You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Hello everyone,
I've got a few new papers to advertise.
First, Protocol Choice and Iteration for the Free Cornering, together with @Niels Voorneveld. This extends the free cornering of a monoidal category to support additional protocol types, in particular branching protocols, in which one of the participants chooses which of two continuations will happen and the other must respond, and iterated protocols, in which the participants repeat all or part of a protocol some number of times, based on choices made during the process. If you have access to JLAMP you can get the "official" version, but frankly I think the preprint is more legible.
Second, my PhD thesis, Partial and Relational Algebraic Theories has been successfully defended and is available from the university library. The thesis is a more careful development of the partial algebraic theories and relational algebraic theories from my published work.
I'll mention that only the first ~150 pages of the document on that website are the thesis monograph, with the rest being copies of my papers. It isn't that scary!
Congrats on the nice looking thesis! You should post it to arXiv (minus the copied papers). That will help people find it.
Congrats!
Evan Patterson said:
Congrats on the nice looking thesis! You should post it to arXiv (minus the copied papers). That will help people find it.
I don't know whether or not I should do this. I'm planning to publish journal version of the two associated conference papers at some point using material from the thesis, so it might be better to wait. I don't really know.
What downside are you envisioning?
I don't know what Chad is thinking, but my own thesis is put together from several papers, and I never put my thesis itself on the arXiv because I would rather that people read the papers instead, as they are more polished. However, this only works if the individual papers actually do get written.
Tomorrow at 14:00 EEST (UTC+3) I'm giving a talk in Tallinn titled Elgot Categories and Abacus Programs. The talk will be broadcast via Zoom, and if you're interested in listening in send me a message here and I'll send you the details.
Abstract:
This talk concerns the categorical representation of the partial
recursive functions. Specifically, I will introduce Elgot
categories, which are a sort of structured rig category in which
all partial recursive functions are representable. Specifically, an
Elgot category is a (uniform) traced cocartesian rig category
with a distinguished isomorphism I+N→N for some object N.
I will construct an initial Elgot category, the morphisms of
which will coincide with the abacus programs of Lambek. Finally,
I will show that in this initial Elgot category, the class of
strongly representable partial functions is precisely the class
of partial recursive functions.
I guess the main references are:
Today is my first day at my new job as a Research Fellow at the University of Tartu. It's a permanent position, which is exciting.
Mike Shulman said:
I don't know what Chad is thinking, but my own thesis is put together from several papers, and I never put my thesis itself on the arXiv because I would rather that people read the papers instead, as they are more polished. However, this only works if the individual papers actually do get written.
I do eventually hope to publish the results from my MSc thesis :sweat_smile:
In other news, I'll be attending a few conferences this summer:
If you'll be at one or more of these and would like to talk about something specific, let me know and I can prepare a bit in advance :)
On Monday, July 8th at 14:30 in Tallinn I'll be speaking at MSFP 2024 about my paper "Protocol Choice and Iteration for the Free Cornering" with Niels Voorneveld.
I've put up a new thing on the arXiv: https://arxiv.org/pdf/2503.21434
It's about representing the partial recursive functions in distributive monoidal categories. The short version is that you need to be traced on the coproduct, and to have an isomorphism hanging around somewhere. I think it's fun.
I'd like to ponder something out loud here:
Let be the set of lists with elements in , defined as usual to be the initial algebra of the endofunctor on . In particular let's write the initial algebra as , so that out lists look like the ones from functional programming.
Let be a set, and suppose that we have partial functions and such that . Note as an aside that an immediate consequence of this equation is that is total.
There is a total function defined by
So for each we have a list, . We do not, however, seem to be able to construct an element of for each element of . In particular, it is unclear what we would map to.
Nonetheless, I can do quite a few list-like things with . So long as I'm happy to have my function become undefined the moment it attempts to inspect an "empty" element of (on which is undefined), there would seem to be nothing I cannot do.
Ah: a problem with this is that may in fact be total, in which case is a stream and my isn't well-defined, since I never reach a .
I guess I have the same thing happening, but with instead of . Here by I mean the terminal coalgebra of the endofunctor . So a colist is a "possibly finite stream" or equivalently a "possibly infinite list".
I would imagine that is injective, but not surjective. I think I probably need to work out some examples if I want to understand this...
I presume you meant your to be recursive on tail in the otherwise case.
Yes, thank you. I've fixed it.
Of course, a partial function is the same as a total function , so then your is exactly the unique map from a coalgebra to a terminal coalgebra.
In general that won't be injective or surjection, and I don't see see why the additional presence of push would make it injective.
Hmm, we do know that is injective because it is a section... which doesn't really seem to help here.
Right: if is undefined on two distinct elements of then is not an injection.
I think this might be a good way to talk about what's going on in Lisp with car/cdr and "improper lists"