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• 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.

• 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)

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 $n$Lab, 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 $BoolAlg \to HeytAlg$. The left adjoint makes things equal, $1 = \neg\neg$, by passing to a quotient along an equivalence relation. The right adjoint eliminates opponents of $1 = \neg\neg$, by considering the subobject of elements where $x = \neg \neg x$.

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.