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Stream: learning: questions

Topic: completion, free completion, free, -ization

Leopold Schlicht (May 30 2022 at 17:07):

There's a bunch of names one can give to a left adjoint: completion, free completion, free ..., -ization. This nLab page explains when to use which. To summarize (as far as I understand what the page is trying to say), we call a left adjoint

• completion if it is faithful and its right adjoint is fully faithful;
• -ization or -ification if (it is not faithful and) its right adjoint is fully faithful (i.e., forgets property);
• free completion if its right adjoint is faithful and forgets property-like structure;
• free if its right adjoint is faithful (i.e., forgets structure).

Is that correct? While I read the article rather carefully and most of the examples seem to support these definitions, I am confused about these counterexamples:

1. On the page itself it says:

By contrast, the forgetful functor from the 2-category of monoidal categories to Cat forgets structure, rather than properties, so its left adjoint should be called a “free” construction rather than a “completion.”

But the forgetful functor from monoidal categories to categories is not faithful, since a monoidal functor is not determined by its underlying functor of categories. So why is one allowed to call that construction "free"?

2. Why is one allowed to call the regularization of a cartesian category "-ization"?

Mike Shulman (May 30 2022 at 17:11):

The rambling reflections on that page shouldn't be taken as gospel, but as an attempt to make sense of an existing hodgepodge of terminology. So it's not surprising that it won't encompass all existing terminologies -- it's more surprising that any systematic classification would encompas so many of the existing terminologies, most of which were probably introduced in an ad hoc manner.

Mike Shulman (May 30 2022 at 17:12):

With that said...

Mike Shulman (May 30 2022 at 17:13):

Ad (1), I don't see anywhere on the page that suggests that we should only use the word "free" when the right adjoint is faithful.

Mike Shulman (May 30 2022 at 17:13):

I could imagine someone making such a suggestion, but only in the context of 1-categories. For higher categories, I think the criterion ought to go up with the dimension.

Mike Shulman (May 30 2022 at 17:14):

Ad (2), I don't think I would call it "regularization" -- I would say "free regular completion".

Mike Shulman (May 30 2022 at 17:14):

Where did you see "regularization"?

Leopold Schlicht (May 30 2022 at 17:48):

Thank you very much!

Ad (1), this quote from the nLab page suggests that we should only use the word "free" when the right adjoint is faithful:

By contrast, when we add structure to objects, i.e. we consider an adjoint to a forgetful functor which is faithful but not necessarily full, we usually do not use the word “completion” but rather the word “free”.

What do you mean by "the criterion ought to go up with the dimension", is there a reasonable definition of "faithful" for 2-functors such that the forgetful functor from monoidal categories to categories is faithful?

Ad (2), I read "regularization" at several places in the Elephant.

Mike Shulman (May 30 2022 at 17:51):

That quote says that we use the word "free" when we add structure, but not that we only use the word free when adding structure.

Mike Shulman (May 30 2022 at 17:53):

I don't mean the criterion for the word "faithful" but the one for the word "free". If we're restricting left adjoints between 1-categories to be called "free" when their right adjoint is faithful, then I would say a left adjoint between 2-categories should be allowed to be called "free" when its right adjoint is locally faithful, and so on. This "changing the goalposts" when we go up in dimension is, I think, another argument for not imposing such a requirement on the word "free", since it's better to be able to consider 1-categories as degenerate 2-categories without changing our terminology.

Mike Shulman (May 30 2022 at 17:56):

Re: the Elephant, ah, I see. Well, as I said, I wouldn't say that.

Leopold Schlicht (May 30 2022 at 18:33):

Thanks!

Leopold Schlicht (May 31 2022 at 07:40):

By the way, would you call a functor "forgetful" only in the case it is faithful?

Mike Shulman (May 31 2022 at 07:41):

No.

Mike Shulman (May 31 2022 at 07:41):

For instance, the projection $\rm Set\times Set \to Set$ clearly "forgets one of the sets", but it is not faithful.

Mike Shulman (May 31 2022 at 07:42):

According to the dogma of [[stuff, structure, property]] we would say it "forgets stuff" rather than structure or properties.

Mike Shulman (May 31 2022 at 07:43):

(Huh, does the auto nLab link not work for page names containing commas or something?)

Leopold Schlicht (May 31 2022 at 07:52):

Alright, and what about "underlying", does that require faithfulness? :P
I mentioned that issue also here.

Joe Moeller (May 31 2022 at 14:12):

I don't know Mike's thoughts, but off the top of my head, I'd expect "underlying" to forget at most structure. The pair of sets one is a good example. If you have $(X, Y) \mapsto X$, then it's not really reasonable to say that X is the underlying set of (X,Y), right? Maybe I'd also probably want it to forget at least structure. Forgetting from Ab to Grp, it seems weird to say you're talking about the underlying group of an abelian group.

Joe Moeller (May 31 2022 at 14:16):

In my experience, when you want to talk about the underlying thingy, it's because you want to access more general maps on the same gizmo.

Leopold Schlicht (May 31 2022 at 14:34):

What do you mean by "forget at most structure" and "forget at least structure" precisely?

Mike Shulman (May 31 2022 at 14:38):

I think the reason it sounds weird to say that $X$ is "the underlying set" of $(X,Y)$ is that there are two sets, so it's the definite article that's inappropriate, not the word "underlying". Consider for instance the category whose objects are a pair $(R,M)$ of a ring $R$ and an $R$-module $M$. Wouldn't you say that $R$ is the underlying ring of a pair $(R,M)$?

Mike Shulman (May 31 2022 at 14:38):

And certainly a monoidal category has an underlying category...

Joe Moeller (May 31 2022 at 14:42):

Good points. Good thing I said "off the top of my head". :upside_down:

Leopold Schlicht (May 31 2022 at 15:19):

@Mike Shulman Your example suggests that an "underlying * functor" doesn't have to be faithful. So when is one allowed to call a functor "underlying" or "forgetful" -- is there any requirement that is usually satisfied? Also, are "underlying" and "forgetful" synonymous?

Would you say "underlying preorder of a category"?

@Joe Moeller Just a reminder that I asked a question you ignored. :P

Joe Moeller (May 31 2022 at 15:29):

Sorry, I mean what is said on the nLab page for stuff structure properties that Mike pointed to.

Mike Shulman (May 31 2022 at 15:35):

I don't think it's possible to give a formal and equivalence-invariant definition of when one uses the words "underlying" and "forgetful". Normally one uses it for a functor $U:C\to D$ when the objects of $C$ are defined to be objects of $D$ with some extra stuff, but this notion of "defined to be" is not equivalence-invariant. For instance, we wouldn't normally say that the free group functor $\rm Set\to Grp$ is "forgetful"; but if we replace the category $\rm Set$ with the equivalent category whose objects are groups equipped with a subset that generates them freely and whose morphisms are group homomorphisms that preserve the generating sets, then the corresponding functor does look "forgetful" since it just forgets the generators.

Mike Shulman (May 31 2022 at 15:38):

Similarly, I wouldn't normally say "underlying preorder of a category", but if $\rm Cat$ were replaced by the equivalent category whose objects are preorders together with a category whose preorder reflection is the given one, then it would make sense to talk of the underlying preorder of such a gadget. (This is not as weird as it may appear, actually -- direct categories often are defined to come with a well-founded ordering on their objects, which could be called their "underlying" one, even though it's usually also their preorder reflection.)

Mike Shulman (May 31 2022 at 15:39):

The most I think could be said is if two categories come with a presentation as models of some formal theory, and the one theory is a syntactic extension of the other, then the induced functor is called "underlying" or "forgetful".

Mike Shulman (May 31 2022 at 15:39):

I don't know if there are situations where I'd use one of "underlying" and "forgetful" but not the other.

Leopold Schlicht (Jun 01 2022 at 12:04):

Thanks, that's enlightening!

Leopold Schlicht (Jun 01 2022 at 12:19):

I think there's a (last?) question remaining: when to use "free"? We already agreed that the right adjoint of a free functor doesn't have to be faithful. So can we use "free" for each left adjoint? Or is "free", like the notions "underlying" and "forgetful", a presentation-dependent notion?

Mike Shulman (Jun 01 2022 at 14:56):

Well, according to the nLab [[free object]], it can be used for any left adjoint. As you said, it would be a little odd to call the abelianization of a group the "free abelian group" on it. More commonly I think it's used for left adjoints of "forgetful" functors, whatever that means.

Leopold Schlicht (Jun 01 2022 at 15:54):

Thanks!

John Baez (Jun 02 2022 at 16:19):

Leopold Schlicht said:

What do you mean by "forget at most structure" and "forget at least structure" precisely?

A functor forgets at most structure iff it's faithful.

John Baez (Jun 02 2022 at 16:21):

I never say "forgets at least structure" but I guess this means not(forgets at most properties). If so, I guess it means "not full and faithful".

John Baez (Jun 02 2022 at 16:24):

The nLab article on "properties, structure and stuff" explains this stuff, as does my paper with @Mike Shulman called "Lectures on n-categories and cohomology".

Leopold Schlicht (Jun 02 2022 at 16:46):

Thanks!

Leopold Schlicht (Jun 05 2022 at 15:55):

Mike Shulman said:

Ad (2), I don't think I would call it "regularization" -- I would say "free regular completion".

FWIW, I also like "free regularization".