It's #CategoryTheory #Caturday time y'all! This week I continued working on Article III of L&S, but got stumped on Exercise 14. Fortunately, Alphonse was being super cute nuzzling Konshu and it gave me an idea for a map which met my requirements! Yay for #math #cats! I'm not quite as confident as I'd like to be, but I'm hoping my confusion will get cleared up later. https://rruff82.github.io/LS-Categories/article3-p2.html

The slides from this talk on #AI and #CategoryTheory have been posted and the recording is now available on YouTube and Vimeo!

In case you're looking for something to do in a bit over an hour, I'm giving a Zoom talk for the University of Nottingham's #MachineLearning seminar at 3pm BST on discrete neural nets and polymorphic learning. You can find a link to the abstract at https://kasprzyk.work/seminars/ml.html and you can find the preprint I'm discussing at https://arxiv.org/abs/2308.00677. In case you'd like to watch later, you'll be able to find a recording on the seminar page.

#AI #CategoryTheory #algebra #combinatorics

Oh man it's #CategoryTheory #Caturday! I wanted to take my time with my #math, so after a few days of convincing myself that the existence of a fixed point in an involution provide a form of parity on maps in L&S Article 3, I decided to take a break. This allowed me to get some other things done, like blogging about this project and the joys of having #CatsOfMastodon on my feed. I felt a bit like my #cats here, with so many toys I don't know what to play with! https://rruff82.github.io/LS-Categories/article3.html

we’ll, okay, I have some issues with how multiplications are defined in (this presentation of) smooth infinitesimal analysis, leading to the field of infinitesimals (instead of the “usual” field of reals).

I think Bell’s approach here may be too brief, and he could do more to distinguish the category theoretic definition from the traditional Dedekind definition.

Of course, he’s trying to keep the resulting calculus familiar and digestible, and not overwhelm the reader with topos needed to generate viable axioms for SIA.

My concern is that if I have to relearn the foundations before reassuring myself that the calculus is well-defined—and only with “nice” functions representable by Taylor series—then that’s a big challenge 🥲

#calculus #analysis #CategoryTheory

It's technically still #Caturday so my #CategoryTheory for the week isn't late! Any confidence I might have potentially built with "Test 1" went out the window as I attempted to prove Brouwer's theorems this week. Seriously though, the level of difficult jump between Sessions 9 and 10 was a little extreme. My only saving grace was that L&S acknowledge this fact in the introduction. I didn't know if I would make it through all 4 exercises this week, but I gave it my best effort. In feline news, Edward has been diagnosed with hyperthyroidism while Alphonse needed his insulin dosage upped to help manage his diabetes. Our vet made a wise crack about doing regular blood transfusions between the two of them. There's some kind of connection between that joke and the lack of retraction from E to I, but I'm too exhausted to make it. To make up for the semi-rushed #math this week, here's both #cats: Edward hiding from his medication and Alphonse mooching for it. https://rruff82.github.io/LS-Categories/session10.html

does abstract interpretation need the full (category-theoretic) generality of adjunction or does (order-theoretic) gallois connection suffice? i guess maybe it depends on what kind of abstraction you're doing? are there good concrete examples where you need the former?

#categoryTheory #abstractInterpretation

#nLab has an entry on #Currying, but it doesn't (seem to?) mention a sum type #dual. https://ncatlab.org/nlab/show/currying #CategoryTheory

@standefer
#logic #categoryTheory
I am curious about the meaning of monotonicity in the face of the tensor-hom adjunction. In the various RMn, we may have\[
((A \otimes B) \Rightarrow C) \cong (A \Rightarrow (B \Rightarrow C))
\]but not\[
((A \otimes B) \le C) \cong (A \le (B \Rightarrow C))
\]where \( \le \) is the ordering on the lattice. Does it matter? In the first case, we *still* have an adjunction, it's just "disconnected" from the lattice. What does that mean?

Although category theory has always been on my list of things to learn I never really bothered to read up on it until i came across this video. It explains it so beautifully. And the fact that such a high quality video is this youtuber's first ever upload makes me excited for all his future videos.

DROP EVERYTHING AND WATCH THIS NOW.

https://www.youtube.com/watch?v=FQYOpD7tv30

#categorytheory #functionalprogramming #mathematics #programming

Inquiry Into Inquiry • Discussion 9
• http://inquiryintoinquiry.com/2023/08/19/inquiry-into-inquiry-discussion-9/

Re: Milo Gardner
• https://www.academia.edu/community/VBqzR5?c=Q4jJVy

MG: ❝Do you agree that Peirce was limited to bivalent logic?❞

Taking classical logic as a basis for reasoning is no more limiting than taking Dedekind cuts as a basis for constructing the real number line. For Peirce's relational approach to logic as semiotics the number of dimensions in a relation is more important than the number of values in each dimension. That is where 3 makes a difference over 2.

#Peirce #Logic #Inquiry #Inference #Information #InformationFusion #Semiotics
#CategoryTheory #Compositionality #RelationTheory #TriadicRelationIrreducibility

Yay! It's another #CategoryTheory #Caturday! Do you ever get the feeling that I'm trying to bribe you into grading my #math with pics of my #cats? Is it working? This week I completed my first "test" in L&S, but I thought it'd be better etiquette if kept it in separate PDF and only shared the link through DMs. If you're willing to look at it, let me know! The hard part of doing math on your own is that sometimes it "looks right" but you're not quite sure. It's kinda like how Edward here is not quite sure about this new toy from our nephew. Is it noms? https://rruff82.github.io/LS-Categories/test1.html

Inquiry Into Inquiry • Discussion 8
• http://inquiryintoinquiry.com/2023/08/18/inquiry-into-inquiry-discussion-8/

Re: Milo Gardner
• https://www.academia.edu/community/Lbxjg5?c=yqXVog

MG: ❝Peirce sensed that bivalent syntax was superceded by trivalent syntax,
but never resolved that nagging question.❞

My Comment —

The main thing is not a question of syntax but a question of the mathematical models we use to cope with object realities and real objectives (pragmata). Signs, syntax, and systems of representation can make a big difference in how well they represent the object domain and how well they serve the purpose at hand but they remain accessory to those objects and purposes.

#Peirce #Logic #Inquiry #Inference #Information #InformationFusion #Semiotics
#CategoryTheory #Compositionality #RelationTheory #TriadicRelationIrreducibility

Inquiry Into Inquiry • Discussion 7
• http://inquiryintoinquiry.com/2023/08/17/inquiry-into-inquiry-discussion-7/

Dan Everett has prompted a number of discussions on Facebook recently which touch on core issues in Peirce's thought — but threads ravel on and fray so quickly in that medium one rarely get a chance to fill out the warp. Not exactly at random, here's a loose thread I think may be worth the candle.

Re: Facebook • Daniel Everett
• https://www.facebook.com/permalink.php?story_fbid=pfbid0be89MXhhCm8rxahRn4PXif6HHSCmkdiUFfMZ3qS1mNqSzRzUWfqej5a8cyz8TcyJl&id=100093271525294

My Comment —

Compositionality started out as a well-defined concept, arising from the composition of mathematical functions, abstracted to the composition of arrows and functors in category theory, and generalized to the composition of binary, two-place, or dyadic relations. In terms of linguistic complexity it's associated with properly context-free languages. That all keeps compositionality on the dyadic side of the border in Peirce's universe. More lately the term has been volatilized to encompass almost any sort of information fusion, which is all well and good so long as folks make it clear what they are talking about, for which use the term “information fusion” would probably be sufficiently vague.

#Peirce #Logic #Inquiry #Inference #Information #InformationFusion #Semiotics
#CategoryTheory #Compositionality #RelationTheory #TriadicRelationIrreducibility

Inquiry Into Inquiry • Discussion 7
• http://inquiryintoinquiry.com/2023/08/17/inquiry-into-inquiry-discussion-7/

Dan Everett has prompted a number of discussions on Facebook recently which touch on core issues in Peirce's thought — but threads ravel on and fray so quickly in that medium one rarely get a chance to fill out the warp. Not exactly at random, here's a loose thread I think may be worth the candle.

Re: Facebook • Daniel Everett
• https://www.facebook.com/permalink.php?story_fbid=pfbid0be89MXhhCm8rxahRn4PXif6HHSCmkdiUFfMZ3qS1mNqSzRzUWfqej5a8cyz8TcyJl&id=100093271525294

My Comment —

Compositionality started out as a well-defined concept, arising from the composition of mathematical functions, abstracted to the composition of arrows and functors in category theory, and generalized to the composition of binary, two-place, or dyadic relations. In terms of linguistic complexity it's associated with properly context-free languages. That all keeps compositionality on the dyadic side of the border in Peirce's universe. More lately the term has been volatilized to encompass almost any sort of information fusion, which is all well and good so long as folks make it clear what they are talking about, for which use the term “information fusion” would probably be sufficiently vague.

#Peirce #Logic #Inquiry #Inference #Information #InformationFusion #Semiotics
#CategoryTheory #Compositionality #RelationTheory #TriadicRelationIrreducibility

Just landed in LA after a friend's wedding, and the fact that everything in sight is labeled "LAX parking" etc. is filling my head with the world's dumbest #categorytheory jokes

I don't know about you, but I'm totally ready for more #CategoryTheory #Caturday! I've been out of town for the last two weeks and it's so good to see my boys again! I reviewed my solutions from the quiz on my flight home, and worked through the "opposed maps" stuff in L&S in preparation for the upcoming "test". Now if you'll excuse me, it's time for a much needed "hide in makeshift cave day". #math #cats https://rruff82.github.io/LS-Categories/quiz2.html

This was the one from last time #categorytheory #meme (repost from twitter)

has anyone written up an explanation of Yoneda in terms of logic? in particular it seems like a “metatheorem” about category theory with a similar kind of structure (and implications) to cut and identity admissibility as metatheorems about logics. is there anything there?

#categorytheory #logic #typetheory

Pushed a huge update of the second chapter of #categorytheoryillustrated - the one where categories are introduced: rewrote a lot of stuff, corrected many mistakes and added an overview of Lawvere's Elementary theory of the category of sets, right before the concept of a category is introduced, section "Defining the rest of set theory using functions".

Worked pretty great, I think.

https://abuseofnotation.github.io/category-theory-illustrated/02_category/

#categorytheory #math #mathematics #iamWriting #functionalprogramming