January 29, 2009
New York City Category Seminar
Posted by John Baez
Manhattan has everything. Now it even has a category theory seminar:
- New York City Category Seminar, starting Spring 2009, weekly, Monday evenings 6:00 – 7:00 PM. Room 4421, The Graduate Center, CUNY, 365 Fifth Avenue (at 34th street, diagonally across from the Empire State Building). Organized by Noson Yanofsky.
The Third Time is the Charm
Posted by John Baez
A puzzle:
Name as many instances as you can of mathematicians or physicists who introduced a logical sequence of three concepts, each depending on the previous one, where the third was crucial for the applications they had in mind.
January 28, 2009
Can Particle Physicists Regain Control of Their Journals?
Posted by John Baez
Here’s an important article:
- Jeffrey R. Young, Physicists set plan in motion to change publishing system, Chronicle of Higher Education, January 30, 2009.
I think the article should be visible for 5 days from the above link. Later, only subscribers to the Chronicle will be able to view it here. So, read it now!
January 27, 2009
Bruce Bartlett’s Thesis
Posted by John Baez
Bruce has left Sheffield and returned to South Africa… but his thesis has hit the arXiv, so if you get lonely for him, read this!
January 26, 2009
Categorified Symplectic Geometry and the String Lie 2-Algebra
Posted by John Baez
Just in time for the workshop in Göttingen, my student Chris Rogers and I have finished a paper that uses 2-plectic geometry to give a new construction of the string Lie 2-algebra:
- John Baez and Chris Rogers, Categorified symplectic geometry and the string Lie 2-algebra.
If you have comments or corrections, I’d love to hear ‘em.
Abstract Stone Duality
Posted by David Corfield
Paul Taylor, perhaps best known to readers for his Practical Foundations of Mathematics and for his macros, has a survey paper out – Foundations for Computable Topology. This provides an introduction to a programme he’s been working on for several years, which offer us
… a purely recursive theory of elementary real analysis in which, unlike in previous approaches, the real closed interval $[0, 1]$ is compact.
January 22, 2009
Limits in the 2-Category of 2-Hilbert Spaces
Posted by John Baez
guest post by Jamie Vicary
I’m interested in understanding the limits that exist in 2Hilb, the 2-category of finite-dimensional 2-Hilbert spaces. (The category 2Vect of finite-dimensional 2-vector spaces over Vect${}_{\mathbb{C}}$ would be just as good.)
Petit Topos, Gros Topos
Posted by John Baez
As I struggle to learn algebraic geometry, I’m running into lots of questions, and I hope some of these will be fun for readers of this blog.
Here’s one: What’s the relation between the topos of sheaves on the little Zariski site and the topos of sheaves on the big Zariski site?
January 14, 2009
Hopf Algebras from Posets
Posted by John Baez
As usual, visiting my friend Bill Schmitt in DC made me think about combinatorics. A long time ago, based on ideas of his advisor Gian-Carlo Rota, he developed some nice ways to get Hopf algebras from collections of partially ordered sets, or ‘posets’:
- Bill Schmitt, Incidence Hopf algebras, Journal of Pure and Applied Algebra 96 (1994), 299–330.
The last example in this paper later became known as the ‘Connes–Kreimer’ Hopf algebra. Why? Because those guys rediscovered it and applied it to renormalization in quantum field theory!
I would like to understand Bill’s constructions in a more category-theoretic way. But this raises some questions about posets of subobjects… and I’m hoping some of you can help me out!
The Space of Robustness
Posted by David Corfield
At Tim Gowers’ blog, there has been a discussion in the ‘somewhat philosophical’ category of How can one equivalent statement be stronger than another?. Gowers gives the example of Hall’s theorem or Hall’s marriage theorem.
In graph theoretic guise,
Let $G$ be a bipartite graph with finite vertex sets $X$ and $Y$ of the same size. A perfect matching in $G$ is a bijection $f: X \to Y$ such that $x$ and $f(x)$ are neighbours for every $x \in X$.
So if members of $X$ are women and members of $Y$ are men, then in this heterosexual universe where women are only prepared to marry men, and vice versa, willingness to marry is represented by an edge between such pairs. A perfect matching sees everyone happily married to a partner. So when does this happy event occur?
Given a subset $A \subset X$, the neighbourhood $\Gamma(A)$ of $A$ is the set of vertices in $Y$ that are joined to at least one vertex in $A$. A trivial necessary condition for the existence of a perfect matching is that for every subset $A \subset X$ the neighbourhood $\Gamma(A)$ is at least as big as $A$, since it must contain $f(A)$, which has the same size as $A$. This condition is called Hall’s condition.
So, if our people can be happily married off, then for any subset of the women, the totality of men at least one of these women is happy to marry is at least as numerous. This is trivially the case, since this totality includes the men they do actually marry.
Much less obviously, Hall’s theorem states that the condition is also sufficient. So long as for every subset of women the subset of men loved by at least one of them is no less numerous, then everyone can be married off happily.
January 12, 2009
Truth as Value and Duty
Posted by David Corfield
Motivation for the Café and nLab:
Mathematical wisdom, if not forgotten, lives as an invariant of all its (re)presentations in a permanently self–renewing discourse.
This is from Yuri Manin’s Truth as value and duty: lessons of mathematics.
January 8, 2009
Ben-Zvi on Geometric Function Theory
Posted by Urs Schreiber
guest post by David Ben-Zvi
Dear Café Patrons,
In this guest post I want to briefly discuss correspondences, integral transforms and their categorification as they apply to representation theory, a topic that might be called geometric function theory.
This has been one of the fundamental paradigms of geometric representation theory (together with localization of representations and, on a much grander scale, the Langlands program) for at least the past twenty years (some key names to mention in this context prior to the last decade are Kazhdan, Lusztig, Beilinson, Bernstein, Drinfeld, Ginzburg, I. Frenkel, Nakajima and Grojnowski).
These ideas are closely related to topics often discussed in this Café (in particular groupoidification and extended topological field theories), and my hope is to facilitate communication between the schools by focussing on some toy examples of geometric function theory and suppressing techincal details.
I will conclude self-centeredly by describing my recent work Integral transforms and Drinfeld centers in derived algebraic geometry (arXiv:0805.0157) with John Francis and David Nadler, in which we prove some basics of categorified function theory using tools from higher category theory and derived algebraic geometry. (Of course this is out of all proportion to its role relative to the seminal works I glancingly mention or omit, but it provides my excuse to be writing here, so please indulge me.)
We were motivated by a desire to understand aspects of one of the more exciting developments of the past five years, namely the convergence of categorified representation theory (in particular the geometric Langlands program), derived algebraic geometry and topological field theory, which was at the center of last year’s special program at the IAS.
I will not attempt to describe these developments here but refer readers to collected lecture notes on my webpage.
I would also like to apologize in advance for the highly informal and imprecise style, inaccuracies and mis- or un-attributions. Finally I want to thank Urs Schreiber and Bruce Bartlett for their encouragement towards the writing of this post.
[The guest post continues in this pdf file:]
David Ben-Zvi, Geometric function theory (pdf, 8 pages).
Categories, Logic and Physics in London
Posted by David Corfield
I and many others braved the British winter and failing rail network to attend the 4th CLP Workshop, held yesterday at Imperial College London.
I was going to say something about Ieke Moerdijk’s talk on ‘Infinity categories and infinity operads’, but I see Urs has already published notes on a very similar talk given in Lausanne last November. (Those operads in the diagram at the top of page 3 are coloured, by the way.) This is part of an effort to provide a new definition of $n$-category.
January 4, 2009
nLab – General Discussion
Posted by Urs Schreiber
The comment section of this entry is the place to post contributions to general discussion concerning the $n$Lab, the wiki associated with this blog.
Discussion previously held at the $n$Lab entry General Discussion should eventually migrate here.
Notice that, when posting a comment here which does not reply to a previous comment but starts a new thread, you can and should choose a new descriptive headline in the little box above the edit pane in which you type your message.