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Stream: learning: questions

Topic: Question about reflector

Paolo Perrone (Sep 09 2024 at 14:25):

Let $I:C\to D$ be a fully faithful functor. Is it true that a reflector (= left adjoint left inverse) for $I$ is the same thing as a terminal retraction (= left inverse) for $I$?
(In general, there seems to be a lot of results in the literature for when a left adjoint is a left inverse, but I can't seem to find much for when a left inverse is a left adjoint. Does anyone know where to look for results of this sort?)

Nathan Corbyn (Sep 09 2024 at 14:42):

I believe this is actually the first lemma proven in the elephant (1.1.1)

Nathan Corbyn (Sep 09 2024 at 14:54):

I just had a look and the answer is not quite—that lemma has the additional assumption that we have an adjunction already

Nathan Corbyn (Sep 09 2024 at 14:56):

However it does relax the notion of reflector to any left adjoint which is separately a left inverse—i.e., the isomorphism witnessing $I$’s left inverse isn’t required to be the counit

Nathan Corbyn (Sep 09 2024 at 15:15):

Is this a counterexample? Take $C = \mathbf{1}$ and $D = \mathbf{2}$ and $I$ the inclusion as an initial segment. This is full and faithful and has a left inverse (the unique map $\mathbf{2} \to \mathbf{1}$), but the (unique) left inverse is not a left adjoint: a unit for the adjunction would have to give a map $1 \to 0 \in \mathbf{2}$.

Paolo Perrone (Sep 09 2024 at 15:25):

Very good, that is a counterexample. Thanks.

John Baez (Sep 09 2024 at 19:41):

Paolo Perrone said:

Let $I:C\to D$ be a fully faithful functor. Is it true that a reflector (= left adjoint left inverse) for $I$ is the same thing as a terminal retraction (= left inverse) for $I$?

I am here to whine slightly about the use of 'reflector' to mean 'left adjoint left inverse'. I use 'reflector', in the sense approved by the nLab, to mean the left adjoint to the inclusion of a reflective subcategory; such a thing is a left inverse of the inclusion up to natural isomorphism. Of course it's possible you are using 'inverse' to mean 'inverse up to natural isomorphism' in the modern HoTT-approved sense, in which case your sin is forgiven. :upside_down:

John Gray used lali as short for 'left adjoint left inverse', where the left inverse was 'on the nose' rather than up to natural isomorphism, and that seems like a fine name for this more technical, evil concept.

There's also a can of worms about the precise definition of 'reflective subcategory', but see the nLab link for more about that.

Nathanael Arkor (Sep 09 2024 at 21:34):

Can we please use terminology like "strict" rather than "evil"? The latter is non-descriptive and misleading.

John Baez (Sep 09 2024 at 22:27):

I am not going to change; I use 'strict' in all sorts of technical senses (e.g 'strict monoidal category') and 'evil' in an informal way to mean 'dangerously strict'.

Rémy Tuyéras (Sep 09 2024 at 23:49):

Not sure if this is useful, but since I saw the word "terminal" in a previous post, I thought it might be helpful to mention that a reflection for a full subcategory $\iota: B \hookrightarrow A$ is the same as saying that the fibers of the domain functor $\mathsf{dom}: A \downarrow \iota \to A$ (i.e., the pullbacks along the functors $\mathbf{1} \to A$) all have an initial object.

We can interpret this as suggesting that the functor $\mathsf{dom}: A \downarrow \iota \to A$ displays a kind of weak initial surjectivity. This intuition can be extended to what might be thought of as constructing a section for $\mathsf{dom}: A \downarrow \iota \to A$ (which is the reflector). This perspective is where the notion of an "initial retraction" might come from.

Nathanael Arkor (Sep 10 2024 at 07:03):

John Baez said:

I am not going to change; I use 'strict' in all sorts of technical senses (e.g 'strict monoidal category') and 'evil' in an informal way to mean 'dangerously strict'.

Could you explain how a lali is "dangerously" strict?

John Baez (Sep 10 2024 at 14:47):

Consider the reflector for the inclusion $\mathsf{AbGp} \to \mathsf{Gp}$. Is it a left adjoint left inverse? According to the John Gray's definition, this requires that the inclusion followed by the reflector be equal to the identity functor on $\mathsf{AbGrp}$.

Let's see. Say I have an abelian group $A$, think of it as a group, and then abelianize it by forming the quotient $A/[A,A]$. Is $A/[A,A]$ equal to $A$? It's obviously isomorphic via the quotient map $A \to A/[A,A]$. But are the elements of $A/[A,A]$ exactly the same as elements of $A$? It seems they might instead be equivalence classes of elements of $A$, each class containing a single element of $A$. Then the quotient map $A \to A/[A,A]$ sends $a$ to $\{a\}$, so it's not the identity.

But of course we shouldn't really have to worry about such things: we are sinking into nitpicky details that are not mathematically interesting. A less strict definition would avoid this problem by requiring merely that the inclusion followed by the reflector be naturally isomorphic to the identity.

Nathanael Arkor (Sep 10 2024 at 15:13):

This doesn't show that lalis are dangerously strict, though. It simply shows there are examples of reflections that are not lalis.

Nathan Corbyn (Sep 10 2024 at 15:16):

I am also interested in the choice of this particular adverb

John Baez (Sep 10 2024 at 15:33):

Well, by

Nathanael Arkor said:

This doesn't show that lalis are dangerously strict, though. It simply shows there are examples of reflections that are not lalis.

The 'danger' is that when you want to apply the concept of lali to examples, you must peer very carefully at nuances of definitions. By replacing the reflector for $\mathsf{AbGp} \to \mathsf{Gp}$ with a naturally isomorphic functor, I believe I can get a lali. But the construction, or even thinking about this issue at all, is waste of time. Category theory was designed to avoid this sort of fiddling about. If we'd defined a lali to be a left adjoint functor that was left inverse up to natural isomorphism, we could avoid all this bureaucratic baloney.

John Baez (Sep 10 2024 at 15:37):

(By the way, for all I know, people now do define a lali to be a left adjoint functor that's a left inverse up to natural isomorphism. I just had the misfortune, when writing a paper about structured cospans with some wonderful coauthors, to need to cite results about lalis and ralis which we could only find in papers that defined these concepts in the original overly strict way.)

Nathanael Arkor (Sep 10 2024 at 15:39):

John Baez said:

The 'danger' is that when you want to apply the concept of lali to examples, you must peer very carefully at nuances of definitions.

But is this not the same as the situation with strict monoidal categories? Isn't this exactly the role the adjective "strict" is playing?

John Baez (Sep 10 2024 at 15:46):

If someone had stated a "strictification theorem for lalis" stating that every "weak lali" is equivalent to a "strict lali", I would tolerate them going ahead and proving lots of theorems about "strict lalis", since I could apply these to weak lalis via this strictification theorem. But instead it seems that people - the sources I was reading, anyway - proved a bunch of theorems about "strict lalis" (which they just called "lalis") without ever proving a strictification theorem (or giving a counterexample to show such a theorem was impossible). This leads to the feeling of disgust that prompts the term "evil": I am being forced to deal with a messy situation.

Rémy Tuyéras (Sep 10 2024 at 16:09):

Maybe it is easier to see @John Baez 's viewpoint if we try to quantify the amount of new information that the reflector provides to us.

For example, if we have a strict left inverse, then we gain no new information, but a left inverse holding up to isomorphism would usually give us rewritting formulas for the "points" of the objects being reflected onto themselves. This new information can be valuable for computation.

Maybe instead of "evil" one could say "worthless" and "ineffective", as in we gain nothing, and we are just stuck in a loop of non-information

Nathanael Arkor (Sep 10 2024 at 16:11):

Given an adjunction $L \dashv R \colon \mathcal D \to \mathcal C$ in which the counit $\varepsilon$ is invertible, we have a family of isomorphisms $\varepsilon_D \colon LRD \to D$ in $\mathcal D$. We may form a category $\mathcal C'$ by adjoining a new isomorphic object $D' \cong RD$ for each object $D \in \mathcal D$, and define a functor $R' \colon \mathcal D \to \mathcal C'$ by $R'D := D'$. Finally, define a functor $L' \colon \mathcal C' \to \mathcal D$ which acts as $L$ on objects of $\mathcal C$, sends each $D'$ to $D$, and sends each freely added isomorphism to the isomorphism defined by $\varepsilon_D$. Then $\mathcal C' \simeq \mathcal C$, and $L' \dashv R'$, which is equivalent as an adjunction to $L \dashv R$, but $L' \dashv R'$ is lali.

Nathanael Arkor (Sep 10 2024 at 16:13):

There's surely a more abstract way to prove reflections can be strictified by exhibiting them as the pseudoalgebras for a 2-monad, but hopefully this at least indicates that it is possible to strictify reflections.

Nathanael Arkor (Sep 10 2024 at 16:14):

@Rémy Tuyéras: exactly the same argument you give could be applied to strict monoidal categories.

Nathanael Arkor (Sep 10 2024 at 16:15):

Which is why I think the adjective "strict" suffices.

Bryce Clarke (Sep 10 2024 at 16:25):

“Strict reflections” (I usually call them split reflections) are the coalgebras for a comonad on $\mathbf{Cat^2}$. The action on objects involves taking comma categories. I suppose it would not be hard to check that this underlies a 2-comonad.

Nathan Corbyn (Sep 10 2024 at 16:46):

Rémy Tuyéras said:

Maybe instead of "evil" one could say "worthless" and "ineffective", as in we gain nothing, and we are just stuck in a loop of non-information

I’m always a little perplexed by the implied ethics of the term “evil” and I think “worthless” / “ineffective” take it a step further. Imagine pointing at someone else’s work and saying “that’s evil”—now image pointing at it and saying “that’s worthless”. I don’t think this is conducive to good science.

The point is not that it’s not important to work up to invariance under equivalence (you’ll note I interpreted left inverse as implicitly up to isomorphism). The point is that this terminology feels unneeded (we already have the word “strict”), uninformative (evil has a very particular flavour as a word which is entirely unrelated to equivalence or invariance thereunder), and unscientific (it introduces a moral dimension to mathematics which, as we know from history, has only ever served to hinder progress—think “you can’t take the root of -1, that’s evil”). Defining evil as “dangerously strict” forces one to ask: in danger of what? If the answer is someone else being unable to use the result, I agree that this is a problem, but not a danger in the moral sense.

Kevin Carlson (Sep 10 2024 at 16:58):

Nathanael Arkor said:

Given an adjunction $L \dashv R \colon \mathcal D \to \mathcal C$ in which the counit $\varepsilon$ is invertible, we have a family of isomorphisms $\varepsilon_D \colon LRD \to D$ in $\mathcal D$. We may form a category $\mathcal C'$ by adjoining a new isomorphic object $D' \cong RD$ for each object $D \in \mathcal D$, and define a functor $R' \colon \mathcal D \to \mathcal C'$ by $R'D := D'$. Finally, define a functor $L' \colon \mathcal C' \to \mathcal D$ which acts as $L$ on objects of $\mathcal C$, sends each $D'$ to $D$, and sends each freely added isomorphism to the isomorphism defined by $\varepsilon_D$. Then $\mathcal C' \simeq \mathcal C$, and $L' \dashv R'$, which is equivalent as an adjunction to $L \dashv R$, but $L' \dashv R'$ is lali.

It’s not possible to do this strictification without changing the categories up to equivalence, though, which is an important detail; I imagine all this was the water in which John Gray was swimming but the ongoing shortage of good books on 2-category theory means these questions still slow a lot of people down. It’s inconvenient.

Nathanael Arkor (Sep 10 2024 at 17:19):

Yes, this is an important thing to note, but the same is also true in general for strictification of monoidal categories.

Rémy Tuyéras (Sep 10 2024 at 17:37):

Nathan Corbyn said:

I’m always a little perplexed by the implied ethics of the term “evil” and I think “worthless” / “ineffective” take it a step further. Imagine pointing at someone else’s work and saying “that’s evil”—now image pointing at it and saying “that’s worthless”. I don’t think this is conducive to good science.

@Nathan Corbyn, I agree with you. I usually try to remain neutral about what should be deemed useful or not, as there are as many reasons to pursue something as there are questions to answer.

I was trying to see if we could make sense of @John Baez's intuition in terms of information (perhaps entropy?). I agree that "worthless" is an inappropriate term to use (mostly if directed toward a colleague's work), but I had hoped other terms could be proposed.

I'm hesitant about "strict" though, not that it is not a perfectly acceptable term, but mainly because John specifically mentioned the encoding of elements. For these kinds of questions, there’s always more than just an isomorphism at play; usually, there is a direction/strategy in how one wants to rewrite the elements, but this is very specific to the rewritting story. Anyway, I personally don't mind using "strict" if the purpose is to talk about the collection of possible reflectors

John Baez (Sep 10 2024 at 18:01):

By the way, I don't know anyone who ever used the term "evil" without smiling: as far as I know it was a joke started by my friend James Dolan, to dramatize the difficulties caused by strict concepts. So all discussions of the "implied ethics", or imagined scenarios of someone pointing at someone's work and saying "that's evil", leaving me scratching my head. I am, however, convinced by now that it's a risky joke to make in a public forum (like many jokes I am tempted to make).

Nathanael Arkor (Sep 10 2024 at 18:04):

I recognise the people who used it first did so in a tongue-in-cheek manner, but most of the people who have come later and have picked up on this terminology do not do so, and don't realise it's tongue-in-cheek, and I do think this causes misunderstanding that would be avoided by using words like "strict" instead.

John Baez (Sep 10 2024 at 18:36):

Really? Wow! Who uses it at a serious piece of "terminology"? Even calling it "terminology" is funny. As if someone said "What an annoying functor!" and someone else looked up [[annoying functor]] on the nLab to figure out what that meant.

John Baez (Sep 10 2024 at 18:46):

I'm not saying I doubt you, by the way. I'm just amazed.

Alex Kreitzberg (Sep 10 2024 at 18:55):

As a beginner I looked up "Evil" on nlab and found a discussion on invariance under equivalence. Google still redirects to the same article even though the name was changed3a9cf198-ffa3-4ee4-9457-02095a99fe2f.jpg

Fwiw when mathematicians use informal language to describe things I often take it almost as seriously as terminology, understanding that it's not.

( Hello! I joined four years ago, and have decided to come back. I was going to lurk a bit, but felt my view might be helpful here XD )

John Baez (Sep 10 2024 at 19:01):

Hi! Yes, the nLab page principle of equivalence currently says

Floating around the web (and maybe the nLab) is the idea of half-jokingly referring to a breaking of equivalence invariance as “evil”. This is probably meant as a pedagogical way of amplifying that it is to be avoided.

I find the cautious word "probably" here quite amusing, as if an anthropologist were studying mathematicians trying to guess what they were up to.

Nathanael Arkor (Sep 10 2024 at 19:03):

John Baez said:

Really? Wow! Who uses it at a serious piece of "terminology"?

A lot of the category theory-adjacent type theorists, for instance.

Alex Kreitzberg (Sep 10 2024 at 19:08):

Riehl also uses "evil" in a very formal way in "Category Theory in context":

Some category theorists go so far as to call a definition "evil" if it is not invariant under equivalence of categories. The only evil definitions we have used so far...

I think what's striking about "evil" as an adjective here is it's reserved for a very specific "sin".

When I'm working with very strict categories I'm worried I'm going to do something very wrong, or that I'm missing the point.

I like that the n-lab calls out the connotation "half-jokingly" though

John Baez (Sep 10 2024 at 19:57):

That is not what I'd call using it in "very formal way": saying that some other people "go so far" as to say it. She's holding the term at arm's length, blaming it on some extreme people she's seen somewhere....

John Baez (Sep 10 2024 at 19:59):

Imagine:

Definition. A monoid where every element has an inverse is called a monoid with inverses, though some go so far as to call it a group.

:laughing:

Alex Kreitzberg (Sep 10 2024 at 20:56):

Well, if "Evil" is loose or metaphorical, then certain folks must be especially annoyed with noninvariant definitions. :face_with_hand_over_mouth:

John Baez (Sep 10 2024 at 21:20):

Sure, like me - they cause endless hassles unless they are carefully justified as technical tools, like the theorem saying every monoidal category is equivalent to a strict one. Then you can prove theorems about strict ones, and as long as you take care that the theorem statements are invariant under equivalence, they'll be true for all monoidal categories.

Homotopy type theory does its best to make noninvariant definitions and theorems impossible.

Matteo Capucci (he/him) (Sep 12 2024 at 13:13):

I don't think 'evil' is to be canceled, though it is easy, as a beginner, to read it as a prescriptivist dogma and not realize the role strictiness has in good CT. Most importantly, it's easy to miss the fact that the principle of equivalence is really parametric on the ambient 2-category! So what's evil in Cat might not be in {cats with a fixed set of objects}. In other words, morality is relative :P

Nathanael Arkor (Sep 12 2024 at 14:49):

@Matteo Capucci (he/him): can you give an example of a situation in which you felt "evil" was helpful terminology, where "strict" would not have been appropriate?

Nathanael Arkor (Sep 12 2024 at 14:51):

though it is easy, as a beginner, to read it as a prescriptivist dogma

Yes, that is exactly my experience, which is one of the reasons I think it is harmful.

John Baez (Sep 12 2024 at 14:56):

Anyone who uses "evil" as a piece of terminology has completely missed the point, in my opinion.

John Baez (Sep 12 2024 at 15:04):

The most similar thing I can think of is this: sometimes a forgetful functor, which is a right adjoint, has its own right adjoint. For example the forgetful functor from Top to Set has a left adjoint sending each set to the discrete space on that set, but also a right adjoint sending each set to the codiscrete space on that set. Lawvere sometimes called right adjoints of forgetful functors 'fascist functors', since they are 'far right' - they're to the right of the right. I think this is a fun joke, but it would be silly to enshrine it as an official piece of terminology.

Matteo Capucci (he/him) (Sep 12 2024 at 15:14):

I agree with John. I wouldn't use 'evil' in a definition. I would use it in commentary though.

Matteo Capucci (he/him) (Sep 12 2024 at 15:15):

I don't think the world would be worse off without 'evil' either

John Baez (Sep 12 2024 at 15:31):

My own thought is this: if anyone here sees anyone running around using the word 'evil' for constructions that thoughtlessly break the principle of equivalence without realizing it's a wry, over-the-top joke, please tell them.

Nathanael Arkor (Sep 12 2024 at 15:35):

I think it's a bit late for that.

John Baez (Sep 12 2024 at 15:37):

It's not too late. I've never seen such people. If I did, I would tell them. If you see such people, you can tell them.

John Baez (Sep 12 2024 at 15:38):

The only way things get cleared up is by people talking to people.

Nathanael Arkor (Sep 12 2024 at 16:10):

John Baez said:

It's not too late. I've never seen such people.

This is precisely my point.

Nathanael Arkor (Sep 12 2024 at 16:11):

The people who use the terminology jokingly won't see all (or even most) of the people using it seriously.

Mike Shulman (Sep 12 2024 at 16:26):

My experience of the Internet is that for anything you care to name, there's always someone out there who will take it too seriously.

Does that mean we should never joke around? Or, at least, be more careful about how we joke around? I don't know.

Alex Kreitzberg (Sep 12 2024 at 16:31):

Y'all got me curious, so I googled "Fascist Functor" and found this exact conversation :rolling_on_the_floor_laughing:

https://nforum.ncatlab.org/discussion/1768/freedom-vs-fascism/?Focus=15404

Calling out Urs' comment:

...I don’t think that jokes in technical terminology work well. it may be fun in pesonal discussion (depending on the participant’s tastes, which may differ), but when it gets ensshrined in terminology it becomes a major nuisance.

Alex Kreitzberg (Sep 12 2024 at 16:37):

It'd be a bit sad if Mathematicians couldn't joke with each other because folks took them too seriously though.

Some of my favorite mathematical discussions are littered with jokes. I think it helps them be more memorable.

Mike Shulman (Sep 12 2024 at 16:38):

I think part of the problem is that when you say something on the Internet it doesn't go away. In face-to-face conversation you can make a joke and only the people who were present hear it, usually realize it was a joke (whether or not they approve of its taste), and that's the end of it. But on the Internet, even in a fairly informal venue like Zulip or blog comments, all our jokes are recorded forever for future readers to stumble across and take too seriously.

John Baez (Sep 12 2024 at 16:44):

This is why the smiley face was the most important invention since the internet. :upside_down:

(By the way: since Mike and Todd have complained about the ambiguity of :upside_down:, I'll hereby explain how I use this piece of terminology: it means don't take this too seriously, I'm kidding around - and I sure hope what I said isn't taken as some sort of attack.)

Peva Blanchard (Sep 12 2024 at 16:46):

Terminology in mathematics either looks like medical terms or words of poets. Now, I am realizing that, indeed, it is rarely funny (at least, from my own limited experience). However, there are names that I find funny:

• flabby sheaf
• petit topos vs gros topos : which would translate to "little (cute) topos" vs "chubby topos"
• evil, which I understand more as "cheating/tricking/deceiving" than "sinning" (I'm french , so it may be idiosyncratic)
• dessin d'enfant
• yoga of motives
• cosmic Galois group

there are probably other that I don't know about, or don't remember.

JR (Sep 12 2024 at 22:26):

John Baez said:

This is why the smiley face was the most important invention since the internet. :upside_down:

(By the way: since Mike and Todd have complained about the ambiguity of :upside_down:, I'll hereby explain how I use this piece of terminology: it means don't take this too seriously, I'm kidding around - and I sure hope what I said isn't taken as some sort of attack.)

Up to symmetry the upside down smiley face is the only different image under reflections, so by right it ought to indicate an adjunction. Instead of finding something funny, this is for funnying something found.

Morgan Rogers (he/him) (Oct 22 2024 at 10:25):

I was just in a lecture on infinity categories where someone stated that "evil" was a technical term... which was so distracting for me that I came straight here to complain :joy:

John Baez (Oct 22 2024 at 16:55):

Don't blame me. :smiling_devil: