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I wrote something short on Categories, Allegories – hopefully not too subjective.
Thanks. I have added a link to this from the list of references at category theory.
I also took the liberty of moving, in Categories, Allegories, the reference item itself up to before the comments on the book, for I think the reader might appreciate seeing that first. Okay? Otherwise I can change it back.
Thanks, Urs (and no problem).
Were there papers or other references about allegories that predate this book?
That’s the question I had too, Mike (and I don’t know the answer, although I have yet to dig around). The wording at the article was a bit waffly on this; you seemed to have picked up the scent.
Too bad the book has no references.
Been digging around, without luck. I don’t have online access to JSTOR or the Journal of Symbolic Logic, but there’s a review by Marta Bunge on Categories, Allegories, and possibly there’s some information there.
I tried looking through the categories list, but I’m not able to access it easily before I get timed out.
Thanks for the pointer; I found it on JSTOR. I can send you a copy if you want, but she doesn’t mention any previous usage of allegories – although she does mention the paper Cartesian bicategories, I which studies a closely related notion and predates the book. Maybe the idea was in the air but the particular notion of allegory originated with the book?
Thanks for looking, Mike! No need to send a copy for now, but yes that’s right about Cartesian Bicategories I (that was 1987, if my memory is correct).
I am slowly becoming convinced that Allegories made their first public appearance with the book, even if it had been gestating in Freyd’s brain for quite some time before then. (He’s the sort of mathematician who always has a lot up his sleeve.) More particularly, my guess is that others had not been considering the modular identity as a launching point for a categorical treatment of relational calculus (even if special cases like modular lattices or Frobenius reciprocity were on people’s minds).
I mean, maybe someone like Tarski realized the importance of the modular identity, but not along particularly categorical lines.
I still don’t feel like I understand the importance of the modular identity — despite having now written a paper myself that depends on it! Its string-diagram derivation in a cartesian bicategory explains “why it’s true” in a certain sense, but why is it that particular identity which provides the essential “shadow of nonexistent finite products” enabling us to reconstruct them?
Mike, I’m maybe not following your question very well. I don’t think I particularly grok the modular identity on a deep level, but as far as reconstructing products go, it seems in their set-up that units and tabulations come into play, which are assumptions on top of the modular law. I.e., to get a product of $X$ and $Y$, tabulate
$X \stackrel{\varepsilon_X}{\to} 1 \stackrel{\varepsilon_{Y}^{o}}{\to} Y$where $1$ is a unit and $\varepsilon_X$ is maximal in $\hom(X, 1)$.
Edit: Just to add something that may be already obvious to you: one of the main examples of allegory Freyd-Scedrov give is a modular lattice as a one-object allegory (where the involution is the identity). This doesn’t have products of $0$-cells, except perhaps in degenerate cases.
Yes, certainly units and tabulations are how you reconstruct products, but why is it the modular law which allows you to reconstruct products using units and tabulations? (I feel I’m not explaining myself particularly well either, so if you still don’t get it, just say so and I’ll try again.)
Maybe what I could do is fill in some missing details in bicategory of relations (which I’ve been meaning to do anyway), regarding this very question, and then maybe we could talk about what I’ve written. I have a sense that you’re not looking so much for a formal proof as a satisfying conceptual explanation, but anyway this would give us something to stare at, and you could tell me if you feel some dissatisfaction – which hopefully would then lead to improvements. :-)
Okay, Mike, I’ve been hammering out some details about allegories, which at least gives us something to stare at. (A lot of details, and rather technical.)
I find Categories, Allegories awfully condensed in places – hard to read and decipher – and so although I say the material is adapted from the book, it would be more accurate to say I’ve done an honest day’s work here. :-)
Now I feel bad about putting you to all of that work! (Although it’s definitely a valuable contribution – I’ve added a link from the nLab page allegory so other people can find it.) I have already read and mostly understood the proof that units and tabulations allow you to reconstruct products in an allegory. (I read it in Sketches of an Elephant, which I find much easier to read than Categories, Allegories.) I was complaining instead that that proof didn’t give me any intuitive feel for why it is the modular law which is important, as opposed to some other axioms which might happen to hold in a bicategory of relations but which don’t explicitly refer to products. Maybe I would feel happier if the notions of tabulation and unit were universally characterized; as it is they seem a bit ad hoc.
I am much happier with the notion of “flat functor” being the “shadow” of left-exactness in the absence of limits, because (1) if limits do exist, then flatness coincides with left-exactness, and (2) the extension of a flat functor to free cocompletions, which have limits, is left-exact. Could there be analogous theorems like “If a locally posetal 2- (or perhaps F-) category has (some universally characterized objects), then it is an allegory if and only if it is a bicategory of relations” and “The free completion of a locally posetal 2/F-category under (some universally characterized objects) is a bicategory of relations if and only if the original category was an allegory”?
Not to worry – the exercise was useful to me personally (and I hazard to say the notes might be useful to anyone who is struggling with that part of C, A)! I had forgotten that Johnstone works through the theory himself. But at least we now have something free and online.
Yes, I suspected you wanted something more “Zen” than a formal proof. I’ll try to think about it (and your questions) more.
Skippable explanation. It may be slightly odd to see a LatestChanges notification over one month after the edit, especially against the backdrop of recent discussion (will do my part there soon, just getting the small stuff out of the way beforehand) but there does not seem to me a way to avoid that: this was not a trivial edit, and back then, for some reason or another I did not notify, so a vital rule was bent. So here goes, keeping it short:
Start of ChangesNote:
I once added to Categories, Allegories a parenthetical remark giving one example, perhaps the most distinctive of all, of the pre-existing assertion
The book, while it covers an extraordinary amount of ground in less than 300 pages, is fairly idiosyncratic, especially in the choice of terminology
namely by mentioning “terminator”. Why I did not notify I do not recall (possibly back then thinking the below-mentioned rule-of-thumb ratio too small), but thinking about it, one should either refrain from such in insertion, or justify it. My reason for inserting this was that back then I read
Diaconesu, Kirby: Models of arithmetic and categories with finiteness conditions. Annals of Pure and Applied Logic 35 (1987) 123-148
wherein the rare term “terminator” instead of “terminal object” is used, admittedly only once, but without explanation. Puzzled by this terminology, and, extrapolating from what I knew of the (slightly illusory, but meaningful) distinction
pentagon : axiom
pentagonator : data
I suspected this may be something like a terminal object with additional data attached, whatever this should mean.
I then turned to the nLab(’s search engine), found precisely 0 occurrences, then turned to the larger web, found the term in “Categories, Allegories”, then turned back to the nLab, and thought it useful and relevant to record this term in Categories, Allegories.
In the wake of this, I added a brief remark to terminal object, recording this rare variant there, with a slightly evaluative comment that this is a less usual synonym.
End of ChangesNote.
[ Skippable explanation:
I need not be told that small edits, especially those which then generate some LatestChanges-trail should be kept at a low volume.
I do this notification to to keep to good practice, and right a (small) wrong: the rule that any nontrivial change, except perhaps by the most experienced, should be rationally justified in writing, was broken by me back then ( small self-rebuke ), unintentionally though and slightly.
This rule evidently seems vital for a collaborative effort like the nLab to function; it automatically reigns-in edits, and keeps one beholden to rational discussion.
Perhaps a good rule of thumb is to try, by and large,
The latter rule of thumb was broken, with reason, in the present comment. ]
Thanks for noting this edit. I think it is good to mention this at terminal object for the benefit of people reading such papers, as you say.
And that’s not a bad rule of thumb. But I actually disagree that there was any reason to break it with your present comment. Very long nForum posts take more time for you to write, and more time for other people to read, and in fact they are less likely to read them at all and more likely to miss the really important content, namely what exactly got changed. In this case I think it would have been sufficient (and better) to write something like
A month ago I added to Categories, Allegories and terminal object a brief mention (and redirect) for the older/idiosyncratic terminology “terminator”.
That would save you, and everyone else, a lot of time and energy.
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