Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty book bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2019

    After being alerted that it was still on the nLab here, I have removed the parenthetical political joke, or what was intended as such.

    diff, v12, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 27th 2019
    • (edited Apr 27th 2019)

    Actually, if I’m not mistaken, that “joke” is due to Mac Lane and appears in technical literature, in which case it might be cited as such.

    Edit: It seems we’ve had the discussion before here. David Corfield tracked down the citation going back to Mac Lane 1950 (Duality in Groups).

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2019
    • (edited Apr 27th 2019)

    Please not.

    Incidentally, we can’t google for MacLane’s use of “fascist group” to see if anyone uses it as a mathematical term, which is the nature of the problem with it.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeApr 27th 2019

    I thought we decided clearly last time that we should mention the terminology in case readers encounter it, but not endorse it, just as we do with any other bad terminology in the literature.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2019

    I’d claim that nobody encounters this except here on the nnLab, where no opportunity is missed to entertain extreme bureaucratic accuracy.

    So according to the single source that was dug out here MacLane spoke, alas, of cofree symmetry groups as ’fascist groups’. Haha.

    We should not proliferate that to “fascist objects” and “fascist functors” as we had, and then to the rest of the cofree world. No.

    We shouldn’t even discuss this any further. This is really beneath.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeApr 27th 2019

    Yes, it’s a fair point that when the terminology isn’t actually used there’s no reason to mention it. So unless someone can find more examples than this single citation, I’m okay with removing it.

    • CommentRowNumber7.
    • CommentAuthorGuest
    • CommentTimeApr 27th 2019
    Not every word out of MacLane's mouth is law.
    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeApr 27th 2019

    Nobody is claiming that anything is special about MacLane, only that when terminology exists in published literature it is a useful service to the reader to explain its relationship to other terminology.

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 27th 2019

    It might be interesting to salvage something of the mnemonic that is associated with the joke, that Finn Lawler brought up in the other thread. Privately, I have found it semi-useful.

    For what it’s worth, I’ve just found out that the terminology originates from Reinhold Baer – not Mac Lane. See page 3 here. I’ve not found convincing evidence that Lawvere ever used this terminology, but I might ask John B. about this.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeApr 27th 2019

    The mnemonic doesn’t really work, though: if the far right is “fascist”, then the far left should be “communist”, which is also the opposite of “free”. Freedom is closer to the middle of the political spectrum. (Depending on what kind of freedom you’re talking about, of course: free markets are on the right, free love is on the left, etc.)

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 28th 2019

    I think ”opressed’ objects would be a better antonym to ’free’ than ’fascist’. In any case, there are a number of weird names in the history of category theory we no longer feel we need to use or reference, and moreover ones that were more widely used and completely neutral in tone.

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 28th 2019
    • (edited Apr 28th 2019)

    I asked John Baez about Lawvere’s usage, and he said he didn’t know of any place in print that Lawvere said anything about “fascist functor”, but that Jim Dolan told him Lawvere used to refer to this in lectures (at SUNY Buffalo where Jim spent some time).

    I really disagree with the formula in #10 “far left = communist”. But maybe it’s better if we don’t continue down that path.

    Part of the mnemonic that resonates with me could be illustrated with the example of the right and left adjoints to the forgetful functor BoolAlgHeytAlgBoolAlg \to HeytAlg. The left adjoint makes things equal, 1=¬¬1 = \neg\neg, by passing to a quotient along an equivalence relation. The right adjoint eliminates opponents of 1=¬¬1 = \neg\neg, by considering the subobject of elements where x=¬¬xx = \neg \neg x.

    I’m trying to think of a way to make this more interesting – the moral is pretty obvious for a category theorist. It could be of interest for beginners, though, who don’t really grok adjoint functors.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeApr 28th 2019

    FWIW, from wikipedia:

    [Far-left politics] has been used to describe ideologies such as: communism, anarchism, anarcho-communism, left-communism, anarcho-syndicalism, Marxism–Leninism, Trotskyism, and Maoism.

    I count 6 different kinds of communism in that list of 8 things.

    The sort of left adjoint that “makes things equal” is not actually the one associated in my mind with the label “free”. I wouldn’t generally speak of “the free Boolean algebra on a Heyting algebra”, instead I would say “the Boolean reflection of a Heyting algebra” or “the Boolean quotient of a Heyting algebra”. In my experience the word “free” is generally used for left adjoints to functors that forget structure, not only properties, and therefore whose left adjoints add structure, e.g. “the free group on a set” or “the free topological space on a set” (although the latter is more commonly called “discrete”), and those left adjoints don’t tend to be the ones that “make things equal”. In fact I might argue that the codiscrete topological space on a set “makes things equal” more than the discrete one does. So I don’t really think this mnemonic is useful.

    Also, do you really need a mnemonic to remember that free functors are left adjoint to forgetful functors rather than right? Just think of the basic example like a free group.

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeApr 28th 2019

    BTW, you are certainly right that we shouldn’t continue very far down the communism road. I felt I had to point out that what I said wasn’t just something I made up but was based on a responsible source; but I won’t say any more about it after this.

    However, I do feel compelled to point out that the fact that the comment leads to this sort of disagreement is itself another argument against propagating it further.

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 28th 2019

    do you really need a mnemonic to remember that free functors are left adjoint to forgetful functors rather than right?

    That’s not really what I was driving at. It was more in the way of what (values of) right adjoints can be expected to look like.

    Re reflections etc.: I was just giving one example that came to mind. As I was trying to say, I’m struggling to find a really good mnemonic to capture the nature of right adjoints to forgetful functors, i.e., what they can generally be expected to “look like”. Clearly the fascist metaphor comes up short in this respect, but it’s a kind of start.

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeApr 28th 2019

    I’m struggling to find a really good mnemonic to capture the nature of right adjoints to forgetful functors, i.e., what they can generally be expected to “look like”

    I think that’s an interesting question; maybe we can focus on that? I can think of several different “kinds” of right adjoints to forgetful functors off the top of my head, and I’m not sure what they all have in common:

    • Coreflections to inclusions
    • Codiscrete functors for cohesive/topological categories
    • Cofree functors for (non-idempotent) comonadic categories

    One intuition that’s sometimes helpful for me is to say that a right adjoint equips things with all possible choices of structure that already exist, in contrast to how a left adjoints adds each needed instance of structure as new stuff that didn’t exist before.

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 29th 2019
    • (edited Apr 29th 2019)

    This kind of characterisation in terms of stuff, structure, property could be very useful. Where would be a good place to gather such thoughts?

    Prop 2.14 of super algebra makes for another good case of left adjoint - quotient, right adjoint - restrict.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)