Posts with this logo use 
November 18, 2018
Modal Types Revisited
Posted by David Corfield
We’ve discussed the prospects for adding modalities to type theory for many a year, e.g., here at the Café back at Modal Types, and frequently at the nLab. So now I’ve written up some thoughts on what philosophy might make of modal types in this preprint. My debt to the people who helped work out these ideas will be acknowledged when I publish the book.
This is to be the fourth chapter of a book which provides reasons for philosophy to embrace modal homotopy type theory. The book takes in order the components: types, dependency, homotopy, and finally modality.
The chapter ends all too briefly with mention of Mike Shulman et al.’s project, which he described in his post – What Is an n-Theory?. I’m convinced this is the way to go.
PS. I already know of the typo on line 8 of page 4.
November 15, 2018
Magnitude: A Bibliography
Posted by Tom Leinster
I’ve just done something I’ve been meaning to do for ages: compiled a bibliography of all the publications on magnitude that I know about. More people have written about it than I’d realized!
This isn’t an exercise in citation-gathering; I’ve only included a paper if magnitude is the central subject or a major theme.
I’ve included works on magnitude of ordinary, un-enriched, categories, in which context magnitude is usually called Euler characteristic. But I haven’t included works on the diversity measures that are closely related to magnitude.
Enjoy! And let me know in the comments if I’ve missed anything.
November 12, 2018
A Well Ordering Is A Consistent Choice Function
Posted by Tom Leinster
Well orderings have slightly perplexed me for a long time, so every now and then I have a go at seeing if I can understand them better. The insight I’m about to explain doesn’t resolve my perplexity, it’s pretty trivial, and I’m sure it’s well known to lots of people. But it does provide a fresh perspective on well orderings, and no one ever taught me it, so I thought I’d jot it down here.
In short: the axiom of choice allows you to choose one element from each nonempty subset of any given set. A well ordering on a set is a way of making such a choice in a consistent way.
November 2, 2018
More Papers on Magnitude
Posted by Simon Willerton
I’ve been distracted by other things for the last few months, but in that time several interesting-looking papers on magnitude (co)homology have appeared on the arXiv. I will just list them here with some vague comments. If anyone (including the author!) would like to write a guest post on any of them then do email me.
For years a standing question was whether magnitude was connected with persistent homology, as both had a similar feel to them. Here Nina relates magnitude homology with persistent homology.
- Magnitude cohomology by Richard Hepworth
In both mine and Richard’s paper on graphs and Tom Leinster and Mike Shulman’s paper on general enriched categories, it was magnitude homology that was considered. Here Richard introduces the dual theory which he shows has the structure of a non-commutative ring.
- Smoothness filtration of the magnitude complex by Kiyonori Gomi
I haven’t looked at this yet as I only discovered it last night. However, when I used to think a lot about gerbes and Deligne cohomology I was a fan of Kiyonori Gomi’s work with Yuji Terashima on higher dimensional parallel transport.
- Graph magnitude homology via algebraic Morse theory by Yuzhou Gu
This is the write-up of some results he announced in a discussion here at the Café. These results answered questions asked by me and Richard in our original magnitude homology for graphs paper, for instance proving the expression for magnitude homology of cyclic graphs that we’d conjectured and giving pairs of graphs with the same magnitude but different magnitude homology.
November 1, 2018
2-Groups in Condensed Matter Physics
Posted by John Baez
This blog was born in 2006 when a philosopher, a physicist and a mathematician found they shared an interest in categorification — and in particular, categorical groups, also known as 2-groups. So it’s great to see 2-groups showing up in theoretical condensed matter physics. From today’s arXiv papers:
- J.P. Ang and Abhishodh Prakash, Higher categorical groups and the classification of topological defects and textures.
Abstract. Sigma models effectively describe ordered phases of systems with spontaneously broken symmetries. At low energies, field configurations fall into solitonic sectors, which are homotopically distinct classes of maps. Depending on context, these solitons are known as textures or defect sectors. In this paper, we address the problem of enumerating and describing the solitonic sectors of sigma models. We approach this problem via an algebraic topological method – combinatorial homotopy, in which one models both spacetime and the target space with algebraic objects which are higher categorical generalizations of fundamental groups, and then counts the homomorphisms between them. We give a self-contained discussion with plenty of examples and a discussion on how our work fits in with the existing literature on higher groups in physics.
The fun will really start when people actually synthesize materials described by these materials! Condensed matter physicists are doing pretty well at realizing theoretically possible phenomena in the lab, so I’m optimistic. But I don’t think it’s happened yet.