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Added some examples to allegory, including that of modular lattice as one-object allegory.
Added a section on syntactic allegories to allegory, mostly to record a result about the interpretation of $\exists$ in unitary pre-tabular allegories.
Presumably the syntactic allegory in turn arises by a standard construction from a syntactic hyperdoctrine?
Re #3: I was thinking the same thing.
We were discussing allegories and such a couple of months ago. Mike asked a question here which is still basically unanswered. Suffice it to say that of all the various categorical machines for discussing first-order theories (including hyperdoctrines, bicategories of relations, and allegories), allegories seem the least well tied-in to the matrix of higher category theory (or anyway the least well-grokked here at the nLab, if Mike’s question and my lack of response are any indication).
Some small edits to allegory. I’ve also added redirects from pre-logos and logos to coherent category and Heyting category respectively.
Re #3: I would expect so, definitely.
Re #4: Later in that thread Mike asked
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”?
We do know, though, that a locally posetal 2-category that is a cartesian bicategory is an allegory iff it is a bicategory of relations, don’t we?
We do know, though, that a locally posetal 2-category that is a cartesian bicategory is an allegory iff it is a bicategory of relations, don’t we?
It’s never occurred to me to wonder until now: is an allegory an extra structure on a locally posetal 2-category, or is it really just a property? In other words, is the dagger structure uniquely determined?
I’ve also added redirects from pre-logos and logos to coherent category and Heyting category respectively.
Are these synonyms? Can you add something to the target pages to say this (or to say whatever is true)?
Yeah, there needs to be some mentioning of XYZ on a page to which XYZ redirects.
From page 12 of
we have the following. Let $\kappa$ be a cardinal. Then
A $\kappa$-geometric category is a regular category with unions for $\kappa$-small families of subobjects, stable under pullback.
Makkai-Reyes called these $\kappa$-logical categories and Freyd-Scedrov called them pre-logoi.
A $\kappa$-Heyting category is a regular category with unions and intersections of $\kappa$-small sets of subobjects and such that pullback of subobjects along any morphism $f$ has a right adjoint $\forall_f$ (the universal quantifier).
In Freyd-Scedrov this is called a logos when $\kappa = \omega$.
I am now moving this into the relevant entries.
By the way, looking again at the entry allegory I find it is missing more of an indication of why we care about allegories. Right in the Idea-section there should be a sentence saying “The theory of allecgories is useful for…” and then probably mention implications for exact completions etc.
why we care about allegories
I suppose someone could write something, but:
There doesn’t seem to be overwhelming enthusiasm for them around here in the first place; they are one way of doing categorical relational calculus, yes, but notions like hyperdoctrines or cartesian bicategories also serve that purpose and seem more flexible or adaptable to categorification. We keep asking ourselves: why this selection of axioms (which look ad hoc to some of us)? I personally would like to understand that better before trying to answer why we care.
You could say, in the manner of a ten-year-old writing up a desultory book report, “the theory of allegories is useful because Freyd and Scedrov (and others) proved a whole bunch of results about them that can now be referred to.” The stuff about regular and exact completions in terms of splitting certain classes of idempotents in bicategories of relations doesn’t particularly need allegories to say it.
Ah, interesting. I didn’t know this. I kept looking at the page “allegories” and asking myself why I should care.
But so this is also a useful piece of information. Why not say it in the entry?
I think such “why-this-definition”-answers are needed also for the general perception of the $n$Lab. It makes a bad impression to happen upon a page that indulges in definitions without telling the reader what the payoff is supposed to be. It makes the impression that somebody is just playing around with definitions instead of doing fruitful mathematics.
Right this moment I cannot, but if you prefer I can later try to distill some remark into the entry from what you just said.
There’s one important way in which the notion of bicategory of relations is less ’flexible’ than that of allegory: a bicategory of relations must have a product. If you want to perform exact completion by adding kleisli objects (i.e. splitting some idempotents, in the locally posetal case) and your input data doesn’t have products of objects yet, then allegories may work where bicategories of relations would fail. This was my situation in my exact completions paper, where after a long time of disparaging allegories I found myself forced to use them!
I like the idea of seeing an allegory as ’a (1,2)-category that would be a bicategory of relations if it had products’. I guess Finn is right that by ’having products’ here we could mean ’being a cartesian bicategory’. Even better would be if we could characterize the ’cartesian’ (1,2)-categories with some universal property, such as being cartesian objects in some 2-category. Then we could ask the other half of my suggestion: is the free completion of an allegory under ’products’ a bicategory of relations, and conversely?
@Todd #6: if the allegory is tabular, or even ’weakly k-tabular’, then the dagger-structure is uniquely determined, but in general I can’t think of any reason why it would be.
@Urs #8: Also, I called those k-geometric categories k-ary regular categories, wanting to emphasize that k is the ’arity’ and not, say, the category dimension.
Mike,
whatever these things are called, the entries need to say it. It’s not sufficient that you tell me here or somewhere out there is some paper that says it. There should be a remark at geometric category saying what you just said, then.
I wrote up something at allegory as per Urs’s suggestion. See what you think.
@Mike: I can see it for tabular categories. But I don’t know what “weakly k-tabular” means (and I’m too lazy or tired now to attempt a guess).
I wrote up something at allegory as per Urs’s suggestion. See what you think.
Thanks, Todd! Very nice, yes, that’s the kind of comment that I was hoping for.
By the way, since it keeps being mentioned, can we say something contentful at relational calculus, at least such as to give a broad orientation?
Re Toby’s #7, Urs’s #8: Yes, sorry, I should have said something about (pre-)logoses on those pages. I’ve added a reference to k-ary regular category and a link to Mike’s paper at geometric category.
Re Mike’s #12: I’m still working on this, so I can’t give you a proof quite yet, but I’m pretty sure that a cartesian bicategory will be the same thing as a ’cartesian equipment’ that is ’functionally complete’/chordate, a cartesian equipment being a cartesian object in the 2-category of equipments, pseudo-functors and lax transformations that are valued in, and pseudo-natural with respect to, tight maps. That is certainly suggested by the material (due to Todd, I think) at cartesian bicategory.
Thanks, Finn!
Re relational calculus, I’d be tempted to try to recall some history, or at least a mathematician’s history, which would involve names like Peirce, Schröder, Tarski, … In the early days there were lots of analogies made between relational calculus and linear algebra, explainable by the fact that $Rel$ is $CMon$-enriched and self-dual. Trouble is that I don’t know the history, really.
I (only) now realize that I pretty much missed that story about “familial regularity and exactness”.
The entries on all the notions unified by this need to point back to that unification. So I have created now a floating TOC and am including it into all the relevant entries:
Please check out that TOC and edit/modify as need be.
Very quick reply: I’m sorry (and surprised) that I didn’t add enough links. I certainly intended to! But there were a lot of pages that needed editing at once, and I guess I missed a bunch. Thanks for the fixes.
@Finn 19 : excellent! I look forward to it.
@Todd 17: It’s in my paper… sorry I don’t have time to write more now, I’m getting up early to go to Montreal tomorrow…
Here is a basic question about allegories that I don’t know the answer to right away: is “allegory” a property or structure one can put on a locally posetal 2-category $B$? The issue is whether there is at most one “opposite” operation $(-)^{op}: \hom(a, b) \to \hom(b, a)$ that makes $B$ an allegory.
You probably do know that if B is tabular, or more generally if every morphism is a join of a composite of maps and their inverses, then its allegory structure is unique, since the opposite of a map in an allegory is its adjoint.
Yeah, I do know that! Strangely, BTW, Freyd-Scedrov define a map in an allegory to be a morphism $R: A \to B$ such that $R^{op}: B \to A$ is its right adjoint, instead of simply as a morphism that possesses a right adjoint (and then proving the right adjoint must be $R^{op}$). Maybe Johnstone proves this in the Elephant; I haven’t checked.
Yeah, he does.
Thanks, Mike! The question of allegories being property-like, while a natural one to ask, is not urgent for me; I just wondered whether you or Finn or someone else happened to know. I can’t tell whether a negative answer would make allegories even more or even less alluring to me, but I suspect “less”.
For what it’s worth: I can show that a (locally posetal) cartesian bicategory carries at most one allegory structure, and this occurs precisely if it’s a bicategory of relations. Meanwhile, cartesian bicategories are property-like with respect to 2-categories. I think these observations suffice for my immediate purpose.
Hopefully the experts here can help a newbie trying to understand allegories. The current definition of distributive allegory says:
A distributive allegory is an allegory whose hom-posets have finite joins that are preserved by composition. Thus a distributive allegory is locally a lattice.
Based on Freyd-Scedrov I wonder whether it should say something like:
A distributive allegory is an allegory whose hom-posets have finite joins that are preserved by composition and that satisfy the distributivity law. Thus a distributive allegory is locally a distributive lattice.
Is this a mistake?
I think you’re right. I’ve fixed it, mentioning also the weaker notion under the name “union allegory” (which is used in the Elephant).
This reminds me that there are still some loose ends in the alternative account of power allegories (“original research”). I should get back to that.
Thanks! I didn’t know about the weaker notion of “union allegory.”
Added a reference to Michael Winter, Goguen Categories. Therein the author develops an application to the construction of fuzzy controllers.
Added missing properties. Moreover, I changes “(1,2)-category” to “locally posetal 2-category” because the former is only stated as a notion depending on a notion of $\infty$-category.
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