March 14, 2019

The Myths of Presentability and the Sharply Large Filter

Posted by Mike Shulman

The theory of locally presentable and accessible categories and functors is an elegant and powerful tool for dealing with size questions in category theory. It contains a lot of powerful results, many found in the standard reference books Locally presentable and accessible categories (by Adamek and Rosicky) and Accessible categories (by Makkai and Pare), that can be quoted without needing to understand their (sometimes quite technical) proofs.

Unfortunately, there are a couple of such “results” that are occasionally quoted, but are actually nowhere to be found in AR or MP, and are in fact false. These are the following claims:

Myth A: “If $\mathcal{C}$ is a locally $\lambda$-presentable category and $\mu$ is a regular cardinal with $\mu\ge\lambda$, then every $\mu$-presentable object of $\mathcal{C}$ can be written as a $\mu$-small $\lambda$-filtered colimit of $\lambda$-presentable objects.”

Myth B: “If $F:\mathcal{C}\to \mathcal{D}$ is an accessible functor between locally presentable categories, then $F$ preserves $\mu$-presentable objects for all sufficiently large regular cardinals $\mu$.”

Below the fold I will recall the definitions of all these words, and then discuss how one might fall into believing these myths, why they are is not true, and what we can do about it.

Definitions: A cardinal number $\lambda$ is regular if the union of $\lt\lambda$ sets of cardinality $\lt\lambda$ still has cardinality $\lt\lambda$. Any successor cardinal (i.e. one of the form $\aleph_{\alpha+1}$) is regular, and ZFC does not prove there exist any others. A category is $\lambda$-small if its set of arrows has cardinality $\lt\lambda$, and $\lambda$-filtered if every $\lambda$-small diagram in it admits some cocone. An object $X$ is $\lambda$-presentable (sometimes called “$\lambda$-compact”) if the covariant hom-functor $Hom(X,-)$ preserves $\lambda$-filtered colimits. A locally $\lambda$-presentable category (sometimes called simply a “$\lambda$-presentable category”, but that properly refers to a $\lambda$-presentable object in $Cat$) is a cocomplete locally small category with a small dense subcategory consisting of $\lambda$-presentable objects. And a functor between locally $\lambda$-presentable categories is $\lambda$-accessible if it preserves $\lambda$-filtered colimits.

Thus a locally presentable category, though large, is “determined by a small amount of information”. So Myth A is an intuitively appealing statement that past some point, the objects of “all higher cardinalities” are “determined” by those of “cardinality” $\lambda$, and similarly Myth B says that past some point accessible functors “preserve size”. And indeed, there are several true statements with similar intuitive meanings; the following facts can be found in AR and MP:

1. If $\lambda\le \mu$, then every $\lambda$-presentable object is $\mu$-presentable, and any $\mu$-small colimit of $\mu$-presentable objects is again $\mu$-presentable. In particular, Myth A has a true converse: any $\mu$-small colimit ($\lambda$-filtered or not) of $\lambda$-presentable objects is $\mu$-presentable.

2. If $\mathcal{C}$ is locally $\lambda$-presentable and $\mu$ is a regular cardinal with $\mu\ge\lambda$, then every $\mu$-presentable object of $\mathcal{C}$ can be written as a $\mu$-small (not necessarily $\lambda$-filtered) colimit of $\lambda$-presentable objects (Remark 1.30 in AR).

3. If $\mathcal{C}$ is $\lambda$-accessible and $\mu$ is a regular cardinal with $\mu\rhd\lambda$ (a “sharp” cardinal inequality — see below), then every $\mu$-presentable object of $\mathcal{C}$ can be written as a $\mu$-small $\lambda$-filtered colimit of $\lambda$-presentable objects (Remark 2.15 in AR).

4. If $F$ is an accessible functor, then there is a regular cardinal $\kappa$ such that $F$ preserves $\mu$-presentable objects for any regular cardinal $\mu$ with $\mu\rhd\kappa$.

Facts (2) and (3) say that if we weaken the conclusion of Myth A by removing $\lambda$-filteredness of the colimit, or strengthen the hypotheses by assuming $\mu\rhd\lambda$ rather than $\mu\ge \lambda$, then it becomes a true statement. (Actually AR proves only that every $\mu$-presentable object is a retract of a colimit of the above forms. In case (3) the retract can be eliminated using Proposition 2.3.11 of MR; eliminating the retract in case (2) is discussed here.) Similarly, fact (4) says that Myth B becomes true if we restrict the cardinals $\mu$ appearing in it to those with $\mu\rhd\kappa$.

(In passing, it’s worth noting that Myth A implies Myth B, and similarly fact (3) implies fact (4). Let $F:\mathcal{C}\to \mathcal{D}$ be an $\lambda$-accessible functor between locally $\lambda$-presentable categories, and let $\kappa$ be such that whenever $X$ is $\lambda$-presentable, $F(X)$ is $\kappa$-presentable. Such a $\kappa$ always exists, since there is only a set of isomorphism classes of $\lambda$-presentable objects in $\mathcal{C}$, and every object of $\mathcal{D}$ is presentable for some cardinal. Then if every $\mu$-presentable object $X$ in $\mathcal{C}$ is a $\mu$-small $\lambda$-filtered colimit of $\lambda$-presentable objects, then $F$ preserves this colimit as it is $\lambda$-accessible. Thus $F(X)$ is a $\mu$-small ($\lambda$-filtered) colimit of $\kappa$-presentable objects, and hence $\mu$-presentable.)

Evidently the difference between the true facts (3) and (4), on one hard, and the false Myths A and B, on the other hand, lies in the difference between $\le$ and $\lhd$. In particular, it’s important that for a given $\lambda$ it is not true that $\lambda\lhd \mu$ for all sufficiently large $\mu$. Saying $\lambda\lhd\mu$ doesn’t mean that $\mu$ is “a lot bigger” than $\lambda$ (indeed it doesn’t have to be very much bigger at all, e.g. $\lambda \lhd \lambda^+$ whenever $\lambda$ is regular, where $\lambda^+$ denotes the smallest cardinal larger than $\lambda$). Instead it is more of a “large cofinality” assertion for $\mu$ with respect to $\lambda$. (The pronunciation of $\lambda\lhd\mu$ as “$\lambda$ is sharply smaller than $\mu$” is, I think, rather unhelpful in this regard, but I haven’t heard any other suggestions either.)

In fact, although I have never seen this in print, I believe that fact (3) above is actually an if-and-only-if. That is, if every $\mu$-presentable object of a locally $\lambda$-presentable category can be written as a $\mu$-small $\lambda$-filtered colimit of $\lambda$-presentable objects, then $\lambda\lhd\mu$. The standard set-theoretic definition of $\lambda\lhd\mu$ is that for any set $X$ with cardinality $|X|\lt\mu$, the poset $P_\lambda(X)$ of subsets of cardinality $\lt\lambda$ has a $\mu$-small cofinal subset. Now consider the category $Set$, in which the $\mu$-presentable objects are the sets of cardinality $\lt\mu$; thus if every $\mu$-presentable object of $Set$ can be written as a $\mu$-small $\lambda$-filtered colimit of $\lambda$-presentable objects, then whenever $|X|\lt\mu$ we have $X\cong \colim_{i\in I} X_i$ where $I$ is $\mu$-small and $\lambda$-filtered and each $|X_i|\lt\lambda$. Let $A$ be the $\mu$-small set of all the images $q_i(X_i) \subseteq X$ of the coprojections $q_i:X_i\to X$, each of which is $\lambda$-small. Then $A$ is cofinal in $P_\lambda(X)$, since for any $\lambda$-small subset $Y\subseteq X$ the inclusion $Y\hookrightarrow X$ must factor through some $X_i$ (since $Y$ is $\lambda$-presentable and the colimit is $\lambda$-filtered), hence $Y \subseteq q_i(X_i)$.

Since it’s known that $\lambda\le\mu$ does not imply $\lambda\lhd\mu$, we can transport such a counterexample (e.g. 2.13(8) in AR) to construct an explicit counterexample to Myth A. Let $\alpha$ be any infinite cardinal, such as $\aleph_0$, and $\lambda = \alpha^+$, which is regular. Let $\beta$ be a cardinal of cofinality $\alpha$ — e.g. if $\alpha=\aleph_0$ then we could take $\beta = \aleph_\omega$ — and set $\mu = \beta^+$. Let $X$ be a set of cardinality $\beta$; then $X$ is $\mu$-presentable in $Set$, but it is not a $\mu$-small $\lambda$-filtered colimit of $\lambda$-presentable objects.

To see this, note that since $\beta$ has cofinality $\alpha$, we can write $X = \bigcup_{n\lt\alpha} X_n$ where each $X_n$ has cardinality $|X_n|\lt \beta$. Suppose $X = \colim_{i\in I} Y_i$ for a $\mu$-small category $I$ and $\lambda$-presentable sets $Y_i$; we will show $I$ is not $\lambda$-filtered. Since $I$ is $\mu$-small, it has cardinality $\le \beta$, so we can choose a surjection $f:X\to ob(I)$. Since each $|X_n|\lt\beta$, the set of objects $\{f(x) \mid x\in X_n\}\subseteq ob(I)$ also has cardinality $\lt\beta$, and since each $Y_i$ also has cardinality $\lt\lambda$, the subset $W_n = \bigcup_{x\in X_n} q_{f(x)}(Y_{f(x)}) \subseteq X$ (where $q_i:Y_{i}\to X$ is the colimit coprojection) has cardinality $\lt \beta\cdot\lambda = \beta$.

Since $X$ has cardinality $\beta$, there is some element of $X\setminus W_n$; define $g:\alpha \to X$ such that $g(n)\in X\setminus W_n$ for all $n$. Since $\alpha$ is a $\lambda$-presentable set, if the colimit were $\lambda$-filtered then $g$ would factor through some $Y_i$. But then there would be an $x\in X$ with $f(x) = i$, and an $n\lt\alpha$ with $x\in X_n$, so that $g$ is contained in $W_n$, contradicting our choice of $g$ such that $g(n) \notin W_n$.

So Myth A is false. Myth B is also false; a counterexample from Remark 3.2(4) of Abstract elementary classes and accessible categories by Beke and Rosicky is the endofunctor of $Set$ defined by $F(X) = X^I$ for any infinite set $I$. Namely, let $\alpha = |I|$, let $\beta$ be any cardinal of cofinality $\alpha$, and let $\mu=\beta^+$. Then $\mu$ is regular, but $F$ does not preserve $\mu$-presentable objects. Specifically, let $X$ be of cardinality $\beta$ (hence is $\mu$-presentable); I claim $|F(X)|\gt \beta$, hence is not $\mu$-presentable.

I guess this is a fairly standard fact in cardinal arithmetic, but not being an expert in that subject, I found it helpful to spell out the relevant “diagonal argument”. Note first that the cofinality assumption implies that $X = \bigcup_{i\in I} X_i$ with $|X_i|\lt \beta$. Suppose for contradiction we have a surjection $f:X\to F(X) = X^I$; define $g:I\to X$ as follows. Given $i$, the set $B_i = \{f(x)(i) \mid x\in X_i \}\subseteq X$ has cardinality $\le |X_i| \lt\beta$. Choose $g(i) \in X \setminus B_i$, which is nonempty since $|X|=\beta \gt |B_i|$. Then $g\in X^I$ cannot be in the image of $f$, since if $f(x) = g$ then $x\in X_i$ for some $i$, hence $g(i) = f(x)(i) \in B_i$, contradicting our choice of $g(i)$.

So far I’ve found half a dozen claims on the Internet and in published papers that amount to Myth A and/or Myth B, sometimes leading to incorrect statements. But I will not point any specific fingers — except at myself! In Theorem 3.1 of arXiv:1307.6248 I claimed that some functor preserves $\kappa$-presentable objects for all cardinal numbers $\kappa\gt |\mathcal{C}|$, which is not in fact the case. This was pointed out to me by Raffael Stenzel, and in trying to figure out exactly what was true in that case, I realized it was an instance of Myth B and started noticing these myths in other places too.

Fortunately, appeals to these myths can almost always be replaced by the true facts (3) and (4) above, at the cost of sometimes changing the statements of theorems. Our first thought about how to phrase them might be “there are arbitrarily large regular cardinals $\mu$ such that…”, but there is a problem with this. Namely, suppose I have a couple lemmas, say Lemma 1 and Lemma 2, that both begin with a quantification like this, and now I want to use them both to prove a theorem. Almost certainly I’ll need to find one regular cardinal $\mu$ that satisfies both Lemma 1 and Lemma 2. But simply knowing that there are arbitrarily large $\mu$ satisfying Lemma 1, and arbitrarily large $\mu$ satisfying Lemma 2, doesn’t tell me that there are arbitrarily large (or indeed even any) $\mu$ that satisfy both.

Now this isn’t a very big problem, because for any set of regular cardinals $\lambda_i$ there are arbitrarily large regular cardinals $\mu$ such that $\lambda_i \lhd \mu$ for all $i$. This follows from the same argument: whenever $\nu$ is such that $\lambda_i \le \nu$ for all $i$, then $\lambda_i \lhd (2^\nu)^+$ for all $i$. But it means that the statements of our lemmas need to be stronger: instead of “there are arbitrarily large regular cardinals $\mu$ such that…” we should say “there is a regular cardinal $\lambda$ such that for any regular cardinal $\mu$ with $\mu\rhd\lambda$, …”.

This phrasing is admittedly a bit unlovely, but I think it can be made slightly nicer. Define a class of regular cardinals to be sharply large if it contains a class of the form $\{ \mu \mid \mu \rhd \lambda \}$ for some $\lambda$. Then we can state our lemmas as “there is a sharply large class of regular cardinals $\mu$ such that …”, or even “the class of regular cardinals $\mu$ such that … is sharply large”. And when putting together multiple lemmas to prove a theorem, we can just use the fact that the intersection of any set of sharply large classes is again sharply large — in other words, the sharply large classes form a “set-complete filter” on the class of regular cardinals.

Posted at March 14, 2019 9:29 PM UTC

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Re: The Myths of Presentability and the Sharply Large Filter

I don’t think you gave a definition of “$\lambda$-accessible category” (as in fact 3) when you defined all the other things. Or maybe you wanted to say “$\lambda$-presentable” there?

Lurie has similar statements in his development of accessible $\infty$-categories in HTT, though (if I remember correctly) his version of the “sharp inequality” is defined a little differently than the one in AR.

If 3 really is an if-and-only-if, then I would want to call $\unlhd$ the “accessibility partial ordering” of regular cardinals, since then $\lambda\unlhd \mu$ if and only if every $\lambda$-accessible category is $\mu$-accessible.

Posted by: Charles Rezk on March 16, 2019 4:17 PM | Permalink | Reply to this

Re: The Myths of Presentability and the Sharply Large Filter

Yes, I neglected to define accessible categories, but I might as well have said “locally $\lambda$-presentable” in fact 3 since the post is mainly about that case. Accessible categories are “like locally presentable ones, but need not be complete and cocomplete”. Facts 1, 3, and 4 are all true about accessible categories, but not I believe fact 2 since an accessible category doesn’t necessarily have enough non-filtered colimits.

And yes, Lurie uses a version of the sharp inequality that’s a priori a bit stronger, but apparently GCH implies that they coincide. I haven’t tried to figure out whether the ordinary version of the sharp inequality would also work in the $\infty$-category case.

“Accessibility partial ordering” isn’t bad. Would you then suggest something like “accessibly large” instead of “sharply large” for the associated filter? (It’s unfortunate, though, that “accessible” and “inaccessible” have very little to do with each other…)

Posted by: Mike Shulman on March 17, 2019 4:30 AM | Permalink | Reply to this

Re: The Myths of Presentability and the Sharply Large Filter

I’m surprised nobody called me on this yet:

A cardinal number $\lambda$ is regular if the union of $\lt\lambda$ sets of cardinality $\lt\lambda$ still has cardinality $\lt\lambda$. Any successor cardinal (i.e. one of the form $\aleph_{\alpha+1}$) is regular, and ZFC does not prove there exist any others.

… except for $\aleph_0$.

… and $1$ and $2$.

Posted by: Mike Shulman on March 22, 2019 12:43 AM | Permalink | Reply to this

Re: The Myths of Presentability and the Sharply Large Filter

Thanks for this, Mike! Personally, I wouldn’t be surprised if I’ve promulgated such myths on the internet in the past – if I’m responsible for any of those misrepresentations you’ve come across, do let me know so I can fix it!

One standard way of saying that a collection of cardinals is “large” is to say that it forms a club set. I wonder if the set of $\mu$ such that $\kappa$ is sharply less than $\mu$ forms a club set for each $\kappa$?

Posted by: Tim Campion on April 3, 2019 2:29 AM | Permalink | Reply to this

Re: The Myths of Presentability and the Sharply Large Filter

I’m glad you appreciated it. But I don’t think any of the claims of these myths that I’ve found were due to you.

I wonder if the set of $\mu$ such that $\kappa$ is sharply less than $\mu$ forms a club set for each $\kappa$?

Good question! I think the answer is no, for the boring reason that the relation $\lhd$ is usually defined only between regular cardinals, and regular cardinals are not a club set (not being closed, for trivial reasons).

More interestingly, we might ask whether $\lhd$ can be meaningfully extended to singular cardinals so that this statement becomes true. The definition “for any set $X$ with cardinality $|X|\lt\mu$, the poset $P_\lambda(X)$ of subsets of cardinality $\lt\lambda$ has a $\mu$-small cofinal subset” makes sense on its face when $\mu$ (or even $\lambda$) is singular, but I haven’t seen any evidence that this is a reasonable thing to do.

Instead we ought really to look at what’s going on at singular cardinals in locally presentable and accessible categories, which is something that Lieberman, Rosicky, and Vasey have recently been studying in papers such as Internal sizes in μ-abstract elementary classes and Set-theoretic aspects of accessible categories. The key definition is the internal size $|X|$ of an object $X$, which is the least cardinal $\lambda$ (not necessarily regular) such that $X$ is $\lambda^+$-presentable. In the category of sets, $|X|$ really is exactly the cardinality of $X$, while in other categories it usually detects the “underlying cardinality” at least when that is sufficiently large.

Now we can define (although I don’t think LRV do exactly this), for an arbitrary cardinal $\mu$, an object $X$ to be $\mu$-small if $|X|\lt \mu$. If $\mu$ is a successor cardinal (hence regular) or a strong inaccessible, then $\mu$-smallness agrees with $\mu$-presentability, and some set-theoretic assumptions imply that the same is true for weak inaccessibles as well (e.g. GCH, which implies that weak inacessibles are strong). Thus, we can ask when and how facts about $\mu$-presentability carry over to $\mu$-smallness.

One relation that I think is interesting is the closure of $\lhd$. That is, if $\lambda$ is regular and $\mu$ is arbitrary, define say $\lambda \blacktriangleleft \mu$ to mean that for any $\nu\lt\mu$ there exists a $\kappa$ with $\nu\lt\kappa\le\mu$ and $\lambda\lhd\kappa$. Then we can prove that if $\lambda\blacktriangleleft\mu$, then any $\mu$-small object in a locally $\lambda$-presentable category is a $\mu$-small $\lambda$-filtered colimit of $\lambda$-presentable objects, and thus for any accessible functor $F$, there is a regular cardinal $\lambda$ such that $F$ preserves $\mu$-small objects whenever $\lambda\blacktriangleleft\mu$. Moreover, for any regular $\lambda$, the class of cardinals $\mu$ such that $\lambda\blacktriangleleft\mu$ is a club, being closed essentially by definition.

A related relation is the one used by Lurie mentioned above: $\lambda\ll\mu$ (which LRV pronounce as “$\mu$ is $\lambda$-closed” — I don’t know whether this has a wider usage in set theory) if $\nu\lt\mu$ implies $\nu^{\lt\lambda}\lt\mu$. When $\lambda$ and $\mu$ are regular, then $\lambda\ll\mu$ implies $\lambda\lhd\mu$, and the converse holds (still when both are regular) under GCH. But the relation $\ll$ is also sensible for singular cardinals, and the class of cardinals $\mu$ such that $\lambda\ll\mu$ is a club. Moreover, at least for $\lambda$ regular we can show that $\lambda\ll\mu$ implies $\lambda\blacktriangleleft\mu$, and hence $\ll$ also has the above properties regarding filtered colimits and accessible functors (I learned this from LRV in an email conversation). Note that if $\lambda\lt\mu$ and $\mu$ is a strong limit, then $\lambda\ll\mu$ and hence $\lambda\blacktriangleleft\mu$, so there is a good supply of these things.

However, what would be really nice would be a relation generalizing $\lhd$ to singular cardinals that has an if-and-only-if characterization in terms of accessible categories. Of course, we could define, say, $\lambda\lessdot\mu$ to mean that any $\mu$-small object in a $\lambda$-accessible category is a $\mu$-small $\lambda$-filtered colimit of $\lambda$-presentable objects. But does this relation have a more explicit set-theoretic characterization? What properties does it have?

(This isn’t just a lark, by the way; I have a reason to be interested in this stuff, although it’s maybe a bit tenuous. In modeling type theory we usually choose inaccessible cardinals to bound the cardinality of objects in the universes, because that makes the universes closed under both $\Sigma$- and $\Pi$-types. The construction of universes, however, works just fine for a sharply large class of regular cardinals, and a universe bounded in size by such a regular cardinal is closed under $\Sigma$-types — but not $\Pi$-types. I think this asymmetry is unpleasing: it would be nice to also construct universes bounded in size by some class of strong limit cardinals, which would hopefully then be closed under $\Pi$-types but not $\Sigma$-types; but this would require some kind of extension of $\lhd$ to singular cardinals. I speculated on the HoTT mailing list about what non-inaccessible universes would look like inside type theory, but didn’t get any bites.)

Posted by: Mike Shulman on April 3, 2019 11:52 AM | Permalink | Reply to this

Re: The Myths of Presentability and the Sharply Large Filter

Phew! I think I might still take a look through some old things I’ve written to check for these kinds of mistakes! A few things:

• It seems to me that if $C$ is locally $\lambda$-presentable, then for $\mu \geq \lambda$ regular, the notion of $\mu$-smallness always coincides with $\mu$-presentability. You’ve already observed that this is true when $\mu$ is a successor. Suppose now that $\mu$ is a regular limit cardinal (i.e. weakly inaccessible). If $X$ is $\mu$-presentable, then $X$ is a $\mu$-small (not necessarily filtered) colimit of $\lambda$-presentable objects, so $X$ is a $\nu$-sized colimit of $\lambda$-presentable objects for some $\lambda \leq \nu \lt \mu$, so $X$ is $\nu^+$-presentable, so $X$ is $\mu$-small. Conversely, if $X$ is $\mu$-small, then $X$ is $\nu^+$-presentable for some $\nu \lt \mu$, so $X$ is $\mu$-presentable.

I’m not sure if this is still true for general $\lambda$-accessible categories. In this case I don’t think we can characterize the $\mu$-presentable objects as $\mu$-small colimits of $\lambda$-presentable objects.

• My wild guess would be that the poset definition of $\lambda \triangleleft \mu$ is still going to correspond to “the correct” notion when $\mu$ is singular. Maybe I’ll give some thought to this.
Posted by: Tim Campion on April 3, 2019 8:39 PM | Permalink | Reply to this

Re: The Myths of Presentability and the Sharply Large Filter

Ah, nice. Yes, in a general accessible category I believe the most we can say is fact 3 from the post, since the proof that a $\mu$-presentable object is a $\mu$-small colimit of $\lambda$-presentable objects depends on the category having all $\mu$-small colimits. If we adapt your argument to use fact 3 instead, we get essentially the result of LRV, Corollary 3.11 that requires certain set-theoretic hypotheses to ensure $\lambda\lhd\mu$ for the weak inaccessible $\mu$.

Posted by: Mike Shulman on April 3, 2019 8:54 PM | Permalink | Reply to this

Re: The Myths of Presentability and the Sharply Large Filter

A couple other thoughts:

• It still may be the case the class of $\mu$ with $\lambda \triangleleft \mu$ is stationary (i.e. intersects every club nontrivially).

• By the way, is there a good analog of the notion of “club” in constructive mathematics / type theory?

• I wonder if there are good reasons that Lurie uses $\lambda \lt \lt \mu$ rather than $\lambda \triangleleft \mu$? Do there exist $\lambda$-accessible $\infty$-categories which are not $\mu$-accessible for some $\lambda \triangleleft \mu$?

Posted by: Tim Campion on April 3, 2019 8:56 PM | Permalink | Reply to this

Re: The Myths of Presentability and the Sharply Large Filter

I don’t believe the class of all regular cardinals is provably stationary either. For instance, the class of all limit cardinals is a club, but does not intersect the class of regular cardinals unless there exists a weak inaccessible, which is unprovable in ZFC.

I don’t recall ever reading about a constructive notion of club. Often in constructive mathematics one wants to talk about families of sets rather than cardinalities, so one could define a class $C$ of sets to be a club if it is closed under arbitrary (set-indexed) unions and is “unbounded” in some sense. I’m not sure what the right notion of unbounded would be; maybe that any set injects into some member of $C$? It might depend on the application.

I have also wondered whether there are good reasons Lurie uses $\ll$ (\ll) instead of $\lhd$. I asked him, and he said he didn’t remember for sure, but probably it would all work for $\lhd$ instead. But I also think I remember there being places in HTT where he uses the actual definition of $\ll$ in ways other than the fact that it implies $\lhd$, so it’s unclear to me.

Posted by: Mike Shulman on April 3, 2019 9:15 PM | Permalink | Reply to this

Re: The Myths of Presentability and the Sharply Large Filter

Btw I tried using the correct latex for $\lt \lt$ and I kept getting an error when I tried to post, so I gave up on it. Probably it was something context-dependent from the first time I tried, though, so I should keep trying in the future.

Posted by: Tim Campion on April 3, 2019 9:40 PM | Permalink | Reply to this

Re: The Myths of Presentability and the Sharply Large Filter

I just took a look at the characterization of $\lambda \triangleleft \mu$ given in AR Thm 2.11, and I think it goes through for $\infty$-categories. That is, I believe the following are equivalent:

1. $\lambda \triangleleft \mu$

2. Every $\lambda$-acessible $\infty$-category is $\mu$-accessible.

To prove this, one just needs to adapt the implication $iv \Rightarrow i$ in AR Thm 2.11, and it looks like the only fact needed for this is that any $\lambda$-filtered $\infty$-category admits a cofinal functor from a $\lambda$-directed poset. This is HTT 5.3.1.16.

So I’d be prepared to believe that $\lambda \triangleleft \mu$ is still the correct notion to work with in $\infty$-categories.

It’s interesting – in my (limited) set-theoretical experience, I have seen the notion $\lambda \lt \lt \mu$ come up. For instance, it’s the usual hypothesis used in the Delta system lemma. That is, if $\lambda \lt \lt \mu$, then any $\mu$-sized set of $\lambda$-small sets contains a $\mu$-sized subset whose pairwise intersections are constant. I wonder if the Delta-system lemma continues to hold for $\lambda \triangleleft \mu$

Posted by: Tim Campion on April 3, 2019 9:37 PM | Permalink | Reply to this

Re: The Myths of Presentability and the Sharply Large Filter

Yes, I agree with that. In fact that’s already how Lurie proves Proposition 5.4.2.9 (in the current (2017) version; it’s 5.4.2.11 in the published one), using Lemma 5.4.2.8 for which $\lhd$ clearly suffices. But there are other uses of $\ll$ in HTT for which I find it’s not so clear; for instance, the proof of Lemma 5.4.4.2 (again, in the current version) seems to be actually using the definition of $\kappa\ll\tau$ rather than just by way of $\kappa\lhd\tau$ or a consequence of it such as 5.4.2.8. I wouldn’t be surprised if there were a way of using $\lhd$ there as well (I haven’t even tried to think about it), but it seems the proof would have to be modified at least a bit.

Posted by: Mike Shulman on April 4, 2019 12:20 AM | Permalink | Reply to this

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