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I gave regular cardinal its own page.
Because I am envisioning readers who know the basic concept of a cardinal, but might forget what “regular” means when they learn, say, about locally representable category. Formerly the Lab would just have pointed them to a long entry cardinal on cardinals in general, where the one-line definition they would be looking for was hidden somewhere. Now instead the link goes to a page where the definition is the first sentence.
Looks better to me, but let me know what you think.
I agree. These are the kinds of little changes that greatly improve readability! =)
I agree. These are the kinds of little changes that greatly improve readability! =)
Okay, good. I have kept the stuff on regular cardinals at cardinal by the way. Because we also had the complaint that too many entries are not self-contained enough. It can be a difficult design decision.
Redundancy is not a priori a bad thing. What would be bad is if the entries contradicted each other. “Recalling” the definition is often helpful, especially for a work like the nLab.
Redundancy is not a priori a bad thing. […] “Recalling” the definition is often helpful, especially for a work like the nLab.
What troubles me is that with whole chunks of material duplicated, all further changes/imrpovements/additions/corrections would need to by done twice and harmonized. That goes against the whole wiki-spirit.
I suppose there should be automated solutions, where one uses an include command to include the content of one entry into another. But then the indluded content needs to b prepared, suitably, for instance in not having its own TOC and things like that. Can get messy.
From regular cardinal:
because a set with cardinality less than aleph-1 is a finite set
which is clearly false :) Friendly lab elf, please bypass the spam filter on my behalf and fix this by replacing this with aleph-0 :)
The way to address the problem that Urs and Harry were discussing is with use of the !include
directive. If the "regular cardinal" section of "cardinal" is truly to be a copy of the content at "regular cardinal", then make it so by putting [[!include regular cardinal]] at the correct point.
(Yay! I managed to type that without using any funny entities for the square brackets.)
I don’t think the text about regular cardinals at cardinal should be the same as (e.g. !included from) regular cardinal. The latter can be arbitrarily detailed, whereas at the former we are mentioning it only as one of many properties of cardinals and thus should not spend a huge amount of time on it. I’m not extremely bothered about the “duplication” here, because it doesn’t seem to me that there should ever be very much content at cardinal to duplicate; most new content should be added to regular cardinal. If the definition of regular cardinal ever changed, then we’d have to change it in both places, but that seems unlikely.
Is there any reason you chose $\pi$ to denote the regular cardinal at regular cardinal? Usually cardinals are denoted by $\kappa$…
Yes, pi is definitely an overloaded letter.
Is there any reason you chose $\pi$ to denote the regular cardinal at regular cardinal?
This was copy-and-pasted from cardinal. I don’t recall who made that choice there, and for what reason.
Added:
Regular cardinals $\lambda$ are used in the definitions of $\lambda$-filtered colimits, $\lambda$-presentable objects, $\lambda$-accessible categories, locally $\lambda$-presentable categories, $\lambda$-ind-completion, and many notions derived from these, e.g., $\lambda$-combinatorial model categories.
Then notions make sense for all cardinals, not necessarily regular. However, the relevant concepts reduce to those for regular cardinals.
Recall that the cofinality $cof(\lambda)$ of a cardinal $\lambda$ is the smallest cardinal $\mu$ such that $\lambda$ is a sum of $\mu$ cardinals smaller than $\lambda$.
A cardinal $\lambda$ is regular if and only if $\lambda=cof(\lambda)$.
A category has $\lambda$-filtered colimits if and only if it has $cof(\lambda)$-filtered colimits.
A category is locally $\lambda$-presentable if and only if it is locally $cof(\lambda)$-presentable.
A category has $\lambda$-filtered colimits if and only if it has $cof(\lambda)$-filtered colimits.
A category is locally $\lambda$-presentable if and only if it is locally $cof(\lambda)$-presentable.
There are arbitrarily large cardinals of cofinality $\omega$, so if this were true then any category with $\lambda$-filtered colimits would have $\omega$-filtered colimits. I think one should replace $cof(\lambda)$ by $\lambda^+$ in these statements (the successor cardinal of $\lambda$, which is always regular). Indeed, if $cof(\lambda)\lt\lambda$, then every set of cardinality $\lt\lambda^+$ is a union of $\lt\lambda$ subsets of cardinality $\lt\lambda$, so every $\lambda$-filtered poset is automatically $\lambda^+$-filtered.
Correction:
If $\lambda$ is not a regular cardinal, then a category has $\lambda$-filtered colimits if and only if it has $\lambda^+$-filtered colimits, and $\lambda^+$ is always a regular cardinal (assuming the axiom of choice).
Re #14: The first part was indeed nonsense.
What is a counterexample to the second part? (A category is locally λ-presentable if and only if it is locally cof(λ)-presentable.)
Note that this second part is the statement of Exercise 1.b(3) in Locally presentable and accessible categories (page 59). But that doesn’t necessarily mean it’s correct, because the immediately preceding Exercise 1.b(2) is definitely wrong (I have verified this with Jiri Rosicky). There is a more extensive and corrected treatment of presentability for non-regular cardinals in section 3 of Internal sizes in μ-abstract elementary classes; perhaps it contains the answer to this question.
Re #17: I must say that the definition of a locally λ-presentable category always confused me by its unmotivated introduction of regular cardinals.
According to #14, if a locally λ-presentable category is locally λ^+-presentable for any nonregular cardinal λ, then it would seem that Exercise 1.b(2) must be false: if μ<ν are regular cardinals then locally μ-presentable categories are always locally ν-presentable, and there are locally ν-presentable categories that are not μ-presentable.
Substituting μ=cf(λ) and ν=λ^+ produces a counterexample.
I agree this is a counterexample.
I was also confused by the restriction to regular cardinals in these definitions. It is explained in the book by Gabriel and Ulmer where presentability is first introduced, but it seems later references never comment on this. On page 2 in the introduction they write (translated from German):
Let $\alpha$ be a cardinal, with $3\leq \alpha\lt\infty$ [sic]. A poset $(N,\leq)$ is called $\alpha$-filtered if for every family $(\nu_i)_{i\in I}$ in $N$ with $card(I)\lt\alpha$ there exists $\mu\in N$ such that $\nu_i\leq \mu$. Let $\beta$ be the smallest regular cardinal $\geq \alpha$. Then every $\alpha$-filtered poset is also $\beta$-filtered. We therefore assume in the sequel that $\alpha$ is regular.
Corrected version:
Regular cardinals $\lambda$ are used in the definitions of $\lambda$-filtered colimits, $\lambda$-presentable objects, $\lambda$-accessible categories, locally $\lambda$-presentable categories, $\lambda$-ind-completion, and many notions derived from these, e.g., $\lambda$-combinatorial model categories.
Then notions make sense for all cardinals, not necessarily regular. However, the relevant concepts reduce to those for regular cardinals.
The relevance of regular cardinals for these concepts was already pointed out by Gabriel and Ulmer in their original treatise on locally presentable categories, where on page 2 we read:
Sei $\alpha$ eine Kardinalzahl, wobei $3 \le \alpha \lt \infty$. Eine geordnete Menge $(N,\le)$ heisst $\alpha$-gerichtet, wenn es für jede Familie $(\nu_i)_{i\in I}$ in $N$ mit $Kard(I) \lt \alpha$ ein $\mu$ gibt derart, dass $\nu_i\le\mu$. Sei $\beta$ die kleinste reguläre Kardinalzahl $\ge \alpha$. Dann ist jede $\alpha$-gerichtete Menge auch $\beta$-gerichtet. Wir setzen deshalb im folgenden zusatzlich voraus, dass $\alpha$ regulär ist (vgl. §0).
If $\lambda$ is not a regular cardinal, then a category has $\lambda$-filtered colimits if and only if it has $\lambda^+$-filtered colimits, and $\lambda^+$ is always a regular cardinal (assuming the axiom of choice). In this case, a category is locally $\lambda$-presentable if and only if it is locally $\lambda^+$-presentable.
The assumption $\alpha\lt\infty$ in the quote looks a bit confusing to me. If $\infty$ means $\aleph_0$ then the smallest cardinal $\geq\alpha$ is necessarily $\infty$. So I assume $\infty$ means “the size of the universe”?
That would actually be consistent with the use of “$\infty$-filtered” in Adamek/Lawvere/Rosicky’s “Continuous categories revisited”.
Added further clarifications:
The relevance of regular cardinals for these concepts was already pointed out by Gabriel and Ulmer in their original treatise on locally presentable categories, where on page 2 we read:
Sei $\alpha$ eine Kardinalzahl, wobei $3 \le \alpha \lt \infty$. Eine geordnete Menge $(N,\le)$ heisst $\alpha$-gerichtet, wenn es für jede Familie $(\nu_i)_{i\in I}$ in $N$ mit $Kard(I) \lt \alpha$ ein $\mu$ gibt derart, dass $\nu_i\le\mu$. Sei $\beta$ die kleinste reguläre Kardinalzahl $\ge \alpha$. Dann ist jede $\alpha$-gerichtete Menge auch $\beta$-gerichtet. Wir setzen deshalb im folgenden zusatzlich voraus, dass $\alpha$ regulär ist (vgl. §0).
where the meaning of $\alpha\lt\infty$ is explained on page 13:
Ausserdem bezeichnen wir mit $\infty$ die kleinste Kardinalzahl, die nicht mehr zu $U$ gehört (die also in unserer Sprache keine Menge ist).
Dieser Arbeit liegt die Mengenlehre von Zermelo-Fraenkel und ein fest gewähltes
Universum $U$ zugrunde. Wir setzen dabei voraus, dass $U$ die Menge $\mathbf{N}$ der natürlichen Zahlen enthält.
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