Added another basic property.
]]>Corrected to 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 -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 such that is its right adjoint, instead of simply as a morphism that possesses a right adjoint (and then proving the right adjoint must be ). 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 ? The issue is whether there is at most one “opposite” operation that makes 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 is -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).
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