Grothendieck's "Tohoku" paper | The n-Category Café
WEBSep 7, 2010 · Grothendieck’s “Tohoku” paper. Posted by John Baez. Michael Barr has spent a lot of time and effort translating Grothendieck’s legendary “ Tohoku ” paper into English. If you don’t mind violating Grothendieck’s stated wishes, you can now read this translation: Alexandre Grothendieck, On certain aspects of homological algebra ...
Timing, Span (Graph) and Cospan (Graph) | The n-Category Café
WEBAug 1, 2022 · This is a category whose objects are graphs, morphisms are cospans X → Z ← Y of graphs and composition is given by pushout, which exist for Graph. Cospan(Graph) is a symmetric monoidal category due to the cocartesian monoidal structure on Graph. For a set A, define the graph 𝒱(A) ≡ (A, ∅) to contain no edges.
How the Simplex is a Vector Space | The n-Category Café
WEBJun 11, 2016 · I think I’ve seen Bayesians say that same thing on the other side of the isomorphism ℝ ≅ (0, ∞) \mathbb{R}\cong (0,\infty).That is, if you take the logarithms of the prior probability and the likelihood ratio to get a “logarithmic measure of odds”, then you can add them together (with the usual notion of “addition”) …
Axioms for Infinitesimal Cohesion | The n-Category Café
WEBApr 12, 2011 · Here is the simple idea of infinitesimal cohesion: any coreflective embedding of categories (and then of cohesive ∞ \infty-toposes) i: H ↪ H th i : \mathbf{H} \hookrightarrow \mathbf{H}_{th} can be understood as providing in H th \mathbf{H}_{th} infinitesimal thickenings of the objects of H \mathbf{H}.We think of the coreflector Red: H th → H Red : \mathbf{H}_{th} \to \mathbf{H} as ...
University of Texas at Austin
WEB% UT Physics bibliographic style, ver. 3.1. Based on: % %X IEEE Transactions bibliography style (29-Jan-88 version) %X numeric labels, order-of-reference, IEEE abbreviations, %X quotes around article titles, commas separate all …
The Binary Octahedral Group | The n-Category Café
WEBDec 25, 2021 · Our friend the binary octahedral group is 2 ⋅ S − 4. But 2 ⋅ S + 4 is also interesting: it’s GL(2, 𝔽3), the group of invertible 2 × 2 matrices with entries in the field with 3 elements! And it’s the symmetry group of the most symmetrical Riemann surface of genus 2: the Bolza curve.
G2 and the Rolling Ball | The n-Category Café - University of Texas …
WEBJun 7, 2013 · I just returned from a month at Hong Kong University, visiting James Fullwood, an algebraic geometer who likes to think about the mathematics of string theory.There, I gave a colloquium on G 2 and the rolling ball, a paper John Baez and I wrote that is due to appear in Transactions of the AMS.This project began over a decade ago in conversations between John and Jim Dolan, later continued ...
Mapping (Co)cylinder Factorizations via the Small Object Argument
WEBThis says that any attaching map from an element of I must factor through a previous stage. So the enriched algebraic small object argument converges at step two. And indeed, the factorization (* *) is exactly the mapping cylinder factorization! By definition (Mf)n = Xn ⊕ Xn − …
Exact Squares | The n-Category Café - University of Texas at Austin
WEBJun 21, 2010 · In order to get used to the calculus of exact squares, the first thing we have to change is to start thinking more in terms of Kan extensions rather than limits and colimits.For instance, it’s usual to think of a pullback as the limit of a diagram of shape B → C ← A B \to C \leftarrow A (a cospan) i.e. an object P P equipped with a natural transformation of cospans: P → P ← P ↓ ↓ ...
On the Bar Construction | The n-Category Café - University of …
WEBMay 31, 2007 · As John said, the bar construction is a wonderful gadget which can be used to compute cohomology for general algebraic theories (groups, algebras, Lie algebras, …), as well as for other things like constructing classifying bundles. Given any monad M M and an M M -algebra X X, the bar construction is an efficient machine that produces a ...