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André Joyal left a comment at evil, presumably sparked by the debate raging on the categories mailing list.
(Don’t remember the exact message that sparked the “debate”, but the archives for the mailing list are here).
I will admit that I’m not too enamoured of the word “evil”, but I don’t find it particularly offensive and indeed it’s “shock” value is something that I would try to retain: if you do something that is “evil” you should be darned sure that you know that you’re doing it and convinced that the final outcome justifies the means. I’m also not convinced by Joyal’s arguments about “choosing a triangulation” or whatever. Sure, we choose a triangulation to compute homology groups, but the homology groups wouldn’t be worth a dime if they actually depended on the choice of triangulation.
I also think that the “subculture” argument is vacuous. Every group that has something in common could be called a “subculture” and every subculture is going to invent shortenings for referring to common terms. And of course there is great confusion when two subcultures choose the same word. My favourite story on this is when I was sitting in a garage whilst my car was being fixed. The mechanic yelled out, “You’ve got a crack in your manifold.”. I was a little confused as to what he meant! (The latest Dr Who puts a different spin on this, I believe).
The thing is not to avoid being a subculture, that’s impossible, but to avoid being a clique. The distinction that I intend to draw is that cliques are defined by who they don’t contain whereas subcultures are defined by who they do. Therefore anyone can join a subculture, but not anyone can join a clique.
Clashes of terminology are inevitable in such a broad subject. What does the word “category” conjure to a functional analyst? Someone not well versed in algebraic geometry might ponder the meaning of a “perverse sheaf”. And the connections between limits and limits seems, if not tenuous, at least to not be all that useful in conveying intuition.
I don’t think the nLab page itself is the most appropriate place to leave a signed comment like that. Perhaps someone qualified could edit the comment in order to incorporate the content as a general opposing view. It is worth having a section dedicated to “A critic”, but maybe not a signed one.
Just thinking out loud…
Two comments:
As far as I am aware the term “evil” as other terms like “cosmic cube” and similar were thought up by John Baez in the context of expositional writing or talking, where they serve their purpose.
While I agree with Andre Joyal on the use of “evil” in the everyday practice of math, what worries me more related to subcultures is that in an age of immense and fascinating progress in catgeory theory, the majority of contributions on the Category Theory mailing list revolve around issues of tiny and tiniest relevance. Our inboxes are filled with discussions about who said what first in 1961, the dislike of Eduardo Dubuc of everything coming from John, and the tremendous subtleties of the word “equal”. We can be sure that any serious mathematician from the outside who stumbles across these discussions will immediately draw the conclusion that this bunch of cateory theorists has nothing of importance to say and has lost contact to all recent and not-so-recent developments. (Be that the case or not, that will be the impression.)
Don’t make me chuckle aloud! I’m in my office trying to pretend to be serious :)
I only occasionally glance at the Cat Theory mailing list, but I tend to come away with almost precisely the same opinion :)
Someone not well versed in algebraic geometry might ponder the meaning of a “perverse sheaf”.
The greatest algebraic geometer of 20th century Grothendieck, considered the term “perverse sheaf” not appropriate. He considered that transfering bad allusions about human imperfections should not contaminate mathematics to which the terminology should agree with its admirable aesthetics. According to Cartier, Grothendieck, who was not born French, had a very subtle feeling for inventing successful mathematical terminology in French.
if you do something that is “evil” you should be darned sure that you know that you’re doing it and convinced that the final outcome justifies the means
The category list discussion had hard time to acknowledge that it is not enough to use word wrong to replace not invariant under equivalence. Now you say in essence that the construction should not be wrong (or purposeles in other case); and there are indeed many ways to be wrong in creating a new definition. OK for that particular warning it is enough to say not wrong.
has lost contact to all recent and not-so-recent developments
There are definitely interesting things happening among pure category theorists and among those at the interface with computer science, and whose motivation and depths are more foreign to people motivated by physics, geometry, topology and higher categories. When I was organizing the conference in Split, I emphasised that the conference should be focused to the physics/geometry aspect what most of the mainstream category theory conferences are not and to skip the computer science, but not because the topics are not interesting in my opinion, but because there were already so many conferences at the interface of computers and categories and because some of the main people (geometers) whom I wanted to see in the first place, would be bored with, say, computer aspects. nlabizants should, in my opinion, not contribute to the isolation of the category field, which is truly often the tendency of the pure 1-category theorists.
But one has to admit that the isolation is partly in the psychological nature of the subject. You see, one of the wonders of the category theory is the simplicity and perfection of understanding which comes with it. This naturaly attracts people who are sharp thinkers but tend to go toward the thourough understanding of basics, of foundations and do not feel comfortable with amazing but also confusing wonders of the mathematics which comes in disordered state, not ready for full digestion, like most of the things coming from string theory.
To illustrate the point, I was told by a prominent category theorists when I gave him hints for some beautiful new developments that this is yet not mathematics and he is largely right. So, in the presence of many problems which such experts do understand deeply (and already dedicated much of their career toward), why would they go to something radically new and feel foolish and childish in, say, physicists’ land ? Some, like Urs and me, are bewildered by the beauty of not understood things in physics and coping with vagueness while getting to understand some things is not an unpleasant endeavor for us. The choice to stand in the land of your own style is not to be criticized (de gustibus non disputandum est), the attitude that only the vista from somebody’s own company is genuinely worthy is (to be criticised).
Many professional 1-category theorists even consider higher category theory as being not yet ready for a true study, or even questionable as a true subject ever, but this view is, it seems to me, largely uninformed.
P.S. applications do not know which math is well understood and which is not. Therefore, going toward the applications, mixes different fields, and for example can not isolate you to 1-categories with full success of understanding the nature. On the opposite, going more to a specialty, even so radical as the category theory is, will always bring the picture which may look self-satisfactory, but from the point of view of nature and width of potential math and potential applications, is restricting. Homological algebra existed before category theory. Discovery of category theory and every new phase in category theory has radically advanced it. One of the newest is with the role of the notion of stable infinity category in promoting a level up the natural home for much of the classical homological algebra. So the application had created examples for higher category theory, in non-manifest form, of course, before the letter has been consciously thought of. This is the advantage which comes with looking at examples and applications, including physics. It gives us examples of things whose full nature will be known only much in future.
I’m also not convinced by Joyal’s arguments
And you shouldn’t be, because they are flat-out wrong, as John Baez explains well in his reply. (I no longer have my copy and it hasn’t yet shown up on the Gmane archive; when it does, I’ll link to it.)
I don’t want to copy debate from the mailing list to the Lab. By all means, let the Lab link to the mailing list, however. If people don’t like the term, then that’s something relevant to talk about, but since the debate is already happening elsewhere, then let us just point to it.
We don’t have debates about the validity of structural foundations, constructivism, the axiom of choice, higher category theory, abstract mathematics at all, etc. Part of the POV is that these are all useful, and we write from that perspective. We don’t censor ideas that only make sense materially, etc, either, but we also consider the question of how to understand them structurally, etc. Writing about fully weak ideas is another part of this.
Added: And just because the POV says that some ideas are useful doesn’t mean that they must be correct. I think that most of use would not consider ourselves constructivists (I may be the only one, and fundamentalist constructivists would disown me); but (I think) we all accept that it useful to say whether some idea is constructive and (if not) how it might be made so.
I agree that most of Joyal’s arguments are wrong, and I think that his comment should, if we keep it at all, be in a query box. However I am sympathetic to the argument that the particular word “evil” is too strong. Perhaps this is correlated with personal beliefs about morality in general, or maybe it’s just idiosyncrasy, but I’d be in favor of switching to a less loaded word like “non-kosher” as was suggested on the list. Even people who personally have no problem with “evil” might think about switching simply because it bothers other people – why annoy people unnecessarily? (I also like that “kosher” and “non-kosher” puts the emphasis on the positive property to be sought after, rather than the negative one to be avoided.)
I’m inclined to agree with Joyal, but I guess that’s just me. For example, a trivial fibration in the natural model structure on is technically an “evil” notion, since it’s required to be a strict surjection on objects, but the only replacement for it that is invariant under equivalence is an equivalence! Surely it would be silly to deny that this is a useful notion, but it only makes sense in a non-invariant way.
Edit: I see that this is one of the issues that sparked the debate on the category theory mailing list =X.
I actually have an idea. Suppose we think of whatever theory we care about as a category with a lluf subcategory of weak equivalences W satisfying 2 of 6 and containing all identities. We call properties or structure that are invariant under W W-invariant or invariant under W. Then in the theory of categories, we can consider a number of weak equivalence structures, isomorphism of categories (resp. equivalence of categories) (resp. thomason-equivalence) (that happen to belong to model structures, but that is irrelevant). We say that a property is isomorphism invariant (resp. equivalence-invariant) (resp. thomason-invariant) if it is preserved by the maps of the right weak-equivalence structure. Everything is relative and nobody gets hurt.
I tried to incorporate Joyal’s comments into something useful, but I’m really not happy with it (and I won’t be at all upset if people delete large swaths of it, or even the whole thing). The problem is that Joyal’s critique is actually one of the poorer ones (at least among those that managed to address the issue at all!). It seems obvious to me that he did not understand the idea. It’s worth writing the evil article to avoid common misunderstandings, and for this reason it’s good that Joyal and others are making comments, but that’s different from incorporating them into the article text.
I also added a bit on other possible terminology. There was some more on the mailing list, which I will try to track down. (The Gmane archives are rather poor for searching or reading threaded commentary.)
Whatever the resolution on 'evil' versus non-kosher versus longer phrases, the tongue breaking compound "equivalence-invariant" found at present in nlab is for my ears a bad substitute for very smooth and far easier phrase "invariant under equivalence".
@ Zoran #11: Agreed, and fixed.
@Zoran: “equivalence-invariant” was just substituting in “equivalence” for W in “W-invariant”.
Harry, you can say invariant under . The -invariant is a legitimate abbreviation meaning the same. In the case of long words like equivalence it may be obscuring. People who did not hear the phrase before in its unabbreviated form can pretty well guess what is meant, and with a bit less success can guess if the form is abbreviated. This is a general rule of thumb.
I’m not comfortable with the current state of evil as of Monday – I don’t like singling out Joyal for opprobrium by quoting his remark and then explaining why he’s wrong. I agree it’s important to explain why that point of view is a misunderstanding, but I think it would be more polite to just link to several comments on the categories list archive (including perhaps Joyal’s) that make similar points.
I’m not comfortable with the current state of evil as of Monday
I also feel the entry deserves improvement. At the very least the “Criticism” needs to be moved to after the general definition, so that one has a chance of knowing what exactly it is that is being criticized.
My real problem, though, is that I don’t understand in which sense you say “Joyal is wrong”. Notably the example of model categories he alludes to seems to be good to me:
in the sense of the entry, a model category is a highly evil way to talk about an -category. In a way this is the point of model categories: that they provide presentations for something more intrinsic. In the model structure it makes sense to ask of an object “is this cofibrant”? Which is a property of course not invariant under weak equivalence. And of course this non-invariance is a feature, not a bug (for if we could already talk about the oo,1-category fully intrinsically, we would not need the model structure to present it in the first place).
So model categories are an example of a highly useful and at the same time highly “evil” concept, in the sense of the entry. I understand Joyal as pointing out that it is unwise to term something very useful “evil”.
Maybe this is debateable. But I don’t see how you can say Joyal is “wrong”.
Another problem with the entry is this: it started out being a crisp simple definition of a catchy phrase.
Now after replying to criticism, it is beginning to evolve into a sophisticated theory. The reader is told that if he sees something in one category that looks “evil” according to the definition but which does not feel “evil” then it’s the reader’s fault for not noticing that he should be applying the entry’s definition to another category instead (paragraphrasing the example of and currently given.)
Before all this I thought I had a good idea of what John meant by “evil”. Namely exactly what Joyal says: “not-invariant under equivalence”, but with the added benefit of carrying Baez-ian connotation appealing to our intuition and highlighting that this might be problematic. I like that. A catchy phrase for an elementary though important technical point.
But now with the entry growing, and following the replies in the CatTheory list, I am beginning to think that maybe I am losing track of what is going on. When I hear you and Toby speak about this now, I am getting the feeling that I should buy a book and learn about the subtle theory of evil some day. I am getting confused. At the same time, I am feeling that nothing of actual relevance is lost in this confusion and I begin to start ignoring all this, as a waste of time.
Ideally, the entry could be rewritten such as to avoid all that.
I very much share the impressions with Urs in 16 and 17.
I think what Toby and I are trying to say is that just the fact that something is defined by using a coordinate system, presentation, model category, etc. does not make it evil. I don’t think the word “evil” (or any replacement for it) should be applied to the coordinate systems, presentations, model categories, etc. themselves. What’s evil is defining a concept in a way that depends on the choice of such a presentation in a non-covariant way.
For instance, the mapping spaces in a model category are invariant under Quillen equivalence and in fact an invariant of the presented (∞,1)-category, so they are not evil, despite being constructed explicitly using model-categorical notions of fibration, cofibration, factorization, etc. which are invisible to the (∞,1)-category. On the other hand, the property of “being left proper” is evil, since a left-proper model category can be Quillen equivalent to a non-left-proper one. Similarly “being a cofibration” is an evil property of a morphism, at least in the sense that it makes no sense to ask whether a morphism in an (∞,1)-category “is a cofibration” without reference to a particular morphism in a particular model category presenting that (∞,1)-category.
I think it is mainly Joyal’s sentence “many things in mathematics are depending on the choice of a representation which is not invariant under equivalences, or under isomorphisms” which sounds wrong to me – the point is that most fundamental concepts are invariant (or covariant) under equivalences. Although we crucially use concepts that are not so covariant in order to talk about the covariant ones. The rest of his quote seems to be right on, though, so perhaps I just misunderstood what he meant by “depending”?
And I’m sorry for complicating things! Yes, by all means let’s try to make the nLab entry simple and easy to understand. I do care about the logical stuff going on in the background, but this page ought to be easy to understand and not confusing.
I still want to change the word “evil.” Right now I am liking the sound of simply saying “covariant,” since that seems to be what is actually meant and is a word that already has the right meaning and even the right connotations. (Whereas “invariant” is strictly speaking only applicable to properties, not structures, and seems to need qualification to make it specific enough for the situation at hand.)
We’re not singling Joyal out; he singled himself out. He wrote an email to the categories mailing list, then copied it to this page for some reason. See my comment #10 above for what I did and why, and to see that I too am not happy with it. Delete the whole thing (his comments and my reply) if you like; none of it is good material.
However, it is not enough to remove misguided criticism; we have to add something to prevent misunderstandings. Merely fiddling about with the names is a distraction, and the discussion on the mailing list is now worse than useless. (People are repeating ideas for the third time while expressing astonishment that nobody has said them before!) I have too little time to write something good at evil now, and I certainly don’t have time to keep up with the mailing list (which is too bad, because there is good discussion hidden amongst the blather).
Well, I wrote a section on Motivation at evil, along with a section on why it’s not always evil to look evil. Hopefully those will help. The Definition section may or may not need to be moved higher, since the Idea section already tells us what we’re talking about. But the Criticism section, while it should be kept, also should be rewritten without the quotation from Joyal, in my opinion.
We’re not singling Joyal out; he singled himself out.
Yes, certainly, but of course that history isn’t evident to a casual reader of the page.
I was going to remove the content from the “Criticism” section and replace it with a link to the cat list discussion, but could not find a good link with Andre’s comment.
I really do not like what is there so if someone provides a link, I’ll remove the content and provide the link.
I added a list of some proposals to the “Terminology” section of the page evil.
25: I vote strongly against new acronyms, especially for general and intuitively well-founded mathematical notions. I also slightly vote against “kosher” expression for appropriateness in math. When I came to United States I saw first time word Kosher on pickled cucumbers which were for to me then incomprehensible reason called Kosher Dills. These cucumbers were usually on sale, while others were not, so in the first 4 years in United States I thought that kosher is a way to pickle cucumbers. Then I learned that the term is a general term for appropriate food in Jewish tradition, hence could be thought n general of appropriateness. Word appropriate is however thought in every basic English course for foreigners, unlike the word kosher which is rather culture specific (some people never visit restaurants: we had such a colloqium speaker in Wisconsin, a world famous mathematician who refused the colloqium dinner as he never ate out the food which is not prepared by him or a close friend or relative; he suggested to go to a local shop instead, buy a packet of strawberries, which he saw on sale, wash them and sit out in a park and enjoy them). Many things which are culturally obvious to some are not to others.
To make few examples which may sound strange to some. I learned first steps about K-theory functor at age 17, but I made my first phone call only at age 19. Or made my first assembly program months before I saw the first computer in my life (age 17) by bare eyes. At 19 and half I worked (in army) on computer simulator and on the phone for transmitting the data from radars. In the army however my intellect visibly degraded by about 25 percent or so because of nonsleeping, brainwashing by “hurrying and then waiting” massage and the (new to me) culture valuing power over reason (it does not to be justified if it is by a more powerful. I was used that everybody around me answers to why question; but I faced now people who would not try to address the why question but threaten instead of an answer: I could not believe it is possible). Having a computer does not make you smarter, sleeping enough and attention to a fellow conversant does. The stereotypes are however different (and dangerous).
Covariant under equivalence for structures and invariant under equivalence for properties sounds good to me.
I was going to remove the content from the “Criticism” section and replace it with a link to the cat list discussion
I want to do this too, but there doesn’t seem to be a good place to link. The categories list no longer has its own archive but directs readers to the Gmane archive. That is up to a week behind, for one thing; but worse, it ignores threading (who replied to what) completely, so it’s impossible to follow a discussion. (Unless Gmane has features that I have not found!)
I don’t feel comfortable removing the Crticism section, since it makes me feel like I am censoring André. However, I would be delighted if somebody else did!
This is not to say that criticism serves no purpose, but rather than copy the criticism here, it is better if we explain the motivation, how to use the concept, what other terminology is available, and where criticism may be found.
I thought that kosher is a way to pickle cucumbers. Then I learned that the term is a general term for appropriate food in Jewish tradition, hence could be thought n general of appropriateness.
I usually see ‘kosher’ translated into English (when it is translated at all) as ‘proper’. Of course, that has many meanings in mathematics already.
I don’t have anything invested, so I removed the material and referred readers to the Cat List (with no specific url).
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