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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeSep 1st 2010

    Added a note about cofree functors to free functor.

    • CommentRowNumber2.
    • CommentAuthorAndrew Stacey
    • CommentTimeSep 1st 2010
    • (edited Sep 1st 2010)

    That page name is crying out for a typo!

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeSep 1st 2010

    Certainly “fascist” is not a proper antonym of free. A cofree functor is bound, no?

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 1st 2010

    Are there any good examples of cofree functors to report?

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 1st 2010

    Lots. For example, the cofree coalgebra on a vector space which applies to other coalgebras of a prop of an operad. The right adjoint to the underlying functor Set CSet/C 0Set^C \to Set/C_0 (and plenty more where that came from). Coreflective subcategories (e.g., the inclusion of kelley spaces in TopTop, where the right adjoint “kelleyfies”).

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeSep 1st 2010

    @ Harry

    No, it’s not a proper antonym. It’s just a joke that Lawvere used.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 1st 2010

    as a public service, I pasted Todd’s reply into cofree functor and coreflective subcategory

    @Toby: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. At least that’s how I feel.

    • CommentRowNumber8.
    • CommentAuthorTobyBartels
    • CommentTimeSep 1st 2010
    • (edited Sep 1st 2010)

    I think that some jokes work well in terminology, but I also think that this isn’t one of them. The fact remains that Lawvere has used it (or at least Jim Dolan and John Baez have used it and attributed it to Lawvere); are we supposed to suppress this? Do you actually have any disagreement with anything that is on the page???

    Edit: I suppose that it’s incumbent on me to give an example of a useful joke in terminology. I submit that it is easy to remember (after you learn it once) what the ham sandwich theorem is about. In contrast, who remembers what the Stone–Tukey theorem is? (Of course, this example does not generalise to cofree/fascist functors.)

    • CommentRowNumber9.
    • CommentAuthorEric
    • CommentTimeSep 1st 2010

    Moral of the story: Jokes are not allowed here :)

    • CommentRowNumber10.
    • CommentAuthorFinnLawler
    • CommentTimeSep 1st 2010

    I think it’s more than just a joke – if I remember rightly the name is given because these functors are ’far right’ adjoints. John Baez used the metaphor (I forget where) that often a ’far left’ adjoint ABA \to B (i.e. the free-BB functor) makes AAs into BBs by giving everything the ’same’ (free) BB-structure, thereby getting rid of any they already had, while a ’far right’ adjoint (the cofree-BB functor) often makes AAs into BBs by simply killing off anything that gets in the way. So the name has a certain mnemonic value in drawing an analogy (admittedly a rather cutesy one) with far-left and far-right governments.

    I do agree, though, that joke-based terminology often starts to grate after a while.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 1st 2010
    • (edited Sep 1st 2010)

    Do you actually have any disagreement with anything that is on the page???

    No. It’s just that if I had written the page, I had played reverse-Bourbaki and not mentioned that terminology, even if it had been used and advocated before.

    By Finn’s explanation I now finally get a glimpse of how this is actually funny, but I think we are already faced with such a mess of unintentionally silly terminology that we do need to intentionally add more silliness to it all.

    Moral of the story: Jokes are not allowed here :)

    To give an example: if you find a community of people who want to write an instruction manual for a nuclear power plant, I guess they can have a great time joking around with each other, but would rather not have too many jokes be included in the book they are writing.

    • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeSep 2nd 2010

    I really don’t understand your last comment, Urs, but since you began it

    Do you actually have any disagreement with anything that is on the page???

    No.

    maybe it doesn’t matter.

    PS: As I try to make interesting subject headers when I report latest changes on the Forum (for which I blame Andrew’s inspiration), does this grate on anybody? I can cut it out, if so.

    • CommentRowNumber13.
    • CommentAuthorAndrew Stacey
    • CommentTimeSep 2nd 2010

    I like jokes and word-play and stuff like that (witness my first comment on this thread), but I don’t like it when it gets exclusive. So I actually like the current wording of the page. Since “fascist functor” is a term that someone may encounter, having it on the page will help someone new to the subject to realise that “fascist functor” is a jokey term for “cofree functor”. But by not making it part of the definition, we don’t perpetuate the joke, thus showing ourselves to be Serious Mathematicians who are Above All That Nonsense Whilst Trying to be Helpful. (Hopefully the capitalisation shows how seriously I take that last part!).

    I’ll echo Toby’s PS: I know that these things can be taken to extremes and that the person taking them to the extreme is often the last person to realise. So a polite “cut it out” would not be taken as rude!

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2010

    I really don’t understand your last comment, Urs

    I am wondering what’s going on. While not important, I’d be interested in sorting this out.

    I am interested in keeping terminology nicely descriptive. Since we are dealing with non-tangible entities in math, their closest to a bodily incarnation is their name, and since I have them around me all day, I am interested in being able to get along with them well. I want to like the math I am dealing with. And I want the terminology to help me think, not to distract me from thinking. If I am after political satire I’ll close the math notebook and open a political magazine. Both these serve their purpose, but not when mixed.

    I don’t think it’s about “Serious Mathematicians who are Above All That Nonsense”. I am all for nonsense, but not in my math. It’s like with a craftsman, who works all day in his work shop, he wants to have all his tools nicely aligned and properly labeled, because that makes the work-flow nice. It can be great fun to give all the tools silly names and have a good laugh, but next morning when work continuous we don’t want the jokes to stand in our way.

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 2nd 2010

    At free functor, I have done some rewording of the “fascist functor” remark, so that hopefully it is clear that we are not endorsing it. I’ve also done some fleshing out in the examples section.

    • CommentRowNumber16.
    • CommentAuthorTobyBartels
    • CommentTimeSep 2nd 2010

    Sometimes silly names not only make me laugh but also help me remember connections. Sometimes jokes are proper labels. Sometimes humour is nicely aligned.

    Even though I don’t use “fascist functor” myself –I included it for the reasons that Andrew explained–, Finn has shown how it too could be useful.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2010
    • (edited Sep 2nd 2010)

    Okay, I have no objections against the entry. I am just afraid of running into situations where people can say things like “We compose this fascist functor with that terrorist functor to find a public enemy adjunction without adding freedom fries. ” or the like in a talk and mean to be speaking about math. It seems inappropriate. But no big deal here, let’s leave it at that. I just wanted to explain myself after you said you didn’t understand my comment.

    • CommentRowNumber18.
    • CommentAuthorTobyBartels
    • CommentTimeSep 2nd 2010

    I would write such a thing only if each term used really helped me to understand the concept that it described. I think that this is unlikely.

    • CommentRowNumber19.
    • CommentAuthorAndrew Stacey
    • CommentTimeSep 2nd 2010

    So when I finally get round to writing that paper about right adjoints for certain functors on LCTVS that are related to holonomy then if I want Urs to read it, I’d better not call it “Fascist functors and fusion in nuclear spaces”.

    • CommentRowNumber20.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 2nd 2010
    • (edited Sep 2nd 2010)

    This discussion reminds me of the discussion about “Ass” and “Boob” over at the Cafe (it’s not the same thing, I know). I’m generally not in favor of terminology which distracts, and I am invariably distracted by AssAss. I guess Urs finds “fascist functor” distracting, and it’s not hard to understand why.

    There was in fact a time when I was writing things like B(O,O,M)B(O, O, M) [two-sided bar construction, where OO is the free operad monad and MM is an operad, i.e., OO-algebra]. So if I were in the habit of using the notation BB for an operad, this could lead to B(O,O,B)B(O, O, B). But I digress…

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2010

    This discussion reminds me of the discussion about “Ass” and “Boob”

    me, too.

    There is a little difference in that AssAss follows a sensible pattern (with ComCom etc) and one may or not may decide to break the pattern in order to avoid awkward terminology. But here it’s about introducing awkward terminology without any need. That’s what bothers me: there is already enough bad or unsuited terminology we have to deal with anyway , it’s a pain to intentionally introduce more of it without any need whatsoever.

    • CommentRowNumber22.
    • CommentAuthorMike Shulman
    • CommentTimeSep 7th 2010

    I’m with Urs here – I find “fascist” at best distracting.

    • CommentRowNumber23.
    • CommentAuthorTobyBartels
    • CommentTimeSep 7th 2010
    • (edited Sep 7th 2010)

    @ Mike

    Is there actually anything in the article now that you don’t like?

    [previous snippy reply redacted]

    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeSep 7th 2010

    No, I don’t object to anything in the article as it stands now. I’m sorry, I was just saying “yes, I agree with what has been decided.” It sounded to me on a cursory reading of this thread (I’ve been away for a bit) as though Urs was the only one with a positive objection, so I wanted to record that he wasn’t alone in that, in case some robot versions of ourselves in the future change their minds and want to start using “fascist.” (-:

    • CommentRowNumber25.
    • CommentAuthorTobyBartels
    • CommentTimeSep 7th 2010

    I apologise for my snippy reply.

    • CommentRowNumber26.
    • CommentAuthorHarry Gindi
    • CommentTimeSep 8th 2010
    • (edited Sep 8th 2010)

    But Todd, Ass(M)Ass(M) is Bourbaki’s original notation in French for the Assassinator of the module M (the set of associated primes).

    • CommentRowNumber27.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 13th 2011

    An early appearance of ’fascist’ as dual to ’free’ is in Saunders Mac Lane’s ’Duality for groups’, Bull. Amer. Math. Soc. 56 (1950), 485-516.

    Call the dual (in this sense) of a free (nonabelian) group a fascist group. R. Baer has shown me a proof of the elegant theorem: every fascist group consists only of the identity element. p. 486