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October 29, 2018

Bar Constructions and Combinatorics of Polyhedra for n-Categories

Posted by John Baez

Samuel Vidal has kindly LaTeXed some notes by Todd Trimble:

Todd wrote these around 1999, as far as I know. I’ve always enjoyed them; they give a clearer introduction to the bar construction than any I’ve seen, and they also suggest a number of fascinating directions for research on the relation between higher categorical structures and polyhedra.

Posted at 3:25 PM UTC | Permalink | Followups (7)

October 18, 2018

Analysis in Higher Gauge Theory

Posted by John Baez

Higher gauge theory has the potential to describe the behavior of 1-dimensional objects and higher-dimensional membranes much as ordinary gauge theory describes the behavior of point particles. But ordinary gauge theory is also a source of fascinating differential equations, which yield interesting results about topology if one uses enough analysis to prove rigorous results about their solutions. What about higher gauge theory?

Andreas Gastel has a new paper studying higher gauge theory using some techniques of analysis that are commonly used in ordinary gauge theory. He’s finding some interesting similarities but also some ways in which higher gauge theory is simpler:

Abstract. We study the problem of finding good gauges for connections in higher gauge theories. We find that, for 2-connections in strict 2-gauge theory and 3-connections in 3-gauge theory, there are local “Coulomb gauges” that are more canonical than in classical gauge theory. In particular, they are essentially unique, and no smallness of curvature is needed in the critical dimensions. We give natural definitions of 2-Yang-Mills and 3-Yang-Mills theory and find that the choice of good gauges makes them essentially linear. As an application, (anti-)selfdual 2-connections over B 6B^6 are always 2-Yang-Mills, and (anti-)selfdual 3-connections over B 8B^8 are always 3-Yang-Mills.

Posted at 9:14 PM UTC | Permalink | Post a Comment

October 14, 2018

Topoi of G-sets

Posted by John Baez

I’m thinking about finite groups these days, from a Klein geometry perspective where we think of a group GG as a source of GG-sets. Since the category of GG-sets is a topos, this lets us translate concepts, facts and questions about groups into concepts, facts and questions about topoi. I’m not at all good at this, so here are a bunch of basic questions.

Posted at 5:36 PM UTC | Permalink | Followups (63)

October 3, 2018

Category Theory 2019

Posted by Tom Leinster

The major annual category theory conference will be held in Edinburgh next year:

Category Theory 2019

University of Edinburgh

7-13 July 2019

Organizing committee: Steve Awodey, Richard Garner, Chris Heunen, Tom Leinster, Christina Vasilakopoulou.

As John has just pointed out, this is followed two days later by the Applied Category Theory conference and school in Oxford, very conveniently for anyone wishing to go to both.

Posted at 11:27 PM UTC | Permalink | Followups (2)

October 2, 2018

Applied Category Theory 2019

Posted by John Baez

I’m helping organize ACT 2019, an applied category theory conference and school at Oxford, July 15-26, 2019. Here’s a ‘pre-announcement’.

More details will come later, but here’s some good news: it’s right after the big annual worldwide category theory conference, which is in Edinburgh in 2019. So, conference-hopping category theorists can attend both!

Posted at 5:00 PM UTC | Permalink | Followups (6)