## Saving Climate Data (Part 2)

16 December, 2016

I want to get you involved in the Azimuth Environmental Data Backup Project, so click on that for more. But first some background.

A few days ago, many scientists, librarians, archivists, computer geeks and environmental activists started to make backups of US government environmental databases. We’re trying to beat the January 20th deadline just in case.

Backing up data is always a good thing, so there’s no point in arguing about politics or the likelihood that these backups are needed. The present situation is just a nice reason to hurry up and do some things we should have been doing anyway.

As of 2 days ago the story looked like this:

Saving climate data (Part 1), Azimuth, 13 December 2016.

A lot has happened since then, but much more needs to be done. Right now you can see a list of 90 databases to be backed up here:

Gov. Climate Datasets (Archive). (Click on the tiny word “Datasets” at the bottom of the page!)

Despite the word ‘climate’, the scope includes other environmental databases, and rightly so. Here is a list of databases that have been backed up:

By going here and clicking “Start Here to Help”:

you can nominate a dataset for rescue, claim a dataset to rescue, let everyone know about a data rescue event, or help in some other way (which you must specify). There’s also other useful information on this page, which was set up by Nick Santos.

The overall effort is being organized by the Penn Program in the Environmental Humanities, or ‘PPEHLab’ for short, headed by Bethany Wiggin. If you want to know what’s going on, it helps to look at their blog:

However, the people organizing the project are currently overwhelmed with offers of help! People worldwide are proceeding to take action in a decentralzed way! So, everything is a bit chaotic, and nobody has an overall view of what’s going on.

I can’t overstate this: if you think that ‘they’ have a plan and ‘they’ know what’s going on, you’re wrong. ‘They’ is us. Act accordingly.

Here’s a list of news articles, a list of ‘data rescue events’ where people get together with lots of computers and do stuff, and a bit about archives and archivists.

### News

Here are some things to read:

• Jason Koebler, Researchers are preparing for Trump to delete government science from the web, Vice, 13 December 2016.

• Brady Dennis, Scientists frantically copying U.S. climate data, fearing it might vanish under Trump, Washington Post, 13 December, 2016. (Also at the Chicago Tribune.)

• Eric Holthaus, Why I’m trying to preserve federal climate data before Trump takes office, Washington Post, 13 December 2016.

• Nicole Mortillaro, U of T heads ‘guerrilla archiving event’ to preserve climate data ahead of Trump presidency, CBC News, 14 December 2016.

• Audie Kornish and Eric Holthaus, Scientists race to preserve climate change data before Trump takes office, All Things Considered, National Public Radio, 14 December 2016.

### Data rescue events

There’s one in Toronto:

Guerrilla archiving event, 10 am – 4 pm EST, Saturday 17 December 2016. Location: Bissell Building, 4th Floor, 140 St. George St. University of Toronto.

There will be one in Philadelphia:

DataRescuePenn Data Harvesting, Friday–Saturday 13–14 January 2017. Location: not determined yet, probably somewhere at the University of Pennsylvania, Philadelphia.

I hear there will also be events in New York City and Los Angeles, but I don’t know details. If you do, please tell me!

### Archives and archivists

Today I helped catalyze a phone conversation between Bethany Wiggin, who heads the PPEHLab, and Nancy Beaumont, head of the Society of American Archivists. Digital archivists have a lot of expertise in saving information, so their skills are crucial here. Big wads of disorganized data are not very useful.

In this conversation I learned that some people are already in contact with the Internet Archive. This archive always tries to save US government websites and databases at the end of each presidential term. Their efforts are not limited to environmental data, and they save not only webpages but entire databases, e.g. data in ftp sites. You can nominate sites to be saved here:

• Internet Archive, End of Presidential Term Harvest 2016.

• Internet Archive blog, Preserving U.S. Government Websites and Data as the Obama Term Ends, 15 December 2016.

## Globular

14 December, 2016

One of my goals is to turn category theory, and even higher category theory, into a practical tool for science. For this we need good scientific ideas—but we also need good software.

My friend Jamie Vicary has been developing some of this software, together with Aleks Kissinger and Krzysztof Bar and others. Jamie demonstrated it at the Simons Institute workshop on compositionality. You can watch his demonstration here:

But since Globular runs on a web browser, you can also try it out yourself here:

Globular.

You can see his talk slides:

• Jamie Vicary, Data structures for quasistrict higher categories. (Talk slides here.)

Abstract. Higher category theory is one of the most general approaches to compositionality, with broad and striking applications across computer science, mathematics and physics. We present a new, simple way to define higher categories, in which many important compositional properties emerge as theorems, rather than axioms. Our approach is amenable to computer implementation, and we present a new proof assistant we have developed, with a powerful graphical calculus. In particular, we will outline a substantial new proof we have developed in our setting.

And in December 2015, he wrote an article about this software on the n-Category Café. It’s been improved since then, but it can’t hurt to read what he wrote—so I append it here!

### Globular: the basic idea

When you’re trying to prove something in a monoidal category, or a higher category, string diagrams are a really useful technique, especially when you’re trying to get an intuition for what you’re doing. But when it comes to writing up your results, the problems start to mount. For a complex proof, it’s hard to be sure your result is correct—a slip of the pen could lead to a false proof, and an error that’s hard to find. And writing up your results can be a huge time-sink, requiring weeks or months using a graphics package, all just for some nice pictures that tell you little about the correctness of the proof, and become useless if you decide to change your approach. Computers should be able help with all these things, in the way that proof assistants like Coq and Agda are allowing us to work with traditional syntactic proofs in a more sophisticated way.

The purpose of this post is to introduce Globular, a new proof assistant for working with higher-categorical proofs using string diagrams. It’s available at http://globular.science, with documentation on the nab. It’s web-based, so everything happens right in your browser: build formal proofs, visualize and step through them; keep your proofs private, share them with collaborators, or make them publicly available.

Before we get into the technical details, here’s a screenshot of Globular in action:

The main part of the screen shows a diagram, which in this case is 2-dimensional. It represents a composite 2-cell in a finitely-presented 2-category, with the blue and red regions representing objects, the lines representing 1-cells, and the vertices representing 2-cells. In fact, this 2d diagram is just an intermediate state of a 3d proof, through which we’re navigating with the ‘Slice’ controls in the top-right. The proof itself has been built up by composing the generators listed in the signature, down the left-hand side of the screen. (If you want to take a look at this proof yourself, you can go straight there; in the top-right, set ‘Project’ to 0, then increment the second ‘Slice’ counter to scroll through the proof.)

Globular has been developed so far in the Quantum Group in
the Oxford Computer Science department, by Krzysztof
Bar
, Katherine Casey, Aleks Kissinger, Jamie Vicary and Caspar Wylie. We haven’t quite got around to it yet, but Globular will be open-source, and we’re really keen for people to get involved and help build the software—there’s a huge amount to do! If you want to help out, get in touch.

### Mathematical foundations

Globular is based on the theory of finitely-presented semistrict n-categories; at the moment, it works up to the level of 3-categories, with an extension to 4-categories actively in development. (You can build cells of any dimension, but from 4-cells and up, some structures are missing.)

Definitions of n-category vary in how strict they are; a definition is semistrict when it’s as strict as possible, while still having the property that every weak n-category satisfies it, up to equivalence. Definitions of semistrict n-category are not unique: in dimension 3, Gray categories put all the weak structure in the interchangers, while Simpson snucategories put it all in the unitors. Globular implements the axioms of a Gray category, because this is the most appropriate for the graphical calculus: the interchangers can be seen graphically, as changes in height of the components of the diagram. By the theory of k-tuply monoidal n-categories, this also lets you build proofs in a monoidal category, or a braided monoidal category, or a monoidal 2-category.

The only things that Globular understands are $k$-cells, for some value of $k$. So if you want to build an n-category where an equation $f=g$ holds between n-cells, you have to do it by adding $(n+1)$-cells $a:f \to g$ and $b:g \to f$. If you then build some composite $C(f)$ involving $f$, you can apply the cell $a$ to obtain $C(g)$, and we interpret this as the equation $C(f) = C(g)$. In a slogan, this is equality via rewriting. This is consistent with the basic premise of homotopy type theory: treat your proofs as first-order structures, which can in turn be reasoned about themselves.

Globular can also handle invertibility in a nice way. For a cell $F:A \to B$ to be invertible, indicated by ticking a box in the signature, means that there also exists an invertible cell $F^{-1}: B \to A$, and invertible cells $\text{id}_A \to F . F^{-1}$ and $\text{id}_B \to F^{-1} . F$. This is a coinductive definition (see Mike Shulman’s nice post on this topic), since we’re defining the notion of invertibility in terms of itself in a higher dimension. This sort of a definition is great for proof assistants to work with, as it allows a lot of structure to be generated from a single compact definition.

### How it works

For a lot more details, take a look at the nLab page. Everything that happens in Globular involves in interaction between the signature on the left-hand side, and the diagram in the main part of the screen. The signature stores the ‘library’ of cells you have available, and the diagram is a particular composite of cells that you have constructed.

To construct a new diagram, clear whatever is currently displayed by clicking the ‘Clear’ button on the right, or pressing ‘c’. Then start by clicking the icon of a n-cell in your signature, which will make a diagram consisting just of that cell. Clicking on the icons of other $k$-generators for $0 < k \leq n$ will display a list of ways the cell can be attached, and when you choose one of these ways, the attachment will be performed, growing your n-diagram. (If you’re starting with a blank workspace you will only have a single 0-cell available, so you won’t be able to do this yet!) Clicking an $(n+1)$-cell $G$ displays a list of ways that your n-diagram $D$ can be rewritten, by identifying the source of $G$ as a subdiagram of $D$. Selecting one of these ways will implement the rewrite, by ‘cutting out’ the chosen subdiagram of $D$, and replacing it with the target of $G$.

Another way to modify the diagram is to click directly on it. Clicking near the edge of the diagram performs an attachment, while clicking in the interior of the diagram performs a rewrite. If more than one attachment or rewrite is consistent with your click, a little menu will pop up for you to choose what you want to do. When you move your mouse pointer over the diagram, a little label pops up to show you what your cursor is hovering over, which is helpful when choosing where to click.

You can also click-and-drag on the diagram. This will attach or rewrite with an interchanger, or naturality for an interchanger, or invertibility for an interchanger, depending on where you have clicked and the direction of the drag. Clicking and dragging is designed to work as if you were really ‘touching’ the strings. So if you want to braid one strand over another, click the strand to ‘grab’ it, and ‘pull’ it to one side. If you want to pull a vertex through a braiding, click the vertex to ‘grab’ it, and ‘pull’ it up or down through its adjacent braiding. Of course, Globular will only carry out the command if the move you are attempting to make is actually valid in that location.

### Example theorems

Here are four worked examples of nontrivial proofs in Globular:

Frobenius implies associative: http://globular.science/1512.004. In a monoidal category, if multiplication and comultiplication morphisms are unital, counital and Frobenius, then they are associative and coassociative.

Strengthening an equivalence: http://globular.science/1512.007. In a 2-category, an equivalence gives rise to an adjoint equivalence, satisfying the snake equations.

Swallowtail comes for free: http://globular.science/1512.006. In a monoidal 2-category, a weakly-dual pair of objects gives rise to a strongly-dual pair, satisfying the swallowtail equations.

Pentagon and triangle implies $\lambda_I = \rho_I$: http://globular.science/1512.002. In a monoidal 2-category, if a pseudomonoid object satisfies pentagon and triangle equations, then it satisfies $\lambda_I = \rho_I$.

We’ll focus on the second example project “Strengthening an equivalence” listed above, and see how it was constructed. This project investigates the factthat every equivalence in a 2-category gives rise to an adjoint equivalence. To start, we therefore need the basic data that exhibits an equivalence in a 2-category: two 0-cells $A$ and $B$, and an invertible 1-cell $F:A \to B$, by the weak definition of ‘invertible’ discussed above. This gives us the following signature:

The 2-cells that witness invertibility of $F$ look like cups and caps in the graphical calculus, but they won’t satisfy the snake equations that define an adjoint equivalence. The idea of this proof is to define a new cap, built out of the invertible structure of $F$, which does satisfy the snake equations with the existing cup.

By starting with a diagram consisting of $F$ alone, pressing ‘i’ to take the identity diagram, and clicking-and-dragging, we build the following 2-diagram, out of the invertible structure associated to $F$:

This is our candidate for our redefined cup. Its source is the identity on $A$, and its target is $F$ composed with $F^{-1}$. It looks a bit like the curved end of a hockey stick.

To store it for later use, we now click the ‘Theorem’ button. Writing $D$ for the 2-diagram we have constructed, this does two things. First, it creates a 2-cell generator that we call “New Cup”, whose source is $s(D)$, and whose target is $t(D)$. This is the redefined cup that we can use in future expressions. Second, it creates an invertible 3-cell generator that we call “New Cup Definition”, with source given by “New Cup”, and with target given by our hockey-stick diagram $D$. This says what “New Cup” means in terms of our original structure. This adds the following cells to our signature:

Because “New Cup Definition” is a 3-cell, by default we see two little icons: one for its source, and one for its target. See how its source is a little picture of “New Cup”, and its target is a little picture of the hockey stick, just as we defined it.

We’re now ready to prove one of the snake equations. We start by building the snake composite, using “New Cup” for the cup, and the invertible structure of $F$ for the cap:

Now have to prove that this equals the identity. Since equality is implemented by rewriting, we must construct a 3-diagram whose source is this snake composite, and whose target is the identity on $F$. To start, we click the ‘Identity’ button to convert our diagram into an identity 3-diagram. The only apparent effect this has is to add a number scroller to the ‘Slice’ area of the controls in the top-right. At the moment we can set this to ‘0’ and ‘1’, representing the source and target of our identity 3-diagram respectively. We set it to ‘1’, because we want to compose things to the target.

We now build up our proof. First, we click on the pink vertex that represents “New Cup”. This will attach our 3-cell “New Cup Definition”, replacing “New Cup” with our hockey-stick picture. By clicking-and-dragging on the diagram, we obtain the following sequence
of pictures:

Pictures 3 to 10 were created by attaching interchangers, and pictures 11 to 15 were created by attaching higher structure generated by the invertibility of $F$. In all cases, this structure was attached just by clicking-and-dragging on the appropriate vertices of the diagram. We’ve turned the snake into the identity, so we’ve finished our proof, which required 14 3-cells. By using the ‘Slice’ control in the top-right, we can navigate through the 15 slices that make up our proof, and review what we just did. Even better, turning the ‘Project’ control to the value ‘1’ tells Globular to project out the lowest dimension. This means that our entire 3-diagram proof can be viewed as a single 2-dimensional diagram, as follows:

This is just like the Morse singularity graphics used by topologists to study the structure of higher-dimensional manifolds. In this picture, the vertices are 3-cells, the lines are 2-cells, and the regions are 1-cells (in fact, every region is the 1-cell $F$.) By moving your mouse pointer over the different parts of the diagram, you can see what the different components are. Interchangers are represented in this projection by braidings.

Now we can do something cool: we can modify our proof, by clicking-and-dragging on the Morse projection. For example, just to the right of centre, there is a crossing, out of which emerge two long vertical lines that travel up a long way before annihilating with one another. Our proof would be much simpler if these two lines just annihilated with each other right after the interchanger. So, we click the vertex at the top of the lines, and drag it down repeatedly, until it gets to where we want it:

We’ve simplified our proof. By clicking-and-dragging some more, you can change the proof in lots of different ways, although you probably won’t get it much simpler than this. Putting the ‘Project’ control back to ‘0’, and navigating through the stages of the proof with the ‘Slice’ control as we were doing before, we can see that our proof has indeed been modified.

This project has been in development for about 18 months, so it feels great to finally launch. We hope the whole community will get clicking-and-dragging, and let us know how easy it is to use, and what other features would be useful. There are certain to still be bugs, so let us know about them too, and we’ll get right on them.

## Saving Climate Data (Part 1)

13 December, 2016

I try to stay out of politics on this website. This post is not mainly about politics. It’s a call to action. We’re trying to do something rather simple and clearly worthwhile. We’re trying to create backups of US government climate data.

The background is, of course, political. Many signs point to a dramatic change in US climate policy:

• Oliver Milman, Trump’s transition: sceptics guide every agency dealing with climate change, The Guardian, 12 December 2016.

So, scientists are now backing up large amounts of climate data, just in case the Trump administration tries to delete it after he takes office on January 20th:

• Brady Dennis, Scientists are frantically copying U.S. climate data, fearing it might vanish under Trump, Washington Post, 13 December 2016.

Of course saving the data publicly available on US government sites is not nearly as good as keeping climate programs fully funded! New data is coming in all the time from satellites and other sources. We need it—and we need the experts who understand it.

Also, it’s possible that the Trump administration won’t go so far as trying to delete big climate science databases. Still, I think it can’t be a bad thing to have backups. Or as my mother always said: better safe than sorry!

Quoting the Washington Post article:

Alarmed that decades of crucial climate measurements could vanish under a hostile Trump administration, scientists have begun a feverish attempt to copy reams of government data onto independent servers in hopes of safeguarding it from any political interference.

The efforts include a “guerrilla archiving” event in Toronto, where experts will copy irreplaceable public data, meetings at the University of Pennsylvania focused on how to download as much federal data as possible in the coming weeks, and a collaboration of scientists and database experts who are compiling an online site to harbor scientific information.

“Something that seemed a little paranoid to me before all of a sudden seems potentially realistic, or at least something you’d want to hedge against,” said Nick Santos, an environmental researcher at the University of California at Davis, who over the weekend began copying government climate data onto a nongovernment server, where it will remain available to the public. “Doing this can only be a good thing. Hopefully they leave everything in place. But if not, we’re planning for that.”

[…]

“What are the most important .gov climate assets?” Eric Holthaus, a meteorologist and self-proclaimed “climate hawk,” tweeted from his Arizona home Saturday evening. “Scientists: Do you have a US .gov climate database that you don’t want to see disappear?”

Within hours, responses flooded in from around the country. Scientists added links to dozens of government databases to a Google spreadsheet. Investors offered to help fund efforts to copy and safeguard key climate data. Lawyers offered pro bono legal help. Database experts offered to help organize mountains of data and to house it with free server space. In California, Santos began building an online repository to “make sure these data sets remain freely and broadly accessible.”

In Philadelphia, researchers at the University of Pennsylvania, along with members of groups such as Open Data Philly and the software company Azavea, have been meeting to figure out ways to harvest and store important data sets.

At the University of Toronto this weekend, researchers are holding what they call a “guerrilla archiving” event to catalogue key federal environmental data ahead of Trump’s inauguration. The event “is focused on preserving information and data from the Environmental Protection Agency, which has programs and data at high risk of being removed from online public access or even deleted,” the organizers said. “This includes climate change, water, air, toxics programs.”

The event is part of a broader effort to help San Francisco-based Internet Archive with its End of Term 2016 project, an effort by university, government and nonprofit officials to find and archive valuable pages on federal websites. The project has existed through several presidential transitions.

I hope that small “guerilla archiving” efforts will be dwarfed by more systematic work, because it’s crucial that databases be copied along with all relevant metadata—and some sort of cryptographic certificate of authenticity, if possible. However, getting lots of people involved is bound to be a good thing, politically speaking.

If you have good computer skills, good understanding of databases, or lots of storage space, please get involved. Efforts are being coordinated by Barbara Wiggin and others at the Data Refuge Project:

• PPEHLab (Penn Program in the Environmental Humanities), DataRefuge.

You can contact them at DataRefuge@ppehlab.org. Nick Santos is also involved, and if you want to get “more plugged into the project” you can contact him here. They are trying to build a climate database mirror website here:

At the help form on this website you can nominate a dataset for rescue, claim a dataset to rescue, let them know about a data rescue event, or help in some other way (which you must specify).

PPEHLab and Penn Libraries are organizing a data rescue event this Thursday:

• PPEHLab, DataRefuge meeting, 14 December 2016.

At the American Geophysical Union meeting in San Francisco, where more than 20,000 earth and climate scientists gather from around the world, there was a public demonstration today starting at 1:30 PST:

Rally to stand up for science, 13 December 2016.

And the “guerilla archiving” hackathon in Toronto is this Saturday—see below. If you know people with good computer skills in Toronto, get them to check it out!

### Guerrilla archiving in Toronto

Here are details on this:

Guerrilla Archiving Hackathon

Date: 10am-4pm, December 17, 2016

Location: Bissell Building, 4th Floor, 140 St. George St. University of Toronto

RSVP and up-to-date information: Guerilla archiving: saving environmental data from Trump.

Bring: laptops, power bars, and snacks. Coffee and pizza provided.

This event collaborates with the Internet Archive’s End of Term 2016 project, which seeks to archive the federal online pages and data that are in danger of disappearing during the Trump administration. Our event is focused on preserving information and data from the Environmental Protection Agency, which has programs and data at high risk of being removed from online public access or even deleted. This includes climate change, water, air, toxics programs. This project is urgent because the Trump transition team has identified the EPA and other environmental programs as priorities for the chopping block.

The Internet Archive is a San Francisco-based nonprofit digital library which aims at preserving and making universally accessible knowledge. Its End of Term web archive captures and saves U.S. Government websites that are at risk of changing or disappearing altogether during government transitions. The Internet Archive has asked volunteers to help select and organize information that will be preserved before the Trump transition.

End of Term web archive: http://eotarchive.cdlib.org/2016.html

New York Times article: “Harvesting Government History, One Web Page at a Time

Activities:

Identifying endangered programs and data

Seeding the End of Term webcrawler with priority URLs

Identifying and mapping the location of inaccessible environmental databases

Hacking scripts to make accessible to the webcrawler hard to reach databases.

Building a toolkit so that other groups can hold similar events

Skills needed: We need all kinds of people — and that means you!

People who can locate relevant webpages for the Internet Archive’s webcrawler

People who can identify data targeted for deletion by the Trump transition team and the organizations they work with

People with knowledge of government websites and information, including the EPA

People with library and archive skills

People who are good at navigating databases

People interested in mapping where inaccessible data is located at the EPA

Hackers to figure out how to extract data and URLs from databases (in a way that Internet Archive can use)

People with good organization and communication skills

People interested in creating a toolkit for reproducing similar events

Contacts: michelle.murphy@utoronto.ca, p.keilty@utoronto.ca

## Modelling Interconnected Systems with Decorated Corelations

9 December, 2016

Here at the Simons Institute workshop on compositionality, my talk on network theory explained how to use ‘decorated cospans’ as a general model of open systems. These were invented by Brendan Fong, and are nicely explained in his thesis:

• Brendan Fong, The Algebra of Open and Interconnected Systems. (Blog article here.)

But he went further: to understand the externally observable behavior of an open system we often want to simplify a decorated cospan and get another sort of structure, which he calls a ‘decorated corelation’.

In this talk, Brendan explained decorated corelations and what they’re good for:

• Brendan Fong, Modelling interconnected systems with decorated corelations. (Talk slides here.)

Abstract. Hypergraph categories are monoidal categories in which every object is equipped with a special commutative Frobenius monoid. Morphisms in a hypergraph category can hence be represented by string diagrams in which strings can branch and split: diagrams that are reminiscent of electrical circuit diagrams. As such they provide a framework for formalising the syntax and semantics of circuit-type diagrammatic languages. In this talk I will introduce decorated corelations as a tool for building hypergraph categories and hypergraph functors, drawing examples from linear algebra and dynamical systems.

## Semantics for Physicists

7 December, 2016

I once complained that my student Brendan Fong said ‘semantics’ too much. You see, I’m in a math department, but he was actually in the computer science department at Oxford: I was his informal supervisor. Theoretical computer scientists love talking about syntax versus semantics—that is, written expressions versus what those expressions actually mean, or programs versus what those programs actually do. So Brendan was very comfortable with that distinction. But my other grad students, coming from a math department didn’t understand it… and he was mentioning it in practically ever other sentence.

In 1963, in his PhD thesis, Bill Lawvere figured out a way to talk about syntax versus semantics that even mathematicians—well, even category theorists—could understand. It’s called ‘functorial semantics’. The idea is that things you write are morphisms in some category $X,$ while their meanings are morphisms in some other category $Y.$ There’s a functor $F \colon X \to Y$ which sends things you write to their meanings. This functor sends syntax to semantics!

But physicists may not enjoy this idea unless they see it at work in physics. In physics, too, the distinction is important! But it takes a while to understand. I hope Prakash Panangaden’s talk at the start of the Simons Institute workshop on compositionality is helpful. Check it out:

## Compositionality in Network Theory

29 November, 2016

I gave a talk at the workshop on compositionality at the Simons Institute for the Theory of Computing next week. I spoke about some new work with Blake Pollard. You can see the slides here:

• John Baez, Compositionality in network theory, 6 December 2016.

and a video here:

Abstract. To describe systems composed of interacting parts, scientists and engineers draw diagrams of networks: flow charts, Petri nets, electrical circuit diagrams, signal-flow graphs, chemical reaction networks, Feynman diagrams and the like. In principle all these different diagrams fit into a common framework: the mathematics of symmetric monoidal categories. This has been known for some time. However, the details are more challenging, and ultimately more rewarding, than this basic insight. Two complementary approaches are presentations of symmetric monoidal categories using generators and relations (which are more algebraic in flavor) and decorated cospan categories (which are more geometrical). In this talk we focus on the latter.

This talk assumes considerable familiarity with category theory. For a much gentler talk on the same theme, see:

## Compositional Frameworks for Open Systems

27 November, 2016

Here are the slides of Blake Pollard’s talk at the Santa Fe Institute workshop on Statistical Physics, Information Processing and Biology:

• Blake Pollard, Compositional frameworks for open systems, 17 November 2016.

He gave a really nice introduction to how we can use categories to study open systems, with his main example being ‘open Markov processes’, where probability can flow in and out of the set of states. People liked it a lot!