Axiomatic Set Theory 9: The Axiom of Choice | The n-Category Café
Nov 15, 2024 · Previously: Part 8.Next: Part 10. It’s the penultimate week of the course, and up until now we’ve abstained from using the axiom of choice.But this week we gorged on it. We proved that all the usual things are equivalent to the axiom of choice: Zorn’s lemma, the well ordering principle, cardinal comparability (given two sets, one must inject into the other), and the souped-up version of ...
Galois Theory | The n-Category Café
Aug 15, 2024 · Galois Theory Posted by Tom Leinster. I’ve just arXived my notes for Edinburgh’s undergraduate Galois theory course, which I taught from 2021 to 2023.. I first shared the notes on my website some time ago. But it took me a while to arXiv them, because I wanted to simultaneously make public most of the other course materials.
Dimensional Analysis in Algebra and Geometry
Aug 8, 2024 · Ancient Greek mathematicians, so I’ve heard, understood multiplication geometrically. To reason about A × B A \times B they would draw line segments of length A A and length B B and form a rectangle with those as sides. This has area A × B A \times B.. Fine and dandy. But then what does an equation like 2 x 2 = x + 3 2x^2 = x + 3 mean? 2 x 2 2x^2 has units of area, x x has units of length ...
Adjoint School 2025 | The n-Category Café
Oct 31, 2024 · The quantum computation topic, and mentor, have changed to this: Structuring Quantum Effects with Monads and Arrows. Mentor: Juliana Kaizer Vizzotto Category theory provides a mathematical framework for understanding structures and their relationships abstractly.
What Is Entropy? | The n-Category Café - University of Texas at …
Jul 20, 2024 · Those other quantities are important, but they’re not called ‘temperature’. They’re called the ‘thermodynamic conjugates’ of the conserved quantities in question. I haven’t heard people discuss the thermodynamic conjugate of angular momentum or momentum, but Landau uses these (perhaps implicitly—I forget) to argue that an object in …
Stirling's Formula from Statistical Mechanics | The n-Category Café
Aug 25, 2024 · Very nice! It’s pleasing to find a real-world interpretation to motivate otherwise obscure calculations. Regarding the point at the end, about how the standard Central Limit Theorem doesn’t imply pointwise convergence of densities, the keyword for the kind of result you need is “local limit theorem”.Here
January 2024 Archives | The n-Category Café - University of Texas …
Jan 29, 2024 · Guest post by Matthew di Meglio. Recently, my PhD supervisor Chris Heunen and I uploaded a preprint to arXiv giving an axiomatic characterisation of the category FCon \mathbf{FCon} of finite-dimensional Hilbert spaces and linear contractions. I thought it might be nice to explain here in a less formal setting the story of how this article came to be, including some of the motivation, ideas ...
ACT 2025 | The n-Category Café
Dec 4, 2024 · Submission Important dates . All deadlines are AoE (Anywhere on Earth). February 26: title and brief abstract submission; March 3: paper submission
Introduction to Categorical Probability | The n-Category Café
Aug 13, 2024 · and the identities as x ↦ {x} x \mapsto \{x\} gives us the Markov category FinSetMulti \mathsf{FinSetMulti} of possibilities!. The same data from the example can be used in a possibilistic way as well; a channel S → W ⊗ H ⊗ T S \to W\otimes H\otimes T can map the sprinkler to all the possible states of weather/climate where the sprinkler has turned on etc.
Random Permutations (Part 14) | The n-Category Café
3 days ago · I want to go back over something from Part 11, but in a more systematic way that I hope is easier to follow.. Namely, I want to prove a wonderful fact about random permutations, the Cycle Length Lemma, using a bit of category theory.