Ph34r my m4d c4t3g0r1c4l sk1llz!

[pozorvlak]

Here is the written-up (and more cautious) version of the talk I'm giving tomorrow, at the 83rd Peripatetic Seminar on Sheaves and Logic. All comments very gratefully received, though I doubt I'll have time to read them before I give the talk.

[What, steerpikelet gets to post her essays and I don't? :-)]

Comments

[antoniabaker]

my dear, you left out the most important P caterfoires, punning, pantomime, promisicuity, pedantry and pgin. Looks incredibly impressive though.

[pozorvlak]

And of course, I've proved that each of those is equivalent to some strict P-category. I have no idea what strict promiscuity would mean, but it sounds like it would be a lot of effort.

[saf2285]

I couldn't understand the abstract, and the opening paragraph finished me off. It sounds suitably complicated though. Well done!

[pozorvlak]

It's not meant to be complicated, it's meant to be elegant and self-evidently true :-( Sadly, it's neither of these things at the moment. The penultimate section is new, however, and that has the potential to simplify things a bit.

[pozorvlak]

Yes, for some unknown reason the theory of monoidal categories is not widely taught at school (though it has been suggested, and someone even wrote a school maths textbook based on category theory). I've written another post which may or may not make things clearer - the document I link to in this post is written research-paper style, ie fairly compressed and assuming you already know roughly what's going on.

[steerpikelet]

Well, it looks very pretty. Now I wish I weren't studying such a transferable subject...
Mdear, what's your phone number? I lost my phone last week and tried to ring you yesterday and couldn't just text me if you don't want to paste on line. my number's still the same.

[pozorvlak]

This is transferable! :-) We had people talking about quantum mechanics, computer security, and all sorts. One guy even mentioned cooking, but I think he might have been joking.

[svintusoid]

Have you any definition for general (not strongly regular)theories now?
I've seen your paper some months ago, but have preferred to use in my work Hu-Kriz-Fiore approach to categorification of algebraic structures(because of its generality and compatibility with theory of Lawvere theories over Cat). :)

P.S.
I'm sorry for my probably bad English. My native language is russian :).

[pozorvlak]

Your English is fine :-)

I have a few candidate definitions for non-strongly-regular theories, but nothing that I'm happy with yet. The important thing to note is that the HKF approach, at least as published currently, is wrong: it gives the Wrong Thing even in quite simple cases (for instance, an HKF-weak commutative monoid is a symmetric monoidal category in which \tau_AA = id_{A*A}, which is not true in most interesting cases). Their definition works in the strongly regular case, when it agrees with (and is very similar to) my definition

[svintusoid]

Shortly it could be seen as "modelling euristics of finding "normal form" in universal algebras".

Let we have an algebra with generators and with unknown relations( for example as subalgebra of free algebra-- situation from, for example, classical invaraint theory, from which this work has grown up). We want to model iterative euristics(and algorythms) of finding some "normal form" for its elements.

Now, i'll try to describe the way of modelling:

I parametrize "level of detalization" by "good" sections(functor from Rel to L-Alg(Cat)) of the following diagram(where L- Lawvere theory, L-Alg(-)-- categories of L-Algebras over Cat or Set, respectively, Rel-- category of surjective morphisms from X-generated free algebra, and arrows are, respectively, forgetting of surjective morphism and decategorification):

L-Rel(X)-->L-Alg(Set)<--L-Alg(Cat

[svintusoid]

The last part of my previous comment could be seen as modification of classical Arbib-Manes approach to systems in categories