The Quintic, the Icosahedron, and Elliptic Curves
WEBApr 18, 2024 · Old-timers here will remember the days when Bruce Bartlett and Urs Schreiber were regularly talking about 2-vector spaces and the like. Later I enjoyed conversations with Bruce and Greg Egan on quintics and the icosahedron.
Ringoids | The n-Category Café - University of Texas at Austin
WEBSep 2, 2006 · Urs writes: But while groups and groupoids have nice arrow-theoretic definitions, the definition of a Lie algebroid (→) is a mess, comparatively.Indeed! On page 43 here I show a hypothesized “periodic table” of Lie n-groupoids, and on page 44 a corresponding periodic table of Lie n-algebroids. The first row of the periodic table of Lie n-algebroids is funny, because they have a smooth ...
Counting Nilpotents | The n-Category Café
WEBDec 3, 2019 · What is the probability that a random linear operator is nilpotent? Recall that a linear operator T T on a vector space V V is called “nilpotent” if T n = 0 T^n = 0 for some n n.Assuming that V V is finite-dimensional and over a finite field, there are only finitely many operators on V V.Choose one at random. It turns out that the probability of it being nilpotent is
Native Type Theory (Part 2) | The n-Category Café
WEBMar 21, 2021 · We can represent a broad class of languages as categories with products and exponents. The idea of native type theory is simply that for any such language T T, we can understand the internal language of 풫 T \mathscr{P}T as the language of T T expanded by predicate logic and dependent type theory.. The type system reasons about abstract syntax trees, the trees of constructors which form terms:
The Brunn-Minkowski Inequality | The n-Category Café
WEBJan 31, 2017 · Thanks for the correction on the measurability of A + B A + B.I think I knew that A A and B B being measurable doesn’t guarantee that A + B A + B is, but I overlooked it when I was writing, and I don’t think I’ve ever seen an explanation.. It’s interesting, because it means that the class of measurable sets isn’t as stable a setting as I’d thought.
Sphere Spectrum Analogue of PGL (2,Z) | The n-Category Café
WEBAug 31, 2020 · But in modern mathematics the sphere spectrum is what Joyal called the “true integers”: it’s the initial ring spectrum just as ℤ \mathbb{Z} is the initial ring. So there should be an enhanced version of PGL (2, ℤ) PGL(2,\mathbb{Z}) with the sphere spectrum taking over the role of the integers, and a lot should be known about it.. What’s it called, and what do people know about it?
Large Sets 9.5 | The n-Category Café - University of Texas at Austin
WEBJul 8, 2021 · Fantastic; thanks! A couple of comments, more for the record than because I think you hadn’t noticed already: In the proof of unboundedness, where we put Z = sup n ∈ ℕ Z n Z = \sup_{n \in \mathbb{N}} Z_n, we’re implicitly using the fact that X X has unbounded cofinality (which it does because it’s inaccessible, hence uncountable and regular).. The proof that L ′ L' is closed is a ...
Optimal Transport and Enriched Categories I | The n-Category Café
WEBJun 6, 2021 · Last year, in an ACT@UCR talk, I spoke about the Fenchel–Legendre transform from a category theoretic perspective and I showed how convex functions arise as a ‘profunctor nucleus’ in the context of categories enriched over the extended real numbers [− ∞, ∞] [-\infty, \infty].At the end of the talk I gave four other examples of things which arise as profunctor nuclei, the final one ...
4d QFT for Khovanov Homology | The n-Category Café
WEBFeb 4, 2011 · Khovanov homology is a knot invariant that is a categorification of the Jones polynomial.. Khovanov homology has long been expected to appear as the observables in a 4-dimensional TQFT in higher analogy of how the Jones polynomial arises as an observable in 3-dimensional Chern-Simons theory.For instance for Σ: K → K ′ \Sigma : K \to K' a cobordism between two knots there is a natural morphism
Convex Spaces | The n-Category Café - University of Texas at Austin
WEBApr 1, 2009 · good question. Another obvious change coefficient changing map is the inclusion ℝ ≥ 0 ↪ ℝ \mathbb{R}_{\geq 0}\hookrightarrow\mathbb{R} This change of coefficients functorially turns every affine space into a convex space; this functor has a left adjoint sending every convex space to its universal enveloping affine space. However due to the classification theorem of convex spaces ...