L-finite limit that is not finite?

Morgan Rogers (he/him) (Apr 29 2021 at 14:49):

Any category equivalent to a finite one, or having a finite initial subcategory

Tim Campion (Apr 29 2021 at 14:49):

One characterization says that the terminal presheaf should be finitely-presentable. https://ncatlab.org/nlab/show/L-finite+category

Tim Campion (Apr 29 2021 at 14:50):

A dumb example would be any category with a terminal object

Nathanael Arkor (Apr 29 2021 at 14:50):

Morgan Rogers (he/him) said:

Any category equivalent to a finite one, or having a finite initial subcategory

Oh, I see, this is very intuitive.

Tim Campion (Apr 29 2021 at 14:51):

I think it even suffices to have a finitely-generated initial subcategory?

Tim Campion (Apr 29 2021 at 14:51):

Oh sure -- so the walking endomorphism would be a natural example!

Tim Campion (Apr 29 2021 at 14:52):

or BN if you roll that way

Tim Campion (Apr 29 2021 at 14:53):

Morgan Rogers (he/him) said:

One is "if $\mathcal{C}$ has limits in $\mathbb{D}$, what is the maximal class of limits it automatically has", whose answer for pullbacks is what you said;
The other is "which limits commute with the colimits commuting with the limits in $\mathbb{D}$"

Yeah, my convention has been to use "saturated" for the first sense (which Albert and Kelly call "closed") and I think I'm realizing I don't have a word for the second sense

Tim Campion (Apr 29 2021 at 14:54):

I keep saying mouthfuls like "closed in the Galois connection given by commutativity of limits and colimits in Set"

Tim Campion (Apr 29 2021 at 14:54):

I think I picked up this convention from a later set of notes by Kelly and Schmitt http://www.tac.mta.ca/tac/volumes/14/17/14-17abs.html

Morgan Rogers (he/him) (Apr 29 2021 at 14:54):

Morgan Rogers (he/him) said:

One is "if $\mathcal{C}$ has limits in $\mathbb{D}$, what is the maximal class of limits it automatically has", whose answer for pullbacks is what you said;
The other is "which limits commute with the colimits commuting with the limits in $\mathbb{D}$"

It seems intuitive that the latter should contain the former, since the former should somehow be the "maximal class of limits constructible from $\mathbb{D}$", but that's not so clear cut because of how these things are typically computed.

Morgan Rogers (he/him) (Apr 29 2021 at 15:00):

Anyway, I hope to gain more insight into this eventually, probably using toposes, and the distributivity direction seems promising.

Tim Campion (Apr 30 2021 at 15:51):

Morgan Rogers (he/him) said:

Tim Campion said:

The doctrine of L-finite simply-connected limits (the saturation of Pullbacks) is saturated but not sound.

It's not clear to me that this is the saturation of pullbacks in the latter sense

Just noticed this comment. It's a theorem of Pare, in his paper "simply-connected limits". I believe this paper is also where L-finiteness was first introduced, or at least first studied systematically.

Tim Campion (Apr 30 2021 at 15:51):

Er -- I see. In the case of finite limits, we have that the L-finite limits are the saturation in both senses.

Tim Campion (Apr 30 2021 at 15:52):

In the case of pullbacks, I believe Pare shows that the L-finite simply-connected limits are the saturation in the sense of Albert and Kelly's "closed" -- i.e. they are the limits you have if you have pullbacks.

Tim Campion (Apr 30 2021 at 15:53):

But it is conceivable that they are not precisely the limits which commute with those colimits which commute with pullbacks in Set.

Tim Campion (Apr 30 2021 at 15:53):

There could in principle be more such limits.

Tim Campion (Apr 30 2021 at 15:54):

Certainly any such limit-indexing diagram must be L-finite, since its limits commute with filtered colimits in Set

Nathanael Arkor (Apr 30 2021 at 15:56):

It's quite surprising to me that something as fundemental as the interaction between limits and colimits hasn't yet been completely understood.

Tim Campion (Apr 30 2021 at 16:00):

Nathanael Arkor said:

It's quite surprising to me that something as fundemental as the interaction between limits and colimits hasn't yet been completely understood.

I think it's a fundamental issue. There are a lot of places in category theory where one can construct a canonical map, but the question of whether the map is an isomorphism is not generally something which can be answered by category theory alone.

Tim Campion (Apr 30 2021 at 16:01):

Like, questions of the form "is the following map an isomorphism" are a natural entrypoint for domain-specific facts to enter.

Tim Campion (Apr 30 2021 at 16:02):

I also suspect personally that in the present case, the 1-categorical questions are probably best understood as shadows of oo-categorical questions.

Tim Campion (Apr 30 2021 at 16:02):

This sense is based on the fact that colimits in Set involve pi_0 truncations.

Tim Campion (Apr 30 2021 at 16:03):

Which I think very often become better-behaved when one moves to the oo-category of spaces.

Tim Campion (Apr 30 2021 at 16:03):

where the pi_0 truncation is replaced by taking a classifying space.

Tim Campion (Apr 30 2021 at 16:07):

So for a start, I guess I would suspect that maybe Foltz's paper on commutativity of limits and colimits might be actually easier to re-do in an oo-categorical setting.

Nathanael Arkor (Apr 30 2021 at 16:07):

Would the 1-categorical results be derivable from the $\infty$-categorical results?

Tim Campion (Apr 30 2021 at 16:08):

I'm thinking probably not directly.

Tim Campion (Apr 30 2021 at 16:10):

But I would suspect that the oo-categorical story, if it were understood, might shed some light indirectly on the 1-categorical story. Maybe suggest a framework to understand it more conceptually rather then just having at it with a bunch of subdivision categories like Foltz.

Morgan Rogers (he/him) (Apr 30 2021 at 16:10):

Tim Campion said:

So for a start, I guess I would suspect that maybe Foltz's paper on commutativity of limits and colimits might be actually easier to re-do in an oo-categorical setting.

Teach me oo-category theory and I'll juggle some limits and colimits for you.

Tim Campion (Apr 30 2021 at 16:13):

There is precedent for 1-categorical statements becoming easier when analogized oo-categorically. I think the classic one is descent for oo-topoi.

Tim Campion (Apr 30 2021 at 16:13):

In an oo-topos, $E$, the functor $E^{op} -> Cat_\infty, X \mapsto E/X$ straight-up preserves limits.

Tim Campion (Apr 30 2021 at 16:14):

Whereas if $E$ is a 1-topos, the functor $E^{op} \to Cat, X \mapsto E/X$ only preserves van Kampen limits. I don't believe this improves if you treat everything as (2,1)-category or something like that.

Tim Campion (Apr 30 2021 at 16:15):

I also have a toy example I'm starting to throw around in these kinds of conversations: the (large, locally large, but locally locally small) $\infty$-category of spaces and spans between them is complete and cocomplete.

Tim Campion (Apr 30 2021 at 16:15):

Whereas the (2,1)-category of sets and spans between them is not.

Tim Campion (Apr 30 2021 at 16:15):

(From the former fact, you can also deduce that the (2,1)-category of sets and spans between them does at least have split idempotents; I don't know a more direct proof!)

Nathanael Arkor (Mar 10 2023 at 13:19):

Morgan Rogers (he/him) said:

Tim Campion said:

Morgan Rogers (he/him) This is fantastic! I would be very interested to read what you have on Foltz.

I wasn't planning to finish drafting it any time soon (I have a thesis to write..!) but if you're willing to wait until, say, December, I can share it!

@Morgan Rogers (he/him): did you find time to continue your translation/correction of Foltz? I'd be very interested to read it.

Morgan Rogers (he/him) (Mar 10 2023 at 15:30):

Ah, no, unfortunately it has yet to reach the top of the priority queue. Depending on how little I procrastinate wrt job applications, I may have time to work on it by the end of the month (knowing someone is interested is certainly a motivator, but it's probably healthiest if I don't make a hard commitment on that for the time being).

Nathanael Arkor (Mar 10 2023 at 17:12):

Completely understandable :) (For what it's worth, I know a couple of other people who are also interested.)