The hand-written notes here are by Christian Williams. They are probably best seen as a reminder to myself as to what I’d like to include in a short book someday.

• Lecture 1: What is pure mathematics all about? The importance of free structures.

• Lecture 2: The natural numbers as a free structure. Adjoint functors.

• Lecture 3: Adjoint functors in terms of unit and counit.

• Lecture 4: 2-Categories. Adjunctions.

• Lecture 5: 2-Categories and string diagrams. Composing adjunctions.

• Lecture 6: The ‘main spine’ of mathematics. Getting a monad from an adjunction.

• Lecture 7: Definition of a monad. Getting a monad from an adjunction. The augmented simplex category.

• Lecture 8: The walking monad, the augmented simplex category and the simplex category.

• Lecture 9: Simplicial abelian groups from simplicial sets. Chain complexes from simplicial abelian groups.

• Lecture 10: The Dold-Thom theorem: the category of simplicial abelian groups is equivalent to the category of chain complexes of abelian groups. The homology of a chain complex.

• Lecture 7: Definition of a monad. Getting a monad from an adjunction. The augmented simplex category.

• Lecture 8: The walking monad, the

augmented simplex category and the simplex category.

• Lecture 9: Simplicial abelian groups from simplicial sets. Chain complexes from simplicial abelian groups.

• Lecture 10: Chain complexes from simplicial abelian groups. The homology of a chain complex.

• Lecture 12: The bar construction: getting a simplicial objects from an adjunction. The bar construction for G-sets, previewed.

• Lecture 13: The adjunction between G-sets and sets.

• Lecture 14: The bar construction for groups.

• Lecture 15: The simplicial set obtained by applying the bar construction to the one-point -set, its geometric realization and the free simplicial abelian group

• Lecture 16: The chain complex coming from the simplicial abelian group its homology, and the definition of group cohomology with coefficients in a -module.

• Lecture 17: Extensions of groups. The Jordan-Hölder theorem. How an extension of a group by an abelian group gives an action of on and a 2-cocycle

• Lecture 18: Classifying abelian extensions of groups. Direct products, semidirect products, central extensions and general abelian extensions. The groups of order 8 as abelian extensions.

• Lecture 19: Group cohomology. The chain complex for the cohomology of with coefficients in , starting from the bar construction, and leading to the 2-cocycles used in classifying abelian extensions. The classification of extensions of by in terms of

• Lecture 20: Examples of group cohomology: nilpotent groups and the fracture theorem. Higher-dimensional algebra and homotopification: the nerve of a category and the nerve of a topological space. as the nerve of the translation groupoid as the walking space with fundamental group

]]>First, remember the story. A subset of the plane has **diameter 1** if the distance between any two points in this set is ≤ 1. A **universal covering** is a convex subset of the plane that can cover a translated, reflected and/or rotated version of every subset of the plane with diameter 1. In 1914, the famous mathematician Henri Lebesgue sent a letter to a fellow named Pál, challenging him to find the universal covering with the least area.

Pál worked on this problem, and 6 years later he published a paper on it. He found a very nice universal covering: a regular hexagon in which one can inscribe a circle of diameter 1. This has area

0.86602540…

But he also found a universal covering with less area, by removing two triangles from this hexagon—for example, the triangles C_{1}C_{2}C_{3} and E_{1}E_{2}E_{3} here:

The resulting universal covering has area

0.84529946…

In 1936, Sprague went on to prove that more area could be removed from another corner of Pál’s original hexagon, giving a universal covering of area

0.8441377708435…

In 1992, Hansen took these reductions even further by removing two more pieces from Pál’s hexagon. Each piece is a thin sliver bounded by two straight lines and an arc. The first piece is tiny. The second is downright microscopic!

Hansen claimed the areas of these regions were 4 · 10^{-11} and 6 · 10^{-18}. This turned out to be wrong. The actual areas are 3.7507 · 10^{-11} and 8.4460 · 10^{-21}. The resulting universal covering had an area of

0.844137708416…

This tiny improvement over Sprague’s work led Klee and Wagon to write:

it does seem safe to guess that progress on [this problem], which has been painfully slow in the past, may be even more painfully slow in the future.

However, in 2015 Philip Gibbs found a way to remove about a million times more area than Hansen’s larger region: a whopping 2.233 · 10^{-5}. This gave a universal covering with area

0.844115376859…

Karine Bagdasaryan and I helped Gibbs write up a rigorous proof of this result, and we published it here:

• John Baez, Karine Bagdasaryan and Philip Gibbs, The Lebesgue universal covering problem, *Journal of Computational Geometry* **6** (2015), 288–299.

Greg Egan played an instrumental role as well, catching various computational errors.

At the time Philip was sure he could remove even more area, at the expense of a more complicated proof. Since the proof was already quite complicated, we decided to stick with what we had.

But this week I met Philip at The philosophy and physics of Noether’s theorems, a wonderful workshop in London which deserves a full blog article of its own. It turns out that he has gone further: he claims to have found a vastly better universal covering, with area

0.8440935944…

This is an improvement of 2.178245 × 10^{-5} over our earlier work—roughly equal to our improvement over Hansen.

You can read his argument here:

• Philip Gibbs, An upper bound for Lebesgue’s universal covering problem, 22 January 2018.

I say ‘claims’ not because I doubt his result—he’s clearly a master at this kind of mathematics!—but because I haven’t checked it and it’s easy to make mistakes, for example mistakes in computing the areas of the shapes removed.

It seems we are closing in on the final result; however, Philip Gibbs believes there is still room for improvement, so I expect it will take at least a decade or two to solve this problem… unless, of course, some mathematicians start working on it full-time, which could speed things up considerably.

]]>• Riverside Mathematics Workshop for Excellence and Diversity, Friday 19 October – Saturday 20 October, 2018. Organized by John Baez, Carl Mautner, José González and Chen Weitao.

This is the first of an annual series of workshops to showcase and celebrate excellence in research by women and other under-represented groups for the purpose of fostering and encouraging growth in the U.C. Riverside mathematical community.

After tea at 3:30 p.m. on Friday there will be two plenary talks, lasting until 5:00. Catherine Searle will talk on “Symmetries of spaces with lower curvature bounds”, and Edray Goins will give a talk called “Clocks, parking garages, and the solvability of the quintic: a friendly introduction to monodromy”. There will then be a banquet in the Alumni Center 6:30 – 8:30 p.m.

On Saturday there will be coffee and a poster session at 8:30 a.m., and then two parallel sessions on pure and applied mathematics, with talks at 9:30, 10:30, 11:30, 1:00 and 2:00. Check out the abstracts here!

(I’m especially interested in Christina Vasilakopoulou’s talk on Frobenius and Hopf monoids in enriched categories, but she’s my postdoc so I’m biased.)

]]>animation by Marius Buliga

I’m helping organize ACT 2019, an applied category theory conference and school at Oxford, July 15-26, 2019.

More details will come later, but here’s the basic idea. If you’re a grad student interested in this subject, you should apply for the ‘school’. Not yet—we’ll let you know when.

]]>Dear all,

As part of a new growing community in Applied Category Theory, now with a dedicated journal

Compositionality, a traveling workshop series SYCO, a forthcoming Cambridge U. Press book series Reasoning with Categories, and several one-off events including at NIST, we launch an annual conference+school series named Applied Category Theory, the coming one being at Oxford, July 15-19 for the conference, and July 22-26 for the school. The dates are chosen such that CT 2019 (Edinburgh) and the ACT 2019 conference (Oxford) will be back-to-back, for those wishing to participate in both.There already was a successful invitation-only pilot, ACT 2018, last year at the Lorentz Centre in Leiden, also in the format of school+workshop.

For the conference, for those who are familiar with the successful QPL conference series, we will follow a very similar format for the ACT conference. This means that we will accept both new papers which then will be published in a proceedings volume (most likely a

Compositionalityspecial Proceedings issue), as well as shorter abstracts of papers published elsewhere. There will be a thorough selection process, as typical in computer science conferences. The idea is that all the best work in applied category theory will be presented at the conference, and that acceptance is something that means something, just like in CS conferences. This is particularly important for young people as it will help them with their careers.Expect a call for submissions soon, and start preparing your papers now!

The school in ACT 2018 was unique in that small groups of students worked closely with an experienced researcher (these were John Baez, Aleks Kissinger, Martha Lewis and Pawel Sobociński), and each group ended up producing a paper. We will continue with this format or a closely related one, with Jules Hedges and Daniel Cicala as organisers this year. As there were 80 applications last year for 16 slots, we may want to try to find a way to involve more students.

We are fortunate to have a number of private sector companies closely associated in some way or another, who will also participate, with Cambridge Quantum Computing Inc. and StateBox having already made major financial/logistic contributions.

On behalf of the ACT Steering Committee,

John Baez, Bob Coecke, David Spivak, Christina Vasilakopoulou

is a good approximation to the number of primes less than or equal to Numerical evidence suggests that is always greater than For example,

and

But in 1914, Littlewood heroically showed that in fact, changes sign infinitely many times!

This raised the question: when does first exceed ? In 1933, Littlewood’s student Skewes showed, assuming the Riemann hypothesis, that it must do so for some less than or equal to

Later, in 1955, Skewes showed *without* the Riemann hypothesis that must exceed for some smaller than

By now this bound has been improved enormously. We now know the two functions cross somewhere near but we don’t know if this is the first crossing!

All this math is quite deep. Here is something less deep, but still fun.

You can show that

and so on.

It’s a nice pattern. But this pattern doesn’t go on forever! It lasts a very, very long time… but not forever.

More precisely, the identity

holds when

but not for all At some point it stops working and never works again. In fact, it definitely fails for all

The integrals here are a variant of the Borwein integrals:

where the pattern continues until

but then fails:

I never understood this until I read Greg Egan’s explanation, based on the work of Hanspeter Schmid. It’s all about convolution, and Fourier transforms:

Suppose we have a rectangular pulse, centred on the origin, with a height of 1/2 and a half-width of 1.

Now, suppose we keep taking moving averages of this function, again and again, with the average computed in a window of half-width 1/3, then 1/5, then 1/7, 1/9, and so on.

There are a couple of features of the original pulse that will persist completely unchanged for the first few stages of this process, but then they will be abruptly lost at some point.

The first feature is that F(0) = 1/2. In the original pulse, the point (0,1/2) lies on a plateau, a perfectly constant segment with a half-width of 1. The process of repeatedly taking the moving average will nibble away at this plateau, shrinking its half-width by the half-width of the averaging window. So, once the sum of the windows’ half-widths exceeds 1, at 1/3+1/5+1/7+…+1/15, F(0) will suddenly fall below 1/2, but up until that step it will remain untouched.

In the animation below, the plateau where F(x)=1/2 is marked in red.

The second feature is that F(–1)=F(1)=1/4. In the original pulse, we have a step at –1 and 1, but if we define F here as the average of the left-hand and right-hand limits we get 1/4, and once we apply the first moving average we simply have 1/4 as the function’s value.

In this case, F(–1)=F(1)=1/4 will continue to hold so long as the points (–1,1/4) and (1,1/4) are surrounded by regions where the function has a suitable symmetry: it is equal to an odd function, offset and translated from the origin to these centres. So long as that’s true for a region wider than the averaging window being applied, the average at the centre will be unchanged.

The initial half-width of each of these symmetrical slopes is 2 (stretching from the opposite end of the plateau and an equal distance away along the x-axis), and as with the plateau, this is nibbled away each time we take another moving average. And in this case, the feature persists until 1/3+1/5+1/7+…+1/113, which is when the sum first exceeds 2.

In the animation, the yellow arrows mark the extent of the symmetrical slopes.

OK, none of this is difficult to understand, but why should we care?

Because this is how Hanspeter Schmid explained the infamous Borwein integrals:

∫sin(t)/t dt = π/2

∫sin(t/3)/(t/3) × sin(t)/t dt = π/2

∫sin(t/5)/(t/5) × sin(t/3)/(t/3) × sin(t)/t dt = π/2…

∫sin(t/13)/(t/13) × … × sin(t/3)/(t/3) × sin(t)/t dt = π/2

But then the pattern is broken:

∫sin(t/15)/(t/15) × … × sin(t/3)/(t/3) × sin(t)/t dt < π/2

Here these integrals are from t=0 to t=∞. And Schmid came up with an even more persistent pattern of his own:

∫2 cos(t) sin(t)/t dt = π/2

∫2 cos(t) sin(t/3)/(t/3) × sin(t)/t dt = π/2

∫2 cos(t) sin(t/5)/(t/5) × sin(t/3)/(t/3) × sin(t)/t dt = π/2

…

∫2 cos(t) sin(t/111)/(t/111) × … × sin(t/3)/(t/3) × sin(t)/t dt = π/2But:

∫2 cos(t) sin(t/113)/(t/113) × … × sin(t/3)/(t/3) × sin(t)/t dt < π/2

The first set of integrals, due to Borwein, correspond to taking the Fourier transforms of our sequence of ever-smoother pulses and then evaluating F(0). The Fourier transform of the sinc function:

sinc(w t) = sin(w t)/(w t)

is proportional to a rectangular pulse of half-width w, and the Fourier transform of a product of sinc functions is the convolution of their transforms, which in the case of a rectangular pulse just amounts to taking a moving average.

Schmid’s integrals come from adding a clever twist: the extra factor of 2 cos(t) shifts the integral from the zero-frequency Fourier component to the sum of its components at angular frequencies –1 and 1, and hence the result depends on F(–1)+F(1)=1/2, which as we have seen persists for much longer than F(0)=1/2.

• Hanspeter Schmid, Two curious integrals and a graphic proof,

Elem. Math.69(2014) 11–17.

I asked Greg if we could generalize these results to give even longer sequences of identities that eventually fail, and he showed me how: you can just take the Borwein integrals and replace the numbers 1, 1/3, 1/5, 1/7, … by some sequence of positive numbers

The integral

will then equal as long as but not when it exceeds 1. You can see a full explanation on Wikipedia:

• Wikipedia, Borwein integral: general formula.

As an example, I chose the integral

which equals if and only if

Thus, the identity holds if

However,

so the identity holds if

or

or

On the other hand, the identity fails if

so it fails if

However,

so the identity fails if

or

or

With a little work one could sharpen these estimates considerably, though it would take more work to find the *exact* value of at which

first fails.

]]>• Tai-Danae Bradley, *What is Applied Category Theory?*

Abstract.This is a collection of introductory, expository notes on applied category theory, inspired by the 2018 Applied Category Theory Workshop, and in these notes we take a leisurely stroll through two themes (functorial semantics and compositionality), two constructions (monoidal categories and decorated cospans) and two examples (chemical reaction networks and natural language processing) within the field.

Check it out!

]]>It’s called the **5/8 theorem**. Randomly choose two elements of a finite group. What’s the probability that they commute? If it exceeds 62.5%, the group must be abelian!

This was probably known for a long time, but the first known proof appears in a paper by Erdös and Turan.

It’s fun to lead up to this proof by looking for groups that are “as commutative as possible without being abelian”. This phrase could mean different things. *One* interpretation is that we’re trying to maximize the probability that two randomly chosen elements commute. But there are two simpler interpretations, which will actually help us prove the 5/8 theorem.

How big can the center of a finite group be, compared to the whole group? If a group is abelian, its center, say is all of But let’s assume is not abelian. How big can be?

Since the center is a subgroup of we know by Lagrange’s theorem that is an integer. To make big we need this integer to be small. How small can it be?

It can’t be 1, since then and would be abelian. Can it be 2?

No! This would force to be abelian, leading to a contradiction! The reason is that the center is always a normal subgroup of , so is a group of size . If this is 2 then has to be But this is generated by one element, so must be generated by its center together with one element. This one element commutes with everything in the center, obviously… but that means is abelian: a contradiction!

For the same reason, can’t be 3. The only group with 3 elements is which is generated by one element. So the same argument leads to a contradiction: is generated by its center and one element, which commutes with everything in the center, so is abelian.

So let’s try There are two groups with 4 elements: and The second, called the Klein four-group, is not generated by one element. It’s generated by two elements! So it offers some hope.

If you haven’t studied much group theory, you could be pessimistic. After all, is still abelian! So you might think this: “If the group is generated by its center and two elements which commute with each other, so it’s abelian.”

But that’s false: even if two elements of commute with each other, this does not imply that the elements of mapping to these elements commute.

This is a fun subject to study, but best way for us to see this right now is to actually find a nonabelian group with . The smallest possible example would have and indeed this works!

Namely, we’ll take to be the 8-element quaternion group

where

and multiplication by works just as you’d expect, e.g.

You can think of these 8 guys as the unit quaternions lying on the 4 coordinate axes. They’re the vertices of a 4-dimensional analogue of the octahedron. Here’s a picture by David A. Richter, where the 8 vertices are projected down from 4 dimensions to the vertices of a cube:

The center of is and the quotient is the Klein four-group, since if we mod out by we get the group

with

So, we’ve found a nonabelian finite group with 1/4 of its elements lying in the center, and this is the maximum possible fraction!

Here’s another way to ask how commutative a finite group can be, without being abelian. Any element has a centralizer consisting of all elements that commute with

How big can be? If is in the center of then is all of So let’s assume is not in the center, and ask how big the fraction can be.

In other words: how large can the fraction of elements of that commute with be, without it being *everything*?

It’s easy to check that the centralizer is a subgroup of So, again using Lagrange’s theorem, we know is an integer. To make the fraction big, we want this integer to be small. If it’s 1, *everything* commutes with So the first real option is 2.

Can we find an element of a finite group that commutes with exactly 1/2 the elements of that group?

Yes! One example is our friend the quaternion group Each non-central element commutes with exactly half the elements. For example, commutes only with its own powers:

So we’ve found a finite group with a non-central element that commutes with 1/2 the elements in the group, and this is maximum possible fraction!

Now let’s tackle the original question. Suppose is a nonabelian group. How can we maximize the probability for two randomly chosen elements of to commute?

Say we randomly pick two elements Then there are two cases. If is in the center of it commutes with with probability 1. But if is not in the center, we’ve just seen it commutes with with probability at most 1/2.

So, to get an upper bound on the probability that our pair of elements commutes, we should make the center as large as possible. We’ve seen that is at most 1/4. So let’s use that.

Then with probability 1/4, commutes with all the elements of while with probability 3/4 it commutes with 1/2 the elements of

So, the probability that commutes with is

Even better, all these bounds are attained by the quaternion group 1/4 of its elements are in the center, while every element not in the center commutes with 1/2 of the elements! So, the probability that two elements in this group commute is 5/8.

So we’ve proved the 5/8 theorem and shown we can’t improve this constant.

I find it very pleasant that the quaternion group is “as commutative as possible without being abelian” in three different ways. But I shouldn’t overstate its importance!

I don’t know the proof, but the website groupprops says the following are equivalent for a finite group :

• The probability that two elements commute is 5/8.

• The inner automorphism group of has 4 elements.

• The inner automorphism group of is

Examining the argument I gave, it seems the probability 5/8 can only be attained if

•

• for every

So apparently any finite group with inner automorphism group must have these other two properties as well!

There are lots of groups with inner automorphism group Besides the quaternion group, there’s one other 8-element group with this property: the group of rotations and reflections of the square, also known as the dihedral group of order 8. And there are six 16-element groups with this property: they’re called the groups of Hall–Senior class two. And I expect that as we go to higher powers of two, there will be vast numbers of groups with this property.

You see, the number of nonisomorphic groups of order grows alarmingly fast. There’s 1 group of order 2, 2 of order 4, 5 of order 8, 14 of order 16, 51 of order 32, 267 of order 64… but 49,487,365,422 of order 1024. Indeed, it seems ‘almost all’ finite groups have order a power of two, in a certain asymptotic sense. For example, 99% of the roughly 50 billion groups of order ≤ 2000 have order 1024.

Thus, if people trying to classify groups are like taxonomists, groups of order a power of 2 are like insects.

In 1964, the amusingly named pair of authors Marshall Hall Jr. and James K. Senior classified all groups of order for They developed some powerful general ideas in the process, like isoclinism. I don’t want to explain it here, but which involves the quotient that I’ve been talking about. So, though I don’t understand much about this, I’m not completely surprised to read that any group of order has commuting probability 5/8 iff it has ‘Hall–Senior class two’.

There’s much more to say. For example, we can define the probability that two elements commute not just for finite groups but also compact topological groups, since these come with a god-given probability measure, called Haar measure. And here again, if the group is nonabelian, the maximum possible probability for two elements to commute is 5/8!

There are also many other generalizations. For example Guralnick and Wilson proved:

• If the probability that two randomly chosen elements of generate a solvable group is greater than 11/30 then itself is solvable.

• If the probability that two randomly chosen elements of generate a nilpotent group is greater than 1/2 then is nilpotent.

• If the probability that two randomly chosen elements of generate a group of odd order is greater than 11/30 then itself has odd order.

The constants are optimal in each case.

I’ll just finish with two questions I don’t know the answer to:

• For exactly what set of numbers can we find a finite group where the probability that two randomly chosen elements commute is If we call this set we’ve seen

But does contain *every* rational number in the interval (0,5/8], or just some? Just some, in fact—but which ones? It should be possible to make some progress on this by examining my proof of the 5/8 theorem, but I haven’t tried at all. I leave it to you!

• For what properties P of a finite group is there a theorem of this form: “if the probability of two randomly chosen elements generating a subgroup of with property P exceeds some value then must itself have property P”? Is there some logical form a property can have, that will guarantee the existence of a result like this?

Here is a nice discussion, where I learned some of the facts I mentioned, including the proof I gave:

• MathOverflow, 5/8 bound in group theory.

Here is an elementary reference, free online if you jump through some hoops, which includes the proof for compact topological groups, and other bits of wisdom:

• W. H. Gustafson, What is the probability that two group elements commute?, *American Mathematical Monthly* **80** (1973), 1031–1034.

For example, if is finite simple and nonabelian, the probability that two elements commute is at most 1/12, a bound attained by

Here’s another elementary article:

• Desmond MacHale, How commutative can a non-commutative group be?, *The Mathematical Gazette* **58** (1974), 199–202.

If you get completely stuck on Puzzle 1, you can look here for some hints on what values the probability of two elements to commute can take… but not a complete solution!

The 5/8 theorem seems to have first appeared here:

• P. Erdös and P. Turán, On some problems of a statistical group-theory, IV, *Acta Math. Acad. Sci. Hung.* **19** (1968) 413–435.

I’ve been spending the last month at the Centre of Quantum Technologies, getting lots of work done. This Friday I’m giving a talk, and you can see the slides now:

• John Baez, Getting to the bottom of Noether’s theorem.

Abstract.In her paper of 1918, Noether’s theorem relating symmetries and conserved quantities was formulated in term of Lagrangian mechanics. But if we want to make the essence of this relation seem as self-evident as possible, we can turn to a formulation in term of Poisson brackets, which generalizes easily to quantum mechanics using commutators. This approach also gives a version of Noether’s theorem for Markov processes. The key question then becomes: when, and why, do observables generate one-parameter groups of transformations? This question sheds light on why complex numbers show up in quantum mechanics.

At 5:30 on Saturday October 6th I’ll talk about this stuff at this workshop in London:

• The Philosophy and Physics of Noether’s Theorems, 5-6 October 2018, Fischer Hall, 1-4 Suffolk Street, London, UK. Organized by Bryan W. Roberts (LSE) and Nicholas Teh (Notre Dame).

This workshop celebrates the 100th anniversary of Noether’s famous paper connecting symmetries to conserved quantities. Her paper actually contains *two* big theorems. My talk is only about the more famous one, Noether’s first theorem, and I’ll change my talk title to make that clear when I go to London, to avoid getting flak from experts. Her second theorem explains why it’s hard to define energy in general relativity! This is one reason Einstein admired Noether so much.

I’ll also give this talk at DAMTP—the Department of Applied Mathematics and Theoretical Physics, in Cambridge—on Thursday October 4th at 1 pm.

The organizers of London workshop on the philosophy and physics of Noether’s theorems have asked me to write a paper, so my talk can be seen as the first step toward that. My talk doesn’t contain any hard theorems, but the main point—that the complex numbers arise naturally from wanting a correspondence between observables and symmetry generators—can be expressed in some theorems, which I hope to explain in my paper.

]]>

It’s an open-access journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline. Topics may concern foundational structures, an organizing principle, or a powerful tool. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition.

*Compositionality* is free of cost for both readers and authors.

We invite you to submit a manuscript for publication in the first issue of Compositionality (ISSN: 2631-4444), a new open-access journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline.

To submit a manuscript, please visit http://www.compositionality-journal.org/for-authors/.

Compositionality refers to complex things that can be built by sticking together simpler parts. We welcome papers using compositional ideas, most notably of a category-theoretic origin, in any discipline. This may concern foundational structures, an organising principle, a powerful tool, or an important application. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition.

Related conferences and workshops that fall within the scope of Compositionality include the Symposium on Compositional Structures (SYCO), Categories, Logic and Physics (CLP), String Diagrams in Computation, Logic and Physics (STRING), Applied Category Theory (ACT), Algebra and Coalgebra in Computer Science (CALCO), and the Simons Workshop on Compositionality.

Submissions should be original contributions of previously unpublished work, and may be of any length. Work previously published in conferences and workshops must be significantly expanded or contain significant new results to be accepted. There is no deadline for submission. There is no processing charge for accepted publications; Compositionality is free to read and free to publish in. More details can be found in our editorial policies at http://www.compositionality-journal.org/editorial-policies/.

John Baez, University of California, Riverside, USA

Bob Coecke, University of Oxford, UK

Kathryn Hess, EPFL, Switzerland

Steve Lack, Macquarie University, Australia

Valeria de Paiva, Nuance Communications, USA

Corina Cirstea, University of Southampton, UK

Ross Duncan, University of Strathclyde, UK

Andree Ehresmann, University of Picardie Jules Verne, France

Tobias Fritz, Max Planck Institute, Germany

Neil Ghani, University of Strathclyde, UK

Dan Ghica, University of Birmingham, UK

Jeremy Gibbons, University of Oxford, UK

Nick Gurski, Case Western Reserve University, USA

Helle Hvid Hansen, Delft University of Technology, Netherlands

Chris Heunen, University of Edinburgh, UK

Aleks Kissinger, Radboud University, Netherlands

Joachim Kock, Universitat Autonoma de Barcelona, Spain

Martha Lewis, University of Amsterdam, Netherlands

Samuel Mimram, Ecole Polytechnique, France

Simona Paoli, University of Leicester, UK

Dusko Pavlovic, University of Hawaii, USA

Christian Retore, Universite de Montpellier, France

Mehrnoosh Sadrzadeh, Queen Mary University, UK

Peter Selinger, Dalhousie University, Canada

Pawel Sobocinski, University of Southampton, UK

David Spivak, MIT, USA

Jamie Vicary, University of Birmingham and University of Oxford, UK

Simon Willerton, University of Sheffield, UK

Sincerely,

The Editorial Board of Compositionality

]]>‘Compositional tasking’ means assigning tasks to networks agents in such a way that you can connect or even overlay such tasked networks and get larger ones. This lets you build up complex plans from smaller pieces.

In my last post in this series, I sketched an approach using ‘commitment networks’. A commitment network is a graph where nodes represent agents and edges represent commitments, like “A should move toward B either for 3 hours or until they meet, whichever comes first”. By overlaying such graphs we can build up commitment networks that describe complex plans of action. The rules for overlaying incorporate ‘automatic deconflicting’. In other words: don’t need to worry about agents being given conflicting duties as you stack up plans… because you’ve decided ahead of time what they should do in these situations.

I still like that approach, but we’ve been asked to develop some ideas more closely connected to traditional methods of tasking, like PERT charts, so now we’ve done that.

‘PERT’ stands for ‘program evaluation and review technique’. PERT charts were developed by the US Navy in 1957, but now they’re used all over industry to help plan and schedule large projects.

Here’s simple example:

The nodes in this graph are different **states**, like “you have built the car but not yet put on the tires”. The edges are different **tasks**, like “put the tires on the car”. Each state is labelled with an arbitrary name: 10, 20, 30, 40 and 50. The tasks also have names: A, B, C, D, E, and F. More importantly, each task is labelled by the amount of time that task requires!

Your goal is to start at state 10 and move all the way to state 50. Since you’re bossing lots of people around, you can make them do tasks simultaneously. However, you can only reach a state after you have done *all* the tasks leading up to that state. For example, you can’t reach state 50 unless you have already done *all* of tasks C, E, and F. Some typical questions are:

• What’s the minimum amount of time it takes to get from state 10 to state 50?

• Which tasks could take longer, without changing the answer to the previous question? How much longer could each task take, without changing the answer? This amount of time is called the **slack** for that task.

There are known algorithms for solving such problems. These help big organizations plan complex projects. So, connecting compositional tasking to PERT charts seems like a good idea.

At first this seemed confusing because in our previous work the nodes represented *agents*, while in PERT charts the nodes represent *states*. Of course graphs can be used for many things, even in the same setup. But the trick was getting everything to fit together nicely.

Now I think we’re close.

John Foley has been working out some nice example problems where a collection of agents need to move along the edges of a graph from specified start locations to specified end locations, taking routes that minimize their total fuel usage. However, there are some constraints. Some edges can only be traversed by specified *teams* of agents: they can’t go alone. Also, no one agent is allowed to run out of fuel.

This is a nice problem because while it’s pretty simple and specific, it’s representative of a large class of problems where a collection of agents are trying to carry out tasks together. ‘Moving along the edge of a graph’ can stand for a task of any sort. The constraint that some edges can only be traversed by specified teams is then a way of saying that certain tasks can only be accomplished by teams.

Furthermore, there are nice software packages for optimization subject to constraints. For example, John likes one called Choco. So, we plan to use one of these as part of the project.

What makes this all *compositional* is that John has expressed this problem using our ‘network model’ formalism, which I began sketching in Part 6. This allows us to assemble tasks for larger collections of agents from tasks for smaller collections.

Here, however, an idea due to my student Joe Moeller turned out to be crucial.

In our first examples of network models, explained earlier in this series, we allowed a *monoid* of networks for any set of agents of different kinds. A monoid has a binary operation called ‘multiplication’, and the idea here was this could describe the operation of ‘overlaying’ networks: for example, laying one set of communication channels, or committments, on top of another.

However, Joe knew full well that a monoid is a category with one object, so he pushed for a generalization that allowed not just a monoid but a *category* of networks for any set of agents of different kinds. I didn’t know what this was good for, but I figured: what the heck, let’s do it. It was a mathematically natural move, and it didn’t make anything harder—in fact it clarified some of our constructions, which is why Joe wanted to do it.

Now that generalization is proving to be crucial! We can take our category of networks to have *states* as objects and *tasks* (ways of moving between states) as morphisms! So, instead of ‘overlaying networks’, the basic operation is now *composing tasks*.

So, we now have a framework where if you specify a collection of agents of different kinds, we can give you the category whose morphisms are tasks those agents can engage in.

An example is John’s setup where the agents are moving around on a graph.

But this framework also handles PERT charts! While the folks who invented PERT charts didn’t think of them this way, one can think of them as describing categories of a certain specific sort, with states as objects and tasks as morphisms.

So, we now have a compositional framework for PERT charts.

I would like to dive deeper into the details, but this is probably enough for one post. I will say, though, that we use some math I’ve just developed with my grad student Jade Master, explained here:

• Open Petri nets (part 3), *Azimuth*, 19 August 2018.

The key is the relation between Petri nets and PERT charts. I’ll have more to say about that soon, I hope!

Some posts in this series:

• Part 1. CASCADE: the Complex Adaptive System Composition and Design Environment.

• Part 2. Metron’s software for system design.

• Part 3. Operads: the basic idea.

• Part 4. Network operads: an easy example.

• Part 5. Algebras of network operads: some easy examples.

• Part 6. Network models.

• Part 7. Step-by-step compositional design and tasking using commitment networks.

• Part 8. Compositional tasking using category-valued network models.

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