@psie Well, that would mean that you need to show that IF $\lim_{x\to a} f(x) = f(x)$, THEN you can always find an appropriate $V$ for any $U$, and vice versa.

Your argument doesn't seem to do that...

Meeh...all the work I put into this :(

have to redo it I guess

heyho. any ideas how to call a monoid whose underlying "set" is a class or collection? I know that "large monoid" is used, but I want something that includes the type of size. so something like "monoid class" or "monoid collection", but that doesn't sound right to me.

for partial orders there is a very good choice: "partially ordered collection"

(a typical example of a large monoid is the collection of all sets with * = union of sets, 1 = empty set).

collection monoid? also a bit weird.

small monoid, medium monoid, large monoid

, monstrous monoid

:D

enormous monoid :)

but seriously, I believe people sometimes use small/medium/large when they can do with 3 universes, which is usually the case

not specifically for monoids but

yes but as I said I need something different :/

(there are reasons too long to be explained here)

monoidal collection?

"collection with monoid structure" is a bit long

@MartinBrandenburg really not a fan

monoidollection

I am out of ideas :D

still I fail to see why small/medium/large doesn't serve you here: it does denote the type of size, under the usual convention that small sets are sets that live in whatever base Grothendieck universe you fix and every escalation in adjective moves you up one universe. If you want something more technical you could talk about $\mathfrak{U}_n$-monoids, I guess?

which is also not great but

well it is a bit too long to explain

that's okay, but it makes the assignment a bit unclear :)

I see

monction and moonoid don't really hit the spot I guess

well I'm out of ideas, in any case

you will end up speaking about sets and large sets, but actually, every set will be large <-- -confusing af

I have removed this language from what I am currently writing

grothendieck universes are only an implementation detail, really

ad "small set," yes. ad "large set," I guess it depends on the audience. If I heard the words "large set," I would think you mean a set that lives at least one Grothendieck universe higher

probably doesn't make sense to you, but yeah, this is basically why I want to speak about collections (or classes, whatever)

U^+

yes collection := set in U^

the only thing I know is that I'd much prefer if people (in my field) universally adopted Grothendieck universes and tried to awkwardly dance around size issues like they still do so often

"Monoidic collection" – A more playful, hybrid term that combines "monoid" with a slightly altered form of "collection."

a friend of mine suggested to write everything capitalized

the field of real numbers, the FIELD of surreal numbers :D

but a wide hat over it like some people do

$\widehat{\text{monoid}}$

@Ben: absolutely. I also use Grothendieck universes now for everything (but only as the "implementation")

imagine having three universes

and after uppercase comes ... bold? underlined? :D

ah, overlined

just more hats: $\widehat{\widehat{\text{monoid}}}$ :P

monoid^^^ (go back three commits from monoid)

@BenSteffan double Fourier Transform of $\text{monoid}$? :D

hmm, why not just a single hat in the end .. that would be less annoying to read

I guess I mean a "check"

monoid$^+$ ?

do it like Lurie and always work relative to a fixed regular cardinal $\kappa$, which may be assumed inaccessible if necessary

@mo-_- sorry, I went away, carried away on the wings of the so called Gustin's sequence space

I generally like the capitalization convention for categories, but it's too awkward for objects

erm, actually I have two inaccessibles ( ~universes)

hence my question

@Thorgott do it like Lurie and explain 3 different ways to handle size issues, one of which is "just ignore them," and then don't make precise which one you're going to use :D

loool

well it's not quite true, he does decide for Grothendieck universes in the end

once category theory is setup correctly with respect to the underlying set theory, lots of things becomes less mysterious and absolutely clear

no issue at all with general functor categories, for example

and I changed my mind about locally small. that's actually a bad general assumption ...

hm

it's almost like defining a top. space to be compact and then wondering why so many constructions don't always work :D

who needs adjoint theorems anyways :^)

usually, I'm also a small, large, very large guy, not coming up with more alternatives :/

well it's a theorem about locally small categories then, which is OK

like there are theorems about compact spaces - you should just not name them spaces.

yeah, I agree

I did not really appreciate these subtleties till I bothered reading proofs of adjoint functor theorems

yes, there you take the equalizer of all endomorphisms (of a weakly initial object)

so you need that there is only a set of these

OR you demand that your category has large equalizers

which is .... rare

(in this sense this theorem is not really a theorem about locally small categories btw)

ok but anyway, no winner for the "monollection" question? :D

actually, how about $\mathcal{U}$-monoid, where $\mathcal{U}$ is the relevant universe

beat you to that :)

oh, sorry, must've missed it