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January 29, 2020

Profunctor Optics: The Categorical View

Posted by John Baez

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.

Posted at 6:28 AM UTC | Permalink | Followups (7)

January 23, 2020

Emily Gets A Huge Prize

Posted by Tom Leinster

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.

Posted at 5:48 PM UTC | Permalink | Followups (5)

January 17, 2020

Random Permutations (Part 13)

Posted by John Baez

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

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

In other words, Perm\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.

Posted at 1:34 AM UTC | Permalink | Followups (22)

January 16, 2020

Codensity Monads

Posted by Tom Leinster

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.

Posted at 2:36 PM UTC | Permalink | Followups (38)

January 11, 2020

Random Permutations (Part 12)

Posted by John Baez

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!

Posted at 4:44 AM UTC | Permalink | Followups (5)

January 10, 2020

Quotienting Out The Degenerate

Posted by Tom Leinster

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.

Posted at 1:11 PM UTC | Permalink | Followups (17)