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January 2025 Archives | The n-Category Café
2 days ago · We’re brought up to say that the dual concept of injection is surjection, and of course there’s a perfectly good reason for this. The monics in the category of sets are the injections, the epics are the surjections, and monics and epics are dual concepts in the usual categorical sense.
Galois Theory | The n-Category Café - University of Texas at Austin
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.
The n-Category Café - University of Texas at Austin
Dec 15, 2024 · Previously: Part 9. The course is over! The grand finale was the theorem that. X × Y ≅ X + Y ≅ max (X, Y) X \times Y \cong X + Y \cong max(X, Y) . for all infinite sets X X and Y Y.Proving this required most of the concepts and results from the second half of the course: well ordered sets, the Cantor–Bernstein theorem, the Hartogs theorem, Zorn’s lemma, and so on.
Random Permutations (Part 14) | The n-Category Café
Dec 17, 2024 · I see; very interesting! So in this kind of situation, it can be easier to work with a Poisson random variable with mean n n than it is to work with the plain old number n n itself. I like it. To implement a similar idea here — that is, to relate the statistics of random permutations and random endomorphisms — we’d surely need to know …
Axiomatic Set Theory 7: Number Systems | The n-Category Café
Previously: Part 6.Next: Part 8. As the course continues, the axioms fade into the background. They rarely get mentioned these days. Much more often, the facts we’re leaning on are theorems that were deduced from theorems that were deduced — at several removes — from the axioms.
Geometry and String Theory - University of Texas at Austin
Feb 3, 2018 · Possible list of Topics for Spring 2025 Liouville Theory. Chatterje and Witten, “Liouville Theory: An Introduction to Rigorous Approaches” We mentioned “7” and “23” which are referred in the above paper. Sheffield, “What is a random surface? Benjamin, Collier, Maloney and Meruliya “Resurgence in Liouville Theory” Gaiotto’s papers
Grothendieck--Galois--Brauer Theory (Part 6) | The n-Category Café
Oct 27, 2023 · Some results I’m not sure if you are aware of, but as a supplement, they are also very interesting. In arXiv:1706.06904, Kong and Zheng discussed separable algebras and semisimple algebras in a more general context.They generalized the definition of division algebras, matrix algebras in a semisimple monoidal category.
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 …
Grothendieck--Galois--Brauer Theory (Part 1) | The n-Category Café
Jun 26, 2023 · Yes, the structure of Spec (k) Spec(k) as a scheme is much more interesting than its structure as a mere topological space; this is one reason I called it a “funny sort of space”.. In particular, the étale topology (which is a Grothendieck topology on the category of schemes) allows us to define interesting covering spaces of Spec …