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I think we should have a page generally about “change of universe” for categories, i.e. starting from a(n n-)category that is large (and often bicomplete, monoidal, etc.) w.r.t. one universe and producing a corresponding one that is large (bicomplete, monoidal) w.r.t. a bigger universe. What we currently call the huge (infinity,1)-sheaf (infinity,1)-topos is a special case of this. I’m not sure what to call such a page, though — change of universe? universe enlargement?
(As a side note, I personally prefer “very large” over “huge,” since “huge” is an adjective that has a separate meaning when applied to cardinal numbers, but that’s not a big deal. Perhaps it would be more future-compatible to say “2-large” with reference to the second universe, anticipating a potential need for 3-large etc. categories, but that might have the undesired implication of some relationship with 2- and 3-categories, so maybe it’s a bad idea.)
It would also be nice to have a standard notation for this operation. I (and many other category theorists, I believe) tend to use levels of capitalization of category names, e.g. Set is the category of small sets and SET the category of large sets. Sometimes people write “set” for the category of small sets, “Set” for the category of large ones, and then maybe SET can be used for the category of very large ones. But obviously this doesn’t work well for a generic category name like $\mathcal{C}$. In Kelly’s “basic concepts” he writes $V'$ for the universe-enlargement of the category $V$, while Lurie seems to put hats on things, but both of those notation have lots of other overloaded meanings. One possible notation that just occurred to me is $C^\uparrow$, which looks a bit like a hat and conveys the sense of “raising up” or “making bigger” while not AFAIK clashing with anything else. Thoughts?
(As a side note, I personally prefer “very large” over “huge,” since “huge” is an adjective that has a separate meaning when applied to cardinal numbers, but that’s not a big deal. Perhaps it would be more future-compatible to say “2-large” with reference to the second universe, anticipating a potential need for 3-large etc. categories, but that might have the undesired implication of some relationship with 2- and 3-categories, so maybe it’s a bad idea.)
Yes, i was wondering about this myself when I named the entry. First I wanted to say “very large” but then somehow I decided against it. I am absolutely not dogmatic about it, as I am not a big fan of either choice.
Capitalization is probaly a good choice of notation for most purposes.
we could use LaTeX syntax for headline sizes
\small Set
\large Set
\Large Set
\huge Set
\Huge Set
(that’s a joke)
change of universe? universe enlargement?
I like the latter. The former sounds too symmetric.
I personally prefer “very large” over “huge”
One possibility is ‘non-moderate’, since ‘moderate’ is already used, although it’s not very nice.
$C^\uparrow$
I’ll be surprised if that’s never been used (for $C$ a category), but if it’s not common, then let’s take it!
$C^\downarrow$ is sometimes used for the arrow category of $C$, along with $C^\to$. I’ve also seen $C^{\downarrow \downarrow}$ for the category of parallel pairs of arrows of $C$. Just to let you know that there is similar completely different notation out there. In a similar vein, why not $\uparrow C$? It’s something like a ’bigger $C$’, which follows English and German adjective-noun order.
$\uparrow C$ looks to me like the principal filter generated by $C$. I’ve seen $C^\to$ for the arrow category, but never $C^\downarrow$; how common do you think it is, and how bad would it be to use $C^\uparrow$ for something completely different?
Hmm, ok. I’m not opposed to $C^\uparrow$, it’s just not ’clicking’ for me. But let me try it for some specific categories: $Top^\uparrow$, $Grp^\uparrow$, $Set^\uparrow$…
Would $^\uparrow C$ be any better? (Hmm, it doesn’t look very good in itex because I can’t get rid of the extra space in between.)
Or we could try $C\uparrow$. And I could live with $\uparrow C$ too—confusion with a principal filter is pretty unlikely. I do see the point that $C^\uparrow$ looks like $C$ raised to the power of something. Anyone else have an opinion?
I share something of David’s reaction, and worry that I’ll misremember $C^\uparrow$ somehow unless a pointer is given in the text (which maybe will have to be done anyway?). A lot of arrows in category theory as it is.
At the moment I don’t have a better suggestion (although I’ve been thinking about it). There’s a slew of symbols out there which look pointy like an arrow (maybe a harpoon, maybe a spade, I don’t know), one of which might be good for our purposes.
Whatever notation we come up with, I wouldn’t use it anywhere without explaining it.
$\spadesuit C$? Hmm. Itex doesn’t seem to have upward-pointing harpoons.
I kind of want to suggest something like $\underline{\underline{C}}$, since double or triple underlines are sometimes used in editing markup to mean “capitalize this”, but probably that would be much too confusing.
And underlines are tricky to iterate, whereas there are ’to the n-th’ versions like $(symbol)^n C$ and $C^{(symbol)n}$.
before discussing this in full generality, could we clarify if we have lots of natural examples where more than one universe enlargement is considered that we want to write about?
For if not, I’d just say we stick to capitalization for indicating the next bigger univserse.
(Do you also always hear John Lennon when thinking about this? )
I’d just say we stick to capitalization for indicating the next bigger univserse.
My issue with capitalization is not that I want to enlarge more than once, but that it’s hard to indicate an enlargement of a category called “C” by capitalizing it. (-: Capitalization only seems to work for “particular” categories with names that are longer than one letter, rather than “arbitrary” categories which are usually denoted by only one (capital) letter.
One possibility is ‘non-moderate’, since ‘moderate’ is already used, although it’s not very nice.
I realized I never replied to this. I don’t think “non-moderate” is the same as “very large”: “non-moderate” means “not a subset of $U_1$” while “very large” means “not an element of $U_2$.” Very large implies non-moderate, but there can be large non-moderate categories that are not very large. In fact, cardinality-wise, most large categories are not moderate, although perhaps curiously, most large categories occurring in practice are moderate.
Funny, for some reason I always associated “Across the Universe” more with George Harrison, because I think of him as being the Beatle most involved with Indian culture, TM, etc. I see I was totally wrong to think that; John Lennon wrote the lyrics. (Harrison wrote “My Sweet Lord” which is maybe influencing my thinking here.)
Quite a lovely song, really.
Mike,
what would be an example of a situation where you would want to enlarge “arbitrary” categories $C$ ? I have currently a bit of trouble thinking about what that would even mean unless I know already what $C$ is like, what it’s objects are like. So it would seem to me that I can only enlarge categories whose “name” I already know.
Have a look at sections 2.6 and 3.11-3.12 of Basic concepts of enriched category theory. My impression was that this is also what the huge (infinity,1)-sheaf (infinity,1)-topos is trying to do.
Have a look at sections 2.6 and 3.11-3.12 of Basic concepts of enriched category theory.
In 2.6 Kelly explicitly says in paraphrase something like “suppose we happen to know that $\mathcal{V}$ is the category of small models of some theory, then we can pass to large models instead”.
That’s what I mean: in order to be able to say what the very large version of some category would be, we first need to assume that we know what kind of small things it is the large category of.
To say more explicitly what Kelly says somewhat implicitly: we may start with $T$ a theory, introduce the notation $\mathcal{V} := T Mod$, then pass to $T MOD$ and then introduce the new notation $\mathcal{V}' := T MOD$.
But without going via $T Mod$ there seems to be no sense in which one can deduce $\mathcal{V}'$ from an “arbitrary” $\mathcal{V}$, it seems to me.
My impression was that this is also what the huge (infinity,1)-sheaf (infinity,1)-topos is trying to do.
But there, too, I happen to know what the small things are (namely small sheaves) in advance to passing to the very large collection of their large siblings. No?
In 3.11-3.12 he describes how to do it for an arbitrary V.
And I think the definition as described at at “huge …” is more like his construction for arbitrary V, than it is like the naive “large models” construction. The naive large models construction would tell you to take large sheaves on the same (small) site, rather than large sheaves on the topos itself.
In 3.11-3.12 he describes how to do it for an arbitrary V.
Oh, i see. Thanks! I had always skipped over that discussion… ;-)
The naive large models construction would tell you to take large sheaves on the same (small) site, rather than large sheaves on the topos itself.
We – or maybe rather I – still need to come back to that discussion in some other thread – which I am almost about to forget (too many other things to do). Wasn’t there, on the contrary, the conclusion that with the definition of large sheaves on a topos, it is just large sheaves on the underlying site? (HTT, top of page 497).
Yes, indeed, so in that case the two approaches agree; I was just trying to give an example of the difference in how the definitions look.
I just wrote a Cafe post about why the two kinds of enlargement should agree for any locally presentable category.
A possibly related paper by Day:
On The Existence Of Category Bicompletions
Authors: Brian J. Day
Abstract: A completeness conjecture is advanced concerning the free small-colimit completion P(A) of a (possibly large) category A. The conjecture is based on the existence of a small generating-cogenerating set of objects in A. We sketch how the validity of the result would lead to the existence of an Isbell-Lambek bicompletion C(A) of such an A, without a “change-of-universe” procedure being necessary to describe or discuss the bicompletion.
Interesting! It is related, certainly, but it isn’t quite “universe enlargement” because he’s specifically trying to get a category of the same size rather than a larger size.
Now we have universe enlargement.
Why does the page universe write $SET$ everywhere instead of $Set$?
Also, can we reach a consensus now regarding what to do about huge (∞,1)-sheaf (∞,1)-topos? What I want to do is (1) change all the hats to capitals (e.g. $\hat{Sh}$ to $SH$), (2) change “huge” to “very large,” and (3) explain how the construction is a special case of universe enlargement (for (∞,1)-categories). But I want to hear from Urs first before I do that.
Hi Mike,
yes, let’s do that. As I tried to indicate on the Café a few days back: it would be good if somebody did that, but I don’t feel I have time for it right now. I can do it later, but if you want to go ahead, please do.
Also we still need to adjust notation at shape of an (infinity,1)-topos and coshape of an (infinity,1)-topos
Sorry, just bending over backwards to avoid stepping on anyone’s toes (how’s that for a mixed metaphor?). (-:
Why does the page universe write $SET$ everywhere instead of $Set$?
I think because $SET$ is large while $Set$ is small, and this is the topic, so we’re drawing it to their attention. I don’t think that I introduced this. However, later I didn’t follow it, since $SET_\alpha$ and $SET_\kappa$ are small, so I’ll change them.
I think because SET is large while Set is small
I don’t really follow… I think of “Set” as the large category of small sets. Are you referring to a different convention.
There is a convention by which anything large is put in all caps. But then there’s also a convention in which anything huge is put in all caps (as you know, since you used it at the top of this thread), or in which anything meta (in the sense of metacategory, where both of these conflicting conventions are used) is put in all caps. So nothing that one can rely on, but I can see the point (on that page) of putting the large universes in all caps.
Looking that the history of universe, it appears that Urs is the one who began this. So maybe he had a reason?
I have moved “huge…” to very large (∞,1)-sheaf (∞,1)-topos and added remarks about universe enlargement, and also reorganized and re-notated coshape of an (∞,1)-topos similarly.
In the process of doing the latter, I got fed up with writing “(∞,1)” in front of “Topos” all the time, so I decided to omit it throughout the page. I was tempted to do the same for “∞Gpd”, but I resisted that temptation. I’d be interested in people’s thoughts. I’ve been wishing for a while that we had some much shorter word for “∞-groupoid”—I can see why some people like “space” (one syllable even!) but I still think that’s too confusing, especially when dealing with topoi.
I have moved “huge…” to very large (∞,1)-sheaf (∞,1)-topos and added remarks about universe enlargement, and also reorganized and re-notated coshape of an (∞,1)-topos similarly.
Thanks, Mike!
In the process of doing the latter, I got fed up with writing “(∞,1)” in front of “Topos” all the time, […] I’d be interested in people’s thoughts. I’ve been wishing for a while that we had some much shorter word for “∞-groupoid”—
I have had this thought many times. Once there was a littly discussion with Harry who had suggested something similar. But I am not sure what we can do.
Of course in 50 years from now we will have dropped all the “$\infty$“s entirely. “Category” will always mean $(\infty,1)$-category, $n$-category will always mean $(\infty,n)$-category. But can we already do this?
Maybe we can. Maybe we should invent a standardized marker (maybe like the floating TOCs or the like) that one can include on a page wich says “This page uses implicit $\infty$-category theory terminology.” Equipped with a link to a page that provides more details on what this means.
I think it is wiser having infinity in the titles and setup and then do constructions without saying infinity etc. I mean that is how Lurie does. He says infinity-category but then does not say infinity functor or infinity presheaf on it but just presheaf and functor, algebraic theory etc (similarly does Joyal). If it is functor on such a thing than the infinity may be superfluous.
But of course, any idiomatics will close the community from the rest of mathematics community, the thing which we complained to old category theory schools.
I created a meta-page to link to: implicit infinity-category theory convention
I’ll try it out when I create the next entry on some $\infty$-category theory topic
I tried it out at coshape of an (∞,1)-topos.
Like for 35. Well done.
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