The n-Category Café

There’s a comprehensive introduction to 2-categories and bicategories now, free on the arXiv:

• Niles Johnson and Donald Yau, 2-Dimensional Categories, 476 pages.

Abstract. This book is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, adjunctions and monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.

Robert Hermann, one of the great expositors of mathematical physics, died on Monday February 10th, 2020. I found this out today from Robert Kotiuga, who spent part of Saturday with him, his daughter Gabrielle, and his ex-wife Lana.

• Dr. Robert C. Hermann, Boston Globe.

The previous discussion on category mistakes got me reading Ofra Magidor’s SEP article on the subject. Magidor was the right choice to produce this article as the author in 2013 of an OUP book Category Mistakes. She is the Waynflete Professor of Metaphysical Philosophy at the University of Oxford (website), a chair once held by one of my favourite British philosophers, R. G. Collingwood.

Now Collingwood came up before in a post of mine as someone who thought that the representation of propositions in Russell’s logic was totally misguided. Rather than freestanding statements, for Collingwood, propositions only make sense in the context of a series of questions and answers. In part, it was thinking through his insights in terms of type theory that got me started on the idea of proposing the latter as a new logic for philosophy of language and metaphysics.

One hoped for effect of my book is that some day philosophers will look to the resources of type theory rather than the standard (untyped) first-order formalisms that are the common currency at the moment. Having been taught first-order logic in a mathematical fashion on my Masters degree many years ago, it struck me how ill-suited it was to represent ordinary language. And yet still our undergraduates are asked to translate from natural language into first-order logic, e.g. Oxford philosophers here. This amusing attempt to translate famous quotations rather proves the point.

To the extent that first-order logic works here, it tends to lean heavily on the supply of a reasonable domain. But when quantification occurs over a variety of domains, as in

Everyone has at some time seen some event that shocked them,

we are asked to imagine some vast pool of individuals to pull out variously people, times and events. Small wonder computer science has looked to control programs via the discipline of types. Just as we want a person in response to Who?, and a place in response to Where?, programs need to compute with terms of the right type.

Type theories come with different degrees of sophistication. I’m advocating dependent type theory. In the Preface to his book, Type-theoretic Grammar (OUP, 1994), Aarne Ranta recounts how the idea of studying natural language in constructive (dependent) type theory occurred to him in 1986:

In Stockholm, when I first discussed the project with Per Martin-Löf, he said that he had designed type theory for mathematics, and than natural language is something else. I said that similar work had been done within predicate calculus, which is just a part of type theory, to which he replied that he found it equally problematic. But his general attitude was far from discouraging: it was more that he was so serious about natural language and saw the problems of my enterprise more clearly than I, who had already assumed the point of view of logical semantics. His criticism was penetrating but patient, and he was generous in telling me about his own ideas. So we gradually developed a view that satisfied both of us, that formal grammar begins with what is well understood formally, and then tries to see how this formal structure is manifested in natural language, instead of starting with natural language in all it unlimitedness and trying to force it into some given formalism.

My book Modal Homotopy Type Theory appears today with Oxford University Press.

As the subtitle – ‘The prospect of a new logic for philosophy’ – suggests, I’m looking to persuade readers that the kinds of things philosophers look to do with the predicate calculus, set theory and modal logic are better achieved by modal homotopy (dependent) type theory.

Since dependent type theories are thoroughly interrelated with category theory, in a sense then, all these years later, I’m still trying to get philosophers interested in the latter. But this book marks a shift in strategy in making the case not only for the philosophy of mathematics, but also for metaphysics and the philosophy of language.

The book explains in order: Why types? Why dependent types? Why homotopy types? Why modal types? I’ll discuss some such issues in forthcoming posts.

Tim Hosgood, Ryan Keleti and others have finished an English translation of this classic:

• Alexander Grothendieck, EGA1: The Language of Schemes

and the LaTeX files are now open-source on GitHub. They are working on the rest of EGA, and they could use help!

guest post by Emily Pillmore and Mario Román

In functional programming, optics are a compositional representation of bidirectional data accessors provided by libraries such as Kmett’s lens, or O’Connor’s mezzolens. Optics were originally called accessors or generalized functional references and appeared as a compositional solution to accessing fields of nested product types. As the understanding of functional references grew, different families of optics were introduced for a variety of different types (e.g. prisms for tagged unions or traversals for containers), culminating in a whole “glassery” of intercomposable accessors.

More recently, Milewski explained the profunctor formulation of these optics in terms of Tambara modules: profunctors endowed with additional structure that makes it interplay nicely with a monoidal action. Milewski explains how a unified profunctor representation of optics can be obtained following Pastro and Street’s “Doubles for monoidal categories”.

In this twelfth post for the Applied Category Theory School we will be presenting Bartosz Milewski’s post “Profunctor optics: the categorical view”. We will give an introduction to the topic of optics, and discuss some of the ideas developed during the course of the ACT Adjoint School 2019.

Café host Emily Riehl has just been awarded a $250,000 prize by her university!! Johns Hopkins gives one President’s Frontier Award every year across the whole university, and the 2020 one has gone to Emily. Up to now it’s usually been given to biological and medical researchers, but when Emily came along they had to make an exception and give it to a mathematician. The award has the goal of “supporting exceptional scholars … who are on the cusp of transforming their fields.”.

Congratulations, Emily! Obviously you’re too modest to announce it yourself here, but someone had to.

You can read all about it here, including the delightful description of how the news was sprung on her:

When Riehl arrived at what she thought was a meeting with a department administrator, she says it was “a complete shock” to find JHU President Ronald J. Daniels, Provost Sunil Kumar, other university leaders, and many colleagues poised to surprise her.

Last time I started talking about the groupoid of ‘finite sets equipped with permutation’, $\mathsf{Perm}$. Remember:

• an object $(X,\sigma)$ of $\mathsf{Perm}$ is a finite set $X$ with a bijection $\sigma \colon X \to X$;
• a morphism $f \colon (X,\sigma) \to (X',\sigma')$ is a bijection $f \colon X \to X'$ such that $\sigma' = f \sigma f^{-1}$.

In other words, $\mathsf{Perm}$ is the groupoid of finite $\mathbb{Z}$-sets. It’s also equivalent to the groupoid of covering spaces of the circle having finitely many sheets!

Today I’d like to talk about another slightly bigger groupoid. It’s very pretty, and I think it will shed light on a puzzle we saw earlier: the mysterious connection between random permutations and Poisson distributions.

I’ll conclude with a question for homotopy theorists.

Yesterday I gave a seminar at the University of California, Riverside, through the magic of Skype. It was the first time I’ve given a talk sitting down, and only the second time I’ve done it in my socks.

The talk was on codensity monads, and that link takes you to the slides. I blogged about this subject lots of times in 2012 (1, 2, 3, 4), and my then-PhD student Tom Avery blogged about it too. In a nutshell, the message is:

Whenever you meet a functor, ask what its codensity monad is.

This should probably be drilled into learning category theorists as much as better-known principles like “whenever you meet a functor, ask what adjoints it has”. But codensity monads took longer to be discovered, and are saddled with a forbidding name — should we just call them “induced monads”?

In any case, following this principle quickly leads to many riches, of which my talk was intended to give a taste.

This time I’d like to repackage some of the results in Part 11 in a prettier way. I’ll describe the groupoid of ‘finite sets equipped with a permutation’ in terms of Young diagrams and cyclic groups. Taking groupoid cardinalities, this description will give a well-known formula for the probability that a random permutation belongs to any given conjugacy class!

This is a quick, off-the-cuff, conceptual question. Hopefully, it has an easy answer.

Often in algebra, we want to quotient out by a set of elements that we regard as trivial or degenerate. That’s almost a tautology: any time we take a quotient, the elements quotiented out are by definition treated as negligible. And often the situation is mathematically trivial too, as when we quotient by the kernel of a homomorphism.

But some examples of quotienting by degenerates are slightly more subtle. The two I have in mind are:

• the definition of exterior power;

• the definition of normalized chain complex.

I’d like to know whether there’s a thread connecting the two.

Inspired by John’s recent series of posts on random permutations, I started thinking about random operators on vector spaces, and nilpotent operators, and Cayley’s tree formula, and, especially, Joyal’s wonderful proof of Cayley’s formula that led him (I guess) to create the equally wonderful theory of species.

Blog posts and comments are often rambling and discursive. That’s part of the fun of it: we think out loud, we try out ideas, we stumble ignorantly through things that others have done better before us, we make mistakes, we refine our ideas, and we learn how to communicate those ideas more efficiently. My own posts on this topic (1, 2, 3) are no exception.

But short sharp accounts are also good! So I wrote a 4.5-page paper containing the thing I think is new. It’s a new proof of the old theorem that when you choose at random a linear operator on a vector space of finite cardinality $N$, the probability of it being nilpotent is $1/N$. And this proof is a linear analogue of Joyal’s proof of Cayley’s formula.

Yay! The first volume of Compositionality has been published! You can read it here:

https://compositionality-journal.org

“Compositionality” is about how complex things can be assembled out of simpler parts. Compositionality is a journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition.

Many mathematicians like the drawings of Escher, or sculptures of surfaces, or colourful plots of fractals and other mathematical phenomena. My own taste seems to be a bit different. I’m not sure how to describe it, but over the years I’ve amassed on my computer a small collection of “mathematical” images, in some rather loose sense of the word. Every time I save one I tell myself I might use it in a blog post one day, but every time I write a blog post, I forget.

Since I never use them, and since it’s Christmas, I thought I’d present them all here instead in one big gallery. Some illustrate some mathematical concept, some refer to the experience of being a mathematician, and a couple are silly visual puns. But many just ring a bell in a part of my mind that seems to me to be connected with the activity of doing mathematics.

I’m afraid I have no idea who any of them were created by (not me!); all have been downloaded from the internet at some point. If you want to find out, I’d suggest a reverse image search.

Merry Christmas! And don’t take the images too seriously — this is just for fun.