Added another basic property.

]]>Corrected $g^o f$ to $g f^o$ in definition of tabulation.

]]>Expanded on definition of map, entire morphism, and functional morphism.

]]>Added proof of distributivity of composition over meets.

]]>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.

]]>Added a reference to Michael Winter, Goguen Categories. Therein the author develops an application to the construction of fuzzy controllers.

]]>Thanks! I didn’t know about the weaker notion of “union allegory.”

]]>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.

]]>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).

]]>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?

]]>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.

]]>Yeah, he does.

]]>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.

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.

]]>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.

]]>There was a small mistake at the end of the proof I put at allegory, so I’ve put a lemma on my personal web here and referred to that instead.

]]>@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…

]]>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.

]]>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.

]]>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.

]]>Thanks, Finn!

]]>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.

]]>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?

@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).

]]>