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Language, Statistics, & Category Theory, Part 1
In the previous post I mentioned a new preprint that John Terilla, Yiannis Vlassopoulos, and I recently posted on the arXiv. In it, we ask a question motivated by the recent successes of the world's best large language models: "What's a mice mathematical framework in which to explain the passage from probability distributions on text to syntactic and semantic information in language?" To understand the motivation behind this question, and to recall what a "large language model" is, I'll encourage you to read the opening article from last time. In the next few blog posts — starting today — I'll give a tour of mathematical ideas presented in the paper towards answering the question above. I like the narrative we give, so I'll follow it closely here on the blog. You might think of the next few posts as an informal tour through the formal ideas found in the paper.
Language, Statistics, & Category Theory, Part 1
https://www.math3ma.com/blog/language-statistics-category-theory-part-1
Language, Statistics, & Category Theory, Part 1
Universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property.
Universal property
https://en.wikipedia.org/wiki/Universal_property
Universal property
Category Theory
Category theory for programmers by Bartosz Milewski. Seattle, Summer 2016. Additional material at https://bartoszmilewski.com/2014/10/28/category-theory-for-...
Category Theory
https://www.youtube.com/playlist?list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_
Category Theory

Introducció

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A Diagram is a Functor
Last week was the start of a mini-series on limits and colimits in category theory. We began by answering a few basic questions, including, "What ARE (co)limits?" In short, they are a way to construct new mathematical objects from old ones. For more on this non-technical answer, be sure to check out Limits and Colimits, Part 1. Towards the end of that post, I mentioned that (co)limits aren't really related to limits of sequences in topology and analysis (but see here). There is however one similarity. In analysis, we ask for the limit of a sequence. In category theory, we also ask for the (co)limit OF something. But if that "something" is not a sequence, then what is it? Answer: a diagram.
A Diagram is a Functor
https://www.math3ma.com/blog/a-diagram-is-a-functor
A Diagram is a Functor
What is Category Theory Anyway?
A quick browse through my Twitter or Instagram accounts, and you might guess that I've had category theory on my mind. You'd be right, too! So I have a few category-theory themed posts lined up for this semester, and to start off, I'd like to (attempt to) answer the question, What is category theory, anyway? for anyone who may not be familiar with the subject.
What is Category Theory Anyway?
https://www.math3ma.com/blog/what-is-category-theory-anyway
What is Category Theory Anyway?
Aquest canal de YT té moltes playlists sobre mates.
arisbe, a guess at the riddle
UNIX, Web3, Math, Complex Systems Theory, Psychology, Neuroscience, Western Culture, Meaning, Linguistics, Rhetoric, Research ---------------------------------------------------------------- "A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies." --Stefan Banach ---------------------------------------------------------------- For Natural Science, Engineering, HPC, Algorithms, Business, Social Science, Philosophy, Big History see *tuva or bust!*. For Music Playlists, Music Theory, Sound Production, Music History, etc.. see *songs of innocence and experience*. *NOTE: You have to click on the title of a multiple playlist (playlist of playlists) to see all the playlists if there are a lot.*
arisbe, a guess at the riddle
https://www.youtube.com/@a-guess-at-the-riddle/playlists?view=50&sort=dd&shelf_id=3
arisbe, a guess at the riddle

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