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In large category, a large category is defined as a category that is not small. However, in CWM (second half of p. 23), a large category is defined as what is referred in $n$Lab as a $U$-moderate category for the fixed universe $U$ assumed throughout CWM.
Shouldn’t there be a comment in large category that warns that the same name “large category” may have several meanings in different references?
I didn’t do the change myself since I have a very limited knowledge about this subject, so perhaps I’m missing something.
… and by the way, a similar comment applies to metacategory, especially to the sentence “…and calls sets and categories in U small and categories not in U large.” – I think this is incorrect (that is, this is not what is said in CWM).
I think MacLane’s approach to foundations is similar, but not exactly, what is generally used in practice (and certainly different, as you say, to the nLab conventions). But you are right, as there are several conventions, this should be noted.
Thanks for the answer, David. I added a comment in large category.
Thanks for pointing this out, Yaron. I expanded your remark further into a discussion of the definitions of “large category” in several different foundational systems.
Wait, is this really true that "large" always means the same thing as "moderate"!? Why did anybody every introduce the term "moderate" then? (See moderate category.)
I know that some people use "large" in this way, using "extra-large" etc for something bigger. But my understanding has been that the naive meaning of "large" is just "not small", and one introduces finer distinctions only when one really needs them. Is there an accepted term for "not small"?
Edit: Actually, I guess that the claim being made on large category is that "large" means "moderate but not small" (oxymoronic as that may seem). That doesn't seem like a very useful term to me, but at least it's not the same as "moderate". (Then "moderate" is introduced to mean "small or large".)
What? What I tried to write is that in the context of one Grothendieck universe U, “large” = not necessarily an element of U and “moderate” = a subset of U (with “small” = an element of U). The alternate CWM usage takes “large” = a subset of U and doesn’t use “moderate.” The question of whether small categories are counted as large is completely separate.
The main definition is
A large category is a category which belongs to the “next largest” size category than a small category does.
which I read as meaning that “large” is only one level higher than “small”; two levels higher is no longer “large” (and neither would zero levels higher be).
Then you list variations. I agree, the version with one universe is an exception; then things are as I would use the terms, and as you quote above. But all other versions are as in the main definition (except perhaps the first version, which has only two levels, so one can’t tell).
I submit that while some people have used terms differently, the natural interpretation of “large” requires that if $C$ is large and $C \subseteq D$, then $D$ must also be large.
Well, I guess “next largest size category” is definitely not quite right in view of the usage of small ⊊ moderate ⊊ large. I can’t think of any way to phrase the meaning informally which includes all the different usages, though. I guess that’s not surprisinge, though, since the “small ⊊ moderate ⊊ large” usage with one universe, and the “small ⊊ large ⊊ not even large” usage of CWM (or even the most traditional ZFC usage of “small ⊊ large ⊊ doesn’t even exist”), are mutually contradictory. How would you explain the notion of “large category” in a usage-egalitarian way?
I submit that while some people have used terms differently, the natural interpretation of “large” requires that if C is large and C⊆D, then D must also be large.
I don’t really have a problem with some categories being “too big to be large,” but I guess I can see that it might be a little confusing.
How would you prefer to use words in the context of two universes U∈V, then? I guess you would want an element of U to be “small” and a set not (or not-necessarily) in U to be “large”? Then what do you call an element of V and a set not in V? Or would you want to use “large” for sets not in V, and something like “medium-sized” for sets in V but not (necessarily) in U?
In the context of two completely arbitrary universes $U \in V$, I’d call an element of $U$ “$U$-small” and a non-element of $U$ “$U$-large”. Similarly for $V$.
In the context of two universes $U \in V$, where $V$ is chosen to be closely related to $U$ in a sensible way, then I’d call an element of $U$ “small”, a non-element of $U$ “large”, an element of $V$ “moderate”, and a non-element of $V$ “very large”. (This combines two different systems, however, so maybe it would be confusing.) For example, if $V$ consists of things which are “definable over $U$” in a reasonable way (the most naive being where $V$ is the power set of $U$, although then $V$ is not a universe in the Grothendieck sense), then this terminology would come naturally to me.
In the context of multiple universes where we also have a notion of when something is definable over (but not necessarily in) a universe, then we get terminology like “$U$-moderate” (which shows up at moderate category). However, if we try to iterate “definable over”, then we run out of words, so I guess that we would need terminology with a natural number parameter.
Okay, I did a bit of rewriting; what do you think?
I like it now; I made a few edits.
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