September 27, 2021

Weakly Globular Double Categories: a Model for Bicategories

Posted by Emily Riehl

guest post by Claire Ott and Emma Phillips as part of the Adjoint School for Applied Category Theory 2021.

As anyone who has worked with bicategories can tell you, checking coherence diagrams can hold up the completion of a proof for weeks. Paoli and Pronk have defined weakly globular double categories, a simplicial model of bicategories which is a sub-2-category of the 2-category of double categories. Today, we’ll introduce weakly globular double categories and briefly talk about the advantage of this model. We’ll also take a look at an application of this model: the SIRS model of infectious disease.

Posted at 11:32 PM UTC | Permalink | Followups (25)

September 23, 2021

Axioms for the Category of Hilbert Spaces (bis)

Posted by Tom Leinster

Guest post by Chris Heunen

Dusk. The alley is secreted in mist. The miserly reflection of the single lamp on the cobblestones makes you nervous. The stranger approaches, hands inside trenchcoat. Why on Earth did you agree to meet here? The transfer happens, the stranger walks away without a word. The package, it’s all about the package. You have it now. You’d been promised it was the category of Hilbert spaces. But how can you be sure? You can’t just ask it. It didn’t come with a certificate of authenticity. All you can check is how morphisms compose. You leg it home and verify the Axioms for the category of Hilbert spaces!

Posted at 1:47 PM UTC | Permalink | Followups (16)

Stefan Forcey on Entropy and Diversity

Posted by Tom Leinster

Publishing a book is strange: you spend years working on it, then there’s a period in which the publishers do their thing, and then finally — a long time after you did most of the work — it appears in print. And that moment feels good, but afterwards… not much! Because, of course, there has to be time for people to actually read the thing before they’re going to react. It can be a bit of an anticlimax.

So I was very happy to see Stefan Forcey’s review of my new book Entropy and Diversity: The Axiomatic Approach, in the latest issue of the European Mathematical Society Magazine. I won’t attempt to add anything to what Stefan wrote, except to say that the very first sentence of his review taught me a word I didn’t previously know.

Posted at 1:16 PM UTC | Permalink | Followups (2)

September 19, 2021

Axioms for the Category of Hilbert Spaces

Posted by Tom Leinster

Chris Heunen and Andre Kornell have found an axiomatic characterization of the category of Hilbert spaces!

Chris Heunen and Andre Kornell, Axioms for the category of Hilbert spaces. arXiv:2109.07418, 2021.

A bit more precisely, they axiomatize the monoidal dagger category of Hilbert spaces: $\mathbf{Hilb}$ equipped with the operations of tensor product and taking the adjoint.

Their paper appeared on the arXiv this week, and uses Solèr’s theorem, which John wrote about here in 2010.

I don’t have time to write more, but I wanted to make sure that Café readers don’t miss this treat.

Edit: Chris has now written a more substantial post on his and Kornell’s paper.

Posted at 11:20 AM UTC | Permalink | Followups (13)

September 16, 2021

Solid Rings

Posted by John Baez

“Solid ring” sounds self-contradictory, since a ring should have a hole in it. But mathematicians use words in funny ways.

Posted at 9:51 PM UTC | Permalink | Followups (29)

September 14, 2021

Shulman’s Practical Type Theory for Symmetric Monoidal Categories

Posted by Emily Riehl

guest post by Nuiok Dicaire and Paul Lessard

Many well-known type theories, Martin-Löf dependent type theory or linear type theory for example, were developed as syntactic treatments of particular forms of reasoning. Constructive mathematical reasoning in the case of Martin-Löf type theory and resource dependent computation in the case of linear type theory. It is after the fact that these type theories provide convenient means to reason about locally Cartesian closed categories or $\star$-autonomous ones. In this post, a long overdue part of the Applied Category Theory Adjoint School (2020!?), we will discuss a then recent paper by Mike Shulman, A Practical Type Theory for Symmetric Monoidal Categories, where the author reverses this approach. Shulman starts with symmetric monoidal categories as the intended semantics and then (reverse)-engineers a syntax in which it is practical to reason about such categories.

Posted at 4:57 PM UTC | Permalink | Followups (3)

September 13, 2021

USD is Hiring

Posted by Mike Shulman

The University of San Diego mathematics department is hiring! We have two assistant professor positions open this year; these are starting tenure-track positions. Applications must be through the USD web site; the deadline is November 15, 2021.

The University of San Diego (not to be confused with the University of California, San Diego) is a relatively small private Catholic university. The university as a whole has a few graduate programs in different academic units, but the math department is in the College of Arts and Sciences, which is a purely undergraduate liberal arts college. The standard teaching expectation is 3 courses per semester on average, although internal or external grants can reduce that a bit. It’s not a place to work unless you love undergraduate teaching as well as research. But for those who do, I think it’s a great place that supports both, with an administration and colleagues who care about both.

Because our focus is primarily on teaching, we don’t generally look for any particular research areas when hiring; but of course I would personally be happy to see applications from any $n$-Category Café patrons. In addition, this year all hires in the university are supposed to align with one of three “cluster themes” of “Climate Change & Environmental Justice”, “Technology & the Human Experience”, and “Borders & Social Justice”; applicants must include a cover letter explaining how their work aligns with one or more of these clusters, broadly interpreted.

Feel free to contact me personally with any questions.

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September 10, 2021

Cospans and Computation - Part 3

Posted by John Baez

guest post by Angeline Aguinaldo and Anna Knörr as part of the Adjoint School for Applied Category Theory 2021

Synthesi and Socrates are back! What have they learnt from Part 2 in our series of blog posts on computing with cospans?

Posted at 3:57 AM UTC | Permalink | Followups (2)