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November 17, 2017

Star-autonomous categories are pseudo Frobenius algebras

Posted by Mike Shulman

A little while ago I talked about how multivariable adjunctions naturally form a polycategory: a structure like a multicategory, but in which codomains as well as domains can involve multiple objects. Now I want to talk about some structures we can define inside this polycategory MVarMVar.

What can you define inside a polycategory? Well, to start with, a polycategory has an underlying multicategory, consisting of the arrows with unary target; so anything we can define in a multicategory we can define in a polycategory. And the most basic thing we can define in a multicategory is a monoid object — in fact, there are some senses in which this is the canonical thing we can define in a multicategory.

So what is a monoid object in MVarMVar?

Posted at 7:28 PM UTC | Permalink | Followups (4)

November 13, 2017

HoTT at JMM

Posted by Mike Shulman

At the 2018 U.S. Joint Mathematics Meetings in San Diego, there will be an AMS special session about homotopy type theory. It’s a continuation of the HoTT MRC that took place this summer, organized by some of the participants to especially showcase the work done during and after the MRC workshop. Following is the announcement from the organizers.

Posted at 10:03 PM UTC | Permalink | Followups (2)

November 11, 2017

Topology Puzzles

Posted by John Baez

Let’s say the closed unit interval [0,1][0,1] maps onto a metric space XX if there is a continuous map from [0,1][0,1] onto XX. Similarly for the Cantor set.

Puzzle 0. Does the Cantor set map onto the closed unit interval, and/or vice versa?

Puzzle 1. Which metric spaces does the closed unit interval map onto?

Puzzle 2. Which metric spaces does the Cantor set map onto?

The first one is easy; the second two are well-known… but still, perhaps, not well-known enough!

Posted at 4:38 AM UTC | Permalink | Followups (25)

November 9, 2017

The 2-Chu Construction

Posted by Mike Shulman

Last time I told you that multivariable adjunctions (“polyvariable adjunctions”?) form a polycategory MVarMVar, a structure like a multicategory but in which codomains as well as domains can involve multiple objects. This time I want to convince you that MVarMVar is actually (a subcategory of) an instance of an exceedingly general notion, called the Chu construction.

As I remarked last time, in defining multivariable adjunctions we used opposite categories. However, we didn’t need to know very much about the opposite of a category AA; essentially all we needed is the existence of a hom-functor hom A:A op×ASethom_A : A^{op}\times A \to Set. This enabled us to define the representable functors corresponding to multivariable morphisms, so that we could then ask them to be isomorphic to obtain a multivariable adjunction. We didn’t need any special properties of the category SetSet or the hom-functor hom Ahom_A, only that each AA comes equipped with a map hom A:A op×ASethom_A : A^{op}\times A \to Set. (Note that this is sort of “half” of a counit for the hoped-for dual pair (A,A op)(A,A^{op}), or it would be if SetSet were the unit object; the other half doesn’t exist in CatCat, but it does once we pass to MVarMVar.)

Furthermore, we didn’t need any cartesian properties of the product ×\times; it could just as well have been any monoidal structure, or even any multicategory structure! Finally, if we’re willing to end up with a somewhat larger category, we can give up the idea that each AA should be equipped with A opA^{op} and hom Ahom_A, and instead allow each objects of our “generalized MVarMVar” to make a free choice of its “opposite” and “hom-functor”.

Posted at 12:09 PM UTC | Permalink | Followups (4)

November 7, 2017

The Polycategory of Multivariable Adjunctions

Posted by Mike Shulman

Adjunctions are well-known and fundamental in category theory. Somewhat less well-known are two-variable adjunctions, consisting of functors f:A×BCf:A\times B\to C, g:A op×CBg:A^{op}\times C\to B, and h:B op×CAh:B^{op}\times C\to A and natural isomorphisms

C(f(a,b),c)B(b,g(a,c))A(a,h(b,c)). C(f(a,b),c) \cong B(b,g(a,c)) \cong A(a,h(b,c)).

These are also ubiquitous in mathematics, for instance in the notion of closed monoidal category, or in the hom-power-copower situation of an enriched category. But it seems that only fairly recently has there been a wider appreciation that it is worth defining and studying them in their own right (rather than simply as a pair of parametrized adjunctions f(a,)g(a,)f(a,-)\dashv g(a,-) and f(,b)h(b,)f(-,b) \dashv h(b,-)).

Now, ordinary adjunctions are the morphisms of a 2-category AdjAdj (with an arbitrary choice of direction, say pointing in the direction of the left adjoint), whose 2-cells are compatible pairs of natural transformations (a fundamental result being that either uniquely determines the other). It’s obvious to guess that two-variable adjunctions should be the binary morphisms in a multicategory of “nn-ary adjunctions”, and this is indeed the case. In fact, Eugenia, Nick, and Emily showed that multivariable adjunctions form a cyclic multicategory, and indeed even a cyclic double multicategory.

In this post, however, I want to argue that it’s even better to regard multivariable adjunctions as forming a slightly different structure called a polycategory.

Posted at 10:45 AM UTC | Permalink | Followups (31)

November 3, 2017

Applied Category Theory Papers

Posted by John Baez

In preparation for the Applied Category Theory special session at U.C. Riverside this weekend, my crew dropped three papers on the arXiv.

Posted at 11:28 PM UTC | Permalink | Post a Comment

Magnitude Homology is Hochschild Homology

Posted by Mike Shulman

Magnitude homology, like magnitude, was born on this blog. Now there is a paper about it on the arXiv:

  • Tom Leinster and Mike Shulman, Magnitude homology of enriched categories and metric spaces, arXiv:1711.00802

I’m also giving a talk about magnitude homology this Saturday at the AMS sectional meeting at UC Riverside (this is the same meeting where John is running a session about applied category theory, but my talk will be in the Homotopy Theory session, 3 pm on Saturday afternoon). Here are my slides.

This paper contains basically everything that’s been said about magnitude homology so far on the blog (somewhat cleaned up), plus several new things. Below the fold I’ll briefly summarize what’s new, for the benefit of a (hypothetical?) reader who remembers all the previous posts. But if you don’t remember the old posts at all, then I suggest just starting directly with the preprint (or the slides for my talk).

I also have a request for help with terminology at the end.

Posted at 6:24 AM UTC | Permalink | Followups (27)

October 28, 2017

The Adjoint School

Posted by John Baez

The deadline for applying to this ‘school’ on applied category theory is Wednesday November 1st. I hear they are still looking for a few really good applicants. Hurry up—this is a great opportunity!

Posted at 7:10 AM UTC | Permalink | Post a Comment

October 26, 2017

Categorification and the Cosmic Cube

Posted by David Corfield

I see that Tobias Dyckerhoff’s A categorified Dold-Kan correspondence has just appeared, looking, as its title suggests, to categorify the nLab: Dold-Kan correspondence. As it says there, the latter

interpolates between homological algebra and general simplicial homotopy theory.

So with Dyckerhoff’s paper we seem to be dipping down to the lower layer of the flamboyantly named ‘cosmic cube’, see slide 10 of these notes by John, and discussed at nLab: cosmic cube. Via chain complexes of stable (,1)(\infty, 1)-categories Dyckerhoff speaks of a ‘categorified homological algebra’, and also through a categorified Eilenberg-Mac Lane spectrum, of a ‘categorified cohomology’.

For old time’s sake, let’s see if anyone is up for the kind of grand vision thing we used to talk about. For one thing we might wonder what plays the role of a categorified homotopy theory, the kind of world where Mike’s suggestions on directed homotopy type theory might find a home.

I see I was raising stratified spaces as relevant back there. In the meantime we now have useful models from A stratified homotopy hypothesis. It turns out that (,1)(\infty, 1)-categories are equivalent to ‘striation sheaves’, a certain kind of sheaf on ‘conically smooth’ stratified spaces. The relevant fundamental (,1)(\infty, 1)-category is the exit-path (,1)(\infty, 1)-category, rather than the entry and exit paths of our older discussions which brought in duals.

In line with Urs’s claim that cohomology concerns mapping spaces in (,1)(\infty, 1)-toposes (nLab: cohomology), perhaps for categorified cohomology we should be looking for parallels in (,2)(\infty, 2)-toposes, an important one of which will be that containing all (,1)(\infty, 1)-categories, or equivalently, all striation sheaves.

Posted at 10:15 AM UTC | Permalink | Followups (25)

October 23, 2017

Unconscious Bias in Recruiting

Posted by Tom Leinster

All of us who work in maths, physics or computer science departments know about the dramatic gender imbalance in our subjects. Many departments and universities have been working hard to make their recruitment processes more inclusive towards under-represented groups — not only for the excellent altruistic reason that it makes the world a better place, but also for the selfish reason that we don’t want to miss out on getting the best people.

There’s research (as well as anecdotal evidence) showing that the wording of job ads can make a big difference to who applies. In particular, it can influence significantly the gender profile of applicants.

My head of department Iain Gordon just pointed out a website by Kat Matfield where you can paste in your ad and get an automatic assessment of the language used. The site matches the ad against lists of “masculine-coded” and “feminine-coded” words and gives you a summary. The first link above is to the academic paper behind the website.

For example, we at Edinburgh are currently advertising a two-year postdoctoral fellowship in any area of mathematics. Try pasting the ad into the site and see what happens!

Posted at 1:22 PM UTC | Permalink | Followups (5)

October 1, 2017

Vladimir Voevodsky, June 4, 1966 - September 30, 2017

Posted by John Baez

Vladimir Voevodsky died this Saturday. He was 51.

Posted at 10:43 PM UTC | Permalink | Followups (30)

September 22, 2017

Lattice Paths and Continued Fractions II

Posted by Simon Willerton

Last time we proved Flajolet’s Fundamental Lemma about enumerating Dyck paths. This time I want to give some examples, in particular to relate this to what I wrote previously about Dyck paths, Schröder paths and what they have to do with reverse Bessel polynomials.

We’ll see that the generating function of the sequence of reverse Bessel polynomials (θ i(R)) i=0 \left(\theta_i(R)\right)_{i=0}^\infty has the following continued fraction expansion.

i=0 θ i(R)t i=11Rtt1Rt2t1Rt3t1 \sum_{i=0}^\infty \theta_i(R) \,t^i = \frac{1}{1-Rt- \frac{t}{1-Rt - \frac{2t}{1-Rt- \frac{3t}{1-\dots}}}}

I’ll even give you a snippet of SageMath code so you can have a play around with this if you like.

Posted at 10:14 AM UTC | Permalink | Post a Comment

Applied Category Theory at UCR (Part 2)

Posted by John Baez

I’m running a special session on applied category theory, and now the program is available:

This is going to be fun.

My former student Brendan Fong is now working with David Spivak at M.I.T., and they’re both coming. My collaborator John Foley at Metron is also coming: we’re working on the CASCADE project for designing networked systems.

Dmitry Vagner is coming from Duke: he wrote a paper with David and Eugene Lerman on operads and open dynamical system. Christina Vaisilakopolou, who has worked with David and Patrick Schultz on dynamical systems, has just joined our group at UCR, so she will also be here. And the three of them have worked with Ryan Wisnesky on algebraic databases. Ryan will not be here, but his colleague Peter Gates will: together with David they have a startup called Categorical Informatics, which uses category theory to build sophisticated databases.

That’s not everyone — for example, most of my students will be speaking at this special session, and other people too — but that gives you a rough sense of some people involved. The conference is on a weekend, but John Foley and David Spivak and Brendan Fong and Dmitry Vagner are staying on for longer, so we’ll have some long conversations… and Brendan will explain decorated corelations in my Tuesday afternoon network theory seminar.

Wanna see what the talks are about?

Posted at 12:08 AM UTC | Permalink | Followups (13)

September 18, 2017

Lattice Paths and Continued Fractions I

Posted by Simon Willerton

In my last post I talked about certain types of lattice paths with weightings on them and formulas for the weighted count of the paths, in particular I was interested in expressing the reverse Bessel polynomials as a certain weighted count of Schröder paths. I alluded to some connection with continued fractions and it is this connection that I want to explain here and in my next post.

In this post I want to prove Flajolet’s Fundamental Lemma. Alan Sokal calls this Flajolet’s Master Theorem, but Viennot takes the stance that it deserves the high accolade of being described as a ‘Fundamental Lemma’, citing Aigner and Ziegler in Proofs from THE BOOK:

“The essence of mathematics is proving theorems – and so, that is what mathematicians do: They prove theorems. But to tell the truth, what they really want to prove, once in their lifetime, is a Lemma, like the one by Fatou in analysis, the Lemma of Gauss in number theory, or the Burnside-Frobenius Lemma in combinatorics.

“Now what makes a mathematical statement a true Lemma? First, it should be applicable to a wide variety of instances, even seemingly unrelated problems. Secondly, the statement should, once you have seen it, be completely obvious. The reaction of the reader might well be one of faint envy: Why haven’t I noticed this before? And thirdly, on an esthetic level, the Lemma – including its proof – should be beautiful!”

Interestingly, Aigner and Ziegler were building up to describing a result of Viennot’s – the Gessel-Lindström-Viennot Lemma – as a fundamental lemma! (I hope to talk about that lemma in a later post.)

Anyway, Flajolet’s Fundamental Lemma that I will describe and prove below is about expressing the weighted count of paths that look like

weighted Motzkhin path

as a continued fraction

11c 0a 1b 11c 1a 2b 21c 2a 3b 31 \frac{1} {1- c_{0} - \frac{a_{1} b_{1}} {1-c_{1} - \frac{a_{2} b_{2}} {1- c_2 - \frac{a_3 b_3} {1-\dots }}}}

Next time I’ll give a few examples, including the connection with reverse Bessel polynomials.

Posted at 12:18 PM UTC | Permalink | Followups (7)

September 12, 2017

Applied Category Theory 2018

Posted by John Baez

We’re having a conference on applied category theory!

The plenary speakers will be:

  • Samson Abramsky (Oxford)
  • John Baez (UC Riverside)
  • Kathryn Hess (EPFL)
  • Mehrnoosh Sadrzadeh (Queen Mary)
  • David Spivak (MIT)

There will be a lot more to say as this progresses, but for now let me just quote from the conference website.

Posted at 2:09 PM UTC | Permalink | Followups (7)