Like for 35. Well done.

]]>I tried it out at coshape of an (∞,1)-topos.

]]>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 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 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 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.

]]>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 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.

]]>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.

Sorry, just bending over backwards to avoid stepping on anyone’s toes (how’s that for a mixed metaphor?). (-:

]]>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

]]>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.

]]>Now we have universe enlargement.

]]>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.

]]>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.

I just wrote a Cafe post about why the two kinds of enlargement should agree for any locally presentable category.

]]>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.

]]>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).

]]>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.

]]>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?

]]>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.

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.

]]>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.

]]>