Biology, Networks and Control Theory

13 September, 2015

The Institute for Mathematics and its Applications (or IMA, in Minneapolis, Minnesota), is teaming up with the Mathematical Biosciences Institute (or MBI, in Columbus, Ohio). They’re having a big program on control theory and networks:

IMA-MBI coordinated program on network dynamics and control: Fall 2015 and Spring 2016.

MBI emphasis semester on dynamics of biologically inspired networks: Spring 2016.

At the IMA

Here’s what’s happening at the Institute for Mathematics and its Applications:

Concepts and techniques from control theory are becoming increasingly interdisciplinary. At the same time, trends in modern control theory are influenced and inspired by other disciplines. As a result, the systems and control community is rapidly broadening its scope in a variety of directions. The IMA program is designed to encourage true interdisciplinary research and the cross fertilization of ideas. An important element for success is that ideas flow across disciplines in a timely manner and that the cross-fertilization takes place in unison.

Due to the usefulness of control, talent from control theory is drawn and often migrates to other important areas, such as biology, computer science, and biomedical research, to apply its mathematical tools and expertise. It is vital that while the links are strong, we bring together researchers who have successfully bridged into other disciplines to promote the role of control theory and to focus on the efforts of the controls community. An IMA investment in this area will be a catalyst for many advances and will provide the controls community with a cohesive research agenda.

In all topics of the program the need for research is pressing. For instance, viable implementations of control algorithms for smart grids are an urgent and clearly recognized need with considerable implications for the environment and quality of life. The mathematics of control will undoubtedly influence technology and vice-versa. The urgency for these new technologies suggests that the greatest impact of the program is to have it sooner rather than later.

First trimester (Fall 2015): Networks, whether social, biological, swarms of animals or vehicles, the Internet, etc., constitute an increasingly important subject in science and engineering. Their connectivity and feedback pathways affect robustness and functionality. Such concepts are at the core of a new and rapidly evolving frontier in the theory of dynamical systems and control. Embedded systems and networks are already pervasive in automotive, biological, aerospace, and telecommunications technologies and soon are expected to impact the power infrastructure (smart grids). In this new technological and scientific realm, the modeling and representation of systems, the role of feedback, and the value and cost of information need to be re-evaluated and understood. Traditional thinking that is relevant to a limited number of feedback loops with practically unlimited bandwidth is no longer applicable. Feedback control and stability of network dynamics is a relatively new endeavor. Analysis and control of network dynamics will occupy mostly the first trimester while applications to power networks will be a separate theme during the third trimester. The first trimester will be divided into three workshops on the topics of analysis of network dynamics and regulation, communication and cooperative control over networks, and a separate one on biological systems and networks.

The second trimester (Winter 2016) will have two workshops. The first will be on modeling and estimation (Workshop 4) and the second one on distributed parameter systems and partial differential equations (Workshop 5). The theme of Workshop 4 will be on structure and parsimony in models. The goal is to explore recent relevant theories and techniques that allow sparse representations, rank constrained optimization, and structural constraints in models and control designs. Our intent is to blend a group of researchers in the aforementioned topics with a select group of researchers with interests in a statistical viewpoint. Workshop 5 will focus on distributed systems and related computational issues. One of our aims is to bring control theorists with an interest in optimal control of distributed parameter systems together with mathematicians working on optimal transport theory (in essence an optimal control problem). The subject of optimal transport is rapidly developing with ramifications in probability and statistics (of essence in system modeling and hence of interest to participants in Workshop 4 as well) and in distributed control of PDE’s. Emphasis will also be placed on new tools and new mathematical developments (in PDE’s, computational methods, optimization). The workshops will be closely spaced to facilitate participation in more than one.

The third trimester (Spring 2016) will target applications where the mathematics of systems and control may soon prove to have a timely impact. From the invention of atomic force microscopy at the nanoscale to micro-mirror arrays for a next generation of telescopes, control has played a critical role in sensing and imaging of challenging new realms. At present, thanks to recent technological advances of AFM and optical tweezers, great strides are taking place making it possible to manipulate the biological transport of protein molecules as well as the control of individual atoms. Two intertwined subject areas, quantum and nano control and scientific instrumentation, are seen to blend together (Workshop 6) with partial focus on the role of feedback control and optimal filtering in achieving resolution and performance at such scales. A second theme (Workshop 7) will aim at control issues in distributed hybrid systems, at a macro scale, with a specific focus the “smart grid” and energy applications.

For more information on individual workshops, go here:

• Workshop 1, Distributed Control and Decision Making Over Networks, 28 September – 2 October 2015.

• Workshop 2, Analysis and Control of Network Dynamics, 19-23 October 2015.

• Workshop 3, Biological Systems and Networks, 11-16 November 2015.

• Workshop 4, Optimization and Parsimonious Modeling, 25-29 January 2016.

• Workshop 5, Computational Methods for Control of Infinite-dimensional Systems, 14-18 March 2016.

• Workshop 6, Quantum and Nano Control, 11-15 April 2016.

• Workshop 7, Control at Large Scales: Energy Markets and Responsive Grids, 9-13 March 2016.

At the MBI

Here’s what’s going on at the Mathematical Biology Institute:

The MBI network program is part of a yearlong cooperative program with IMA.

Networks and deterministic and stochastic dynamical systems on networks are used as models in many areas of biology. This underscores the importance of developing tools to understand the interplay between network structures and dynamical processes, as well as how network dynamics can be controlled. The dynamics associated with such models are often different from what one might traditionally expect from a large system of equations, and these differences present the opportunity to develop exciting new theories and methods that should facilitate the analysis of specific models. Moreover, a nascent area of research is the dynamics of networks in which the networks themselves change in time, which occurs, for example, in plasticity in neuroscience and in up regulation and down regulation of enzymes in biochemical systems.

There are many areas in biology (including neuroscience, gene networks, and epidemiology) in which network analysis is now standard. Techniques from network science have yielded many biological insights in these fields and their study has yielded many theorems. Moreover, these areas continue to be exciting areas that contain both concrete and general mathematical problems. Workshop 1 explores the mathematics behind the applications in which restrictions on general coupled systems are important. Examples of such restrictions include symmetry, Boolean dynamics, and mass-action kinetics; and each of these special properties permits the proof of theorems about dynamics on these special networks.

Workshop 2 focuses on the interplay between stochastic and deterministic behavior in biological networks. An important related problem is to understand how stochasticity affects parameter estimation. Analyzing the relationship between stochastic changes, network structure, and network dynamics poses mathematical questions that are new, difficult, and fascinating.

In recent years, an increasing number of biological systems have been modeled using networks whose structure changes in time or which use multiple kinds of couplings between the same nodes or couplings that are not just pairwise. General theories such as groupoids and hypergraphs have been developed to handle the structure in some of these more general coupled systems, and specific application models have been studied by simulation. Workshop 3 will bring together theorists, modelers, and experimentalists to address the modeling of biological systems using new network structures and the analysis of such structures.

Biological systems use control to achieve desired dynamics and prevent undesirable behaviors. Consequently, the study of network control is important both to reveal naturally evolved control mechanisms that underlie the functioning of biological systems and to develop human-designed control interventions to recover lost function, mitigate failures, or repurpose biological networks. Workshop 4 will address the challenging subjects of control and observability of network dynamics.


Workshop 1: Dynamics in Networks with Special Properties, 25-29 January, 2016.

Workshop 2: The Interplay of Stochastic and Deterministic Dynamics in Networks, 22-26 February, 2016.

Workshop 3: Generalized Network Structures and Dynamics, 21-15 March, 2016.

Workshop 4: Control and Observability of Network Dynamics, 11-15 April, 2016.

You can get more schedule information on these posters:

A Compositional Framework for Markov Processes

4 September, 2015


This summer my students Brendan Fong and Blake Pollard visited me at the Centre for Quantum Technologies, and we figured out how to understand open continuous-time Markov chains! I think this is a nice step towards understanding the math of living systems.

Admittedly, it’s just a small first step. But I’m excited by this step, since Blake and I have been trying to get this stuff to work for a couple years, and it finally fell into place. And we think we know what to do next.

Here’s our paper:

• John C. Baez, Brendan Fong and Blake S. Pollard, A compositional framework for open Markov processes.

And here’s the basic idea…

Open detailed balanced Markov processes

A continuous-time Markov chain is a way to specify the dynamics of a population which is spread across some finite set of states. Population can flow between the states. The larger the population of a state, the more rapidly population flows out of the state. Because of this property, under certain conditions the populations of the states tend toward an equilibrium where at any state the inflow of population is balanced by its outflow.

In applications to statistical mechanics, we are often interested in equilibria such that for any two states connected by an edge, say i and j, the flow from i to j equals the flow from j to i. A continuous-time Markov chain with a chosen equilibrium having this property is called ‘detailed balanced‘.

I’m getting tired of saying ‘continuous-time Markov chain’, so from now on I’ll just say ‘Markov process’, just because it’s shorter. Okay? That will let me say the next sentence without running out of breath:

Our paper is about open detailed balanced Markov processes.

Here’s an example:

The detailed balanced Markov process itself consists of a finite set of states together with a finite set of edges between them, with each state i labelled by an equilibrium population q_i >0, and each edge e labelled by a rate constant r_e > 0.

These populations and rate constants are required to obey an equation called the ‘detailed balance condition’. This equation means that in equilibrium, the flow from i to j equal the flow from j to i. Do you see how it works in this example?

To get an ‘open’ detailed balanced Markov process, some states are designated as inputs or outputs. In general each state may be specified as both an input and an output, or as inputs and outputs multiple times. See how that’s happening in this example? It may seem weird, but it makes things work better.

People usually say Markov processes are all about how probabilities flow from one state to another. But we work with un-normalized probabilities, which we call ‘populations’, rather than probabilities that must sum to 1. The reason is that in an open Markov process, probability is not conserved: it can flow in or out at the inputs and outputs. We allow it to flow both in and out at both the input states and the output states.

Our most fundamental result is that there’s a category \mathrm{DetBalMark} where a morphism is an open detailed balanced Markov process. We think of it as a morphism from its inputs to its outputs.

We compose morphisms in \mathrm{DetBalMark} by identifying the output states of one open detailed balanced Markov process with the input states of another. The populations of identified states must match. For example, we may compose this morphism N:

with the previously shown morphism M to get this morphism M \circ N:

And here’s our second most fundamental result: the category \mathrm{DetBalMark} is actually a dagger compact category. This lets us do other stuff with open Markov processes. An important one is ‘tensoring’, which lets us take two open Markov processes like M and N above and set them side by side, giving M \otimes N:

The so-called compactness is also important. This means we can take some inputs of an open Markov process and turn them into outputs, or vice versa. For example, using the compactness of \mathrm{DetBalMark} we can get this open Markov process from M:

In fact all the categories in our paper are dagger compact categories, and all our functors preserve this structure. Dagger compact categories are a well-known framework for describing systems with inputs and outputs, so this is good.

The analogy to electrical circuits

In a detailed balanced Markov process, population can flow along edges. In the detailed balanced equilibrium, without any flow of population from outside, the flow along from state i to state j will be matched by the flow back from j to i. The populations need to take specific values for this to occur.

In an electrical circuit made of linear resistors, charge can flow along wires. In equilibrium, without any driving voltage from outside, the current along each wire will be zero. The potentials will be equal at every node.

This sets up an analogy between detailed balanced continuous-time Markov chains and electrical circuits made of linear resistors! I love analogy charts, so this makes me very happy:

    Circuits    Detailed balanced Markov processes
potential population
current flow
conductance rate constant
power dissipation

This analogy is already well known. Schnakenberg used it in his book Thermodynamic Network Analysis of Biological Systems. So, our main goal is to formalize and exploit it. This analogy extends from systems in equilibrium to the more interesting case of nonequilibrium steady states, which are the main topic of our paper.

Earlier, Brendan and I introduced a way to ‘black box’ a circuit and define the relation it determines between potential-current pairs at the input and output terminals. This relation describes the circuit’s external behavior as seen by an observer who can only perform measurements at the terminals.

An important fact is that black boxing is ‘compositional’: if one builds a circuit from smaller pieces, the external behavior of the whole circuit can be determined from the external behaviors of the pieces. For category theorists, this means that black boxing is a functor!

Our new paper with Blake develops a similar ‘black box functor’ for detailed balanced Markov processes, and relates it to the earlier one for circuits.

When you black box a detailed balanced Markov process, you get the relation between population–flow pairs at the terminals. (By the ‘flow at a terminal’, we more precisely mean the net population outflow.) This relation holds not only in equilibrium, but also in any nonequilibrium steady state. Thus, black boxing an open detailed balanced Markov process gives its steady state dynamics as seen by an observer who can only measure populations and flows at the terminals.

The principle of minimum dissipation

At least since the work of Prigogine, it’s been widely accepted that a large class of systems minimize entropy production in a nonequilibrium steady state. But people still fight about the the precise boundary of this class of systems, and even the meaning of this ‘principle of minimum entropy production’.

For detailed balanced open Markov processes, we show that a quantity we call the ‘dissipation’ is minimized in any steady state. This is a quadratic function of the populations and flows, analogous to the power dissipation of a circuit made of resistors. We make no claim that this quadratic function actually deserves to be called ‘entropy production’. Indeed, Schnakenberg has convincingly argued that they are only approximately equal.

But still, the ‘dissipation’ function is very natural and useful—and Prigogine’s so-called ‘entropy production’ is also a quadratic function.

Black boxing

I’ve already mentioned the category \mathrm{DetBalMark}, where a morphism is an open detailed balanced Markov process. But our paper needs two more categories to tell its story! There’s the category of circuits, and the category of linear relations.

A morphism in the category \mathrm{Circ} is an open electrical circuit made of resistors: that is, a graph with each edge labelled by a ‘conductance’ c_e > 0, together with specified input and output nodes:

A morphism in the category \mathrm{LinRel} is a linear relation L : U \leadsto V between finite-dimensional real vector spaces U and V. This is nothing but a linear subspace L \subseteq U \oplus V. Just as relations generalize functions, linear relations generalize linear functions!

In our previous paper, Brendan and I introduced these two categories and a functor between them, the ‘black box functor’:

\blacksquare : \mathrm{Circ} \to \mathrm{LinRel}

The idea is that any circuit determines a linear relation between the potentials and net current flows at the inputs and outputs. This relation describes the behavior of a circuit of resistors as seen from outside.

Our new paper introduces a black box functor for detailed balanced Markov processes:

\square : \mathrm{DetBalMark} \to \mathrm{LinRel}

We draw this functor as a white box merely to distinguish it from the other black box functor. The functor \square maps any detailed balanced Markov process to the linear relation obeyed by populations and flows at the inputs and outputs in a steady state. In short, it describes the steady state behavior of the Markov process ‘as seen from outside’.

How do we manage to black box detailed balanced Markov processes? We do it using the analogy with circuits!

The analogy becomes a functor

Every analogy wants to be a functor. So, we make the analogy between detailed balanced Markov processes and circuits precise by turning it into a functor:

K : \mathrm{DetBalMark} \to \mathrm{Circ}

This functor converts any open detailed balanced Markov process into an open electrical circuit made of resistors. This circuit is carefully chosen to reflect the steady-state behavior of the Markov process. Its underlying graph is the same as that of the Markov process. So, the ‘states’ of the Markov process are the same as the ‘nodes’ of the circuit.

Both the equilibrium populations at states of the Markov process and the rate constants labelling edges of the Markov process are used to compute the conductances of edges of this circuit. In the simple case where the Markov process has exactly one edge from any state i to any state j, the rule is this:

C_{i j} = H_{i j} q_j


q_j is the equilibrium population of the jth state of the Markov process,

H_{i j} is the rate constant for the edge from the jth state to the ith state of the Markov process, and

C_{i j} is the conductance (that is, the reciprocal of the resistance) of the wire from the jth node to the ith node of the resulting circuit.

The detailed balance condition for Markov processes says precisely that the matrix C_{i j} is symmetric! This is just right for an electrical circuit made of resistors, since it means that the resistance of the wire from node i to node j equals the resistance of the same wire in the reverse direction, from node j to node i.

A triangle of functors

If you paid careful attention, you’ll have noticed that I’ve described a triangle of functors:

And if you know anything about how category theorists think, you’ll be wondering if this diagram commutes.

In fact, this triangle of functors does not commute! However, a general lesson of category theory is that we should only expect diagrams of functors to commute up to natural isomorphism, and this is what happens here:

The natural transformation \alpha ‘corrects’ the black box functor for resistors to give the one for detailed balanced Markov processes.

The functors \square and \blacksquare \circ K are actually equal on objects. An object in \mathrm{DetBalMark} is a finite set X with each element i \in X labelled a positive populations q_i. Both functors map this object to the vector space \mathbb{R}^X \oplus \mathbb{R}^X. For the functor \square, we think of this as a space of population-flow pairs. For the functor \blacksquare \circ K, we think of it as a space of potential-current pairs. The natural transformation \alpha then gives a linear relation

\alpha_{X,q} : \mathbb{R}^X \oplus \mathbb{R}^X \leadsto \mathbb{R}^X \oplus \mathbb{R}^X

in fact an isomorphism of vector spaces, which converts potential-current pairs into population-flow pairs in a manner that depends on the q_i. I’ll skip the formula; it’s in the paper.

But here’s the key point. The naturality of \alpha actually allows us to reduce the problem of computing the functor \square to the problem of computing \blacksquare. Suppose

M: (X,q) \to (Y,r)

is any morphism in \mathrm{DetBalMark}. The object (X,q) is some finite set X labelled by populations q, and (Y,r) is some finite set Y labelled by populations r. Then the naturality of \alpha means that this square commutes:

Since \alpha_{X,q} and \alpha_{Y,r} are isomorphisms, we can solve for the functor \square as follows:

\square(M) = \alpha_Y \circ \blacksquare K(M) \circ \alpha_X^{-1}

This equation has a clear intuitive meaning! It says that to compute the behavior of a detailed balanced Markov process, namely \square(f), we convert it into a circuit made of resistors and compute the behavior of that, namely \blacksquare K(f). This is not equal to the behavior of the Markov process, but we can compute that behavior by converting the input populations and flows into potentials and currents, feeding them into our circuit, and then converting the outputs back into populations and flows.

What we really do

So that’s a sketch of what we do, and I hope you ask questions if it’s not clear. But I also hope you read our paper! Here’s what we actually do in there. After an introduction and summary of results:

• Section 3 defines open Markov processes and the open master equation.

• Section 4 introduces detailed balance for open Markov

• Section 5 recalls the principle of minimum power
for open circuits made of linear resistors, and explains how to black box them.

• Section 6 introduces the principle of minimum dissipation for open detailed balanced Markov processes, and describes how to black box these.

• Section 7 states the analogy between circuits and detailed balanced Markov processes in a formal way.

• Section 8 describes how to compose open Markov processes, making them into the morphisms of a category.

• Section 9 does the same for detailed balanced Markov processes.

• Section 10 describes the ‘black box functor’ that sends any open detailed balanced Markov process to the linear relation describing its external behavior, and recalls the similar functor for circuits.

• Section 11 makes the analogy between between open detailed balanced Markov processes and open circuits even more formal, by making it into a functor. We prove that together with the two black box functors, this forms a triangle that commutes up to natural isomorphism.

• Section 12 is about geometric aspects of this theory. We show that the linear relations in the image of these black box functors are Lagrangian relations between symplectic vector spaces. We also show that the master equation can be seen as a gradient flow equation.

• Section 13 is a summary of what we have learned.

Finally, Appendix A is a quick tutorial on decorated cospans. This is a key mathematical tool in our work, developed by Brendan in an earlier paper.

Trends in Reaction Network Theory (Part 2)

1 July, 2015

Here in Copenhagen we’ll soon be having a bunch of interesting talks on chemical reaction networks:

Workshop on Mathematical Trends in Reaction Network Theory, 1-3 July 2015, Department of Mathematical Sciences, University of Copenhagen. Organized by Elisenda Feliu and Carsten Wiuf.

Looking through the abstracts, here are a couple that strike me.

First of all, Gheorghe Craciun claims to have proved the biggest open conjecture in this field: the Global Attractor Conjecture!

• Gheorge Craciun, Toric differential inclusions and a proof of the global attractor conjecture.

This famous old conjecture says that for a certain class of chemical reactions, the ones coming from ‘complex balanced reaction networks’, the chemicals will approach equilibrium no matter what their initial concentrations are. Here’s what Craciun says:

Abstract. In a groundbreaking 1972 paper Fritz Horn and Roy Jackson showed that a complex balanced mass-action system must have a unique locally stable equilibrium within any compatibility class. In 1974 Horn conjectured that this equilibrium is a global attractor, i.e., all solutions in the same compatibility class must converge to this equilibrium. Later, this claim was called the Global Attractor Conjecture, and it was shown that it has remarkable implications for the dynamics of large classes of polynomial and power-law dynamical systems, even if they are not derived from mass-action kinetics. Several special cases of this conjecture have been proved during the last decade. We describe a proof of the conjecture in full generality. In particular, it will follow that all detailed balanced mass action systems and all deficiency zero mass-action systems have the global attractor property. We will also discuss some implications for biochemical mechanisms that implement noise filtering and cellular homeostasis.

Manoj Gopalkrishnan wrote a great post explaining the concept of complex balanced reaction network here on Azimuth, so if you want to understand the conjecture you could start there.

Even better, Manoj is talking here about a way to do statistical inference with chemistry! His talk is called ‘Statistical inference with a chemical soup’:

Abstract. The goal is to design an “intelligent chemical soup” that can do statistical inference. This may have niche technological applications in medicine and biological research, as well as provide fundamental insight into the workings of biochemical reaction pathways. As a first step towards our goal, we describe a scheme that exploits the remarkable mathematical similarity between log-linear models in statistics and chemical reaction networks. We present a simple scheme that encodes the information in a log-linear model as a chemical reaction network. Observed data is encoded as initial concentrations, and the equilibria of the corresponding mass-action system yield the maximum likelihood estimators. The simplicity of our scheme suggests that molecular environments, especially within cells, may be particularly well suited to performing statistical computations.

It’s based on this paper:

• Manoj Gopalkrishnan, A scheme for molecular computation of maximum likelihood estimators for log-linear models.

I’m not sure, but this idea may exploit existing analogies between the approach to equilibrium in chemistry, the approach to equilibrium in evolutionary game theory, and statistical inference. You may have read Marc Harper’s post about that stuff!

David Doty is giving a broader review of ‘Computation by (not about) chemistry’:

Abstract. The model of chemical reaction networks (CRNs) is extensively used throughout the natural sciences as a descriptive language for existing chemicals. If we instead think of CRNs as a programming language for describing artificially engineered chemicals, what sorts of computations are possible for these chemicals to achieve? The answer depends crucially on several formal choices:

1) Do we treat matter as infinitely divisible (real-valued concentrations) or atomic (integer-valued counts)?

2) How do we represent the input and output of the computation (e.g., Boolean presence or absence of species, positive numbers directly represented by counts/concentrations, positive and negative numbers represented indirectly by the difference between counts/concentrations of a pair of species)?

3) Do we assume mass-action rate laws (reaction rates proportional to reactant counts/concentrations) or do we insist the system works correctly under a broader class of rate laws?

The talk will survey several recent results and techniques. A primary goal of the talk is to convey the “programming perspective”: rather than asking “What does chemistry do?”, we want to understand “What could chemistry do?” as well as “What can chemistry provably not do?”

I’m really interested in chemical reaction networks that appear in biological systems, and there will be lots of talks about that. For example, Ovidiu Radulescu will talk about ‘Taming the complexity of biochemical networks through model reduction and tropical geometry’. Model reduction is the process of simplifying complicated models while preserving at least some of their good features. Tropical geometry is a cool version of algebraic geometry that uses the real numbers with minimization as addition and addition as multiplication. This number system underlies the principle of least action, or the principle of maximum energy. Here is Radulescu’s abstract:

Abstract. Biochemical networks are used as models of cellular physiology with diverse applications in biology and medicine. In the absence of objective criteria to detect essential features and prune secondary details, networks generated from data are too big and therefore out of the applicability of many mathematical tools for studying their dynamics and behavior under perturbations. However, under circumstances that we can generically denote by multi-scaleness, large biochemical networks can be approximated by smaller and simpler networks. Model reduction is a way to find these simpler models that can be more easily analyzed. We discuss several model reduction methods for biochemical networks with polynomial or rational rate functions and propose as their common denominator the notion of tropical equilibration, meaning finite intersection of tropical varieties in algebraic geometry. Using tropical methods, one can strongly reduce the number of variables and parameters of biochemical network. For multi-scale networks, these reductions are computed symbolically on orders of magnitude of parameters and variables, and are valid in wide domains of parameter and phase spaces.

I’m talking about the analogy between probabilities and quantum amplitudes, and how this makes chemistry analogous to particle physics. You can see two versions of my talk here, but I’ll be giving the ‘more advanced’ version, which is new:

Probabilities versus amplitudes.

Abstract. Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, the master equation for a chemical reaction network describes the interactions of molecules in a stochastic rather than quantum way. If we look at it from the perspective of quantum theory, this formalism turns out to involve creation and annihilation operators, coherent states and other well-known ideas—but with a few big differences.

Anyway, there are a lot more talks, but if I don’t have breakfast and walk over to the math department, I’ll miss those talks!

You can learn more about individual talks in the comments here (see below) and also in Matteo Polettini’s blog:

• Matteo Polettini, Mathematical trends in reaction network theory: part 1 and part 2, Out of Equilibrium, 1 July 2015.

PROPs for Linear Systems

18 May, 2015

Eric Drexler likes to say: engineering is dual to science, because science tries to understand what the world does, while engineering is about getting the world to do what you want. I think we need a slightly less ‘coercive’, more ‘cooperative’ approach to the world in order to develop ‘ecotechnology’, but it’s still a useful distinction.

For example, classical mechanics is the study of what things do when they follow Newton’s laws. Control theory is the study of what you can get them to do.

Say you have an upside-down pendulum on a cart. Classical mechanics says what it will do. But control theory says: if you watch the pendulum and use what you see to move the cart back and forth correctly, you can make sure the pendulum doesn’t fall over!

Control theorists do their work with the help of ‘signal-flow diagrams’. For example, here is the signal-flow diagram for an inverted pendulum on a cart:

When I take a look at a diagram like this, I say to myself: that’s a string diagram for a morphism in a monoidal category! And it’s true. Jason Erbele wrote a paper explaining this. Independently, Bonchi, Sobociński and Zanasi did some closely related work:

• John Baez and Jason Erbele, Categories in control.

• Filippo Bonchi, Paweł Sobociński and Fabio Zanasi, Interacting Hopf algebras.

• Filippo Bonchi, Paweł Sobociński and Fabio Zanasi, A categorical semantics of signal flow graphs.

I’ll explain some of the ideas at the Turin meeting on the categorical foundations of network theory. But I also want to talk about this new paper that Simon Wadsley of Cambridge University wrote with my student Nick Woods:

• Simon Wadsley and Nick Woods, PROPs for linear systems.

This makes the picture neater and more general!

You see, Jason and I used signal flow diagrams to give a new description of the category of finite-dimensional vector spaces and linear maps. This category plays a big role in the control theory of linear systems. Bonchi, Sobociński and Zanasi gave a closely related description of an equivalent category, \mathrm{Mat}(k), where:

• objects are natural numbers, and

• a morphism f : m \to n is an n \times m matrix with entries in the field k,

and composition is given by matrix multiplication.

But Wadsley and Woods generalized all this work to cover \mathrm{Mat}(R) whenever R is a commutative rig. A rig is a ‘ring without negatives’—like the natural numbers. We can multiply matrices valued in any rig, and this includes some very useful examples… as I’ll explain later.

Wadsley and Woods proved:

Theorem. Whenever R is a commutative rig, \mathrm{Mat}(R) is the PROP for bicommutative bimonoids over R.

This result is quick to state, but it takes a bit of explaining! So, let me start by bringing in some definitions.

Bicommutative bimonoids

We will work in any symmetric monoidal category, and draw morphisms as string diagrams.

A commutative monoid is an object equipped with a multiplication:

and a unit:

obeying these laws:

For example, suppose \mathrm{FinVect}_k is the symmetric monoidal category of finite-dimensional vector spaces over a field k, with direct sum as its tensor product. Then any object V \in \mathrm{FinVect}_k is a commutative monoid where the multiplication is addition:

(x,y) \mapsto x + y

and the unit is zero: that is, the unique map from the zero-dimensional vector space to V.

Turning all this upside down, cocommutative comonoid has a comultiplication:

and a counit:

obeying these laws:

For example, consider our vector space V \in \mathrm{FinVect}_k again. It’s a commutative comonoid where the comultiplication is duplication:

x \mapsto (x,x)

and the counit is deletion: that is, the unique map from V to the zero-dimensional vector space.

Given an object that’s both a commutative monoid and a cocommutative comonoid, we say it’s a bicommutative bimonoid if these extra axioms hold:

You can check that these are true for our running example of a finite-dimensional vector space V. The most exciting one is the top one, which says that adding two vectors and then duplicating the result is the same as duplicating each one, then adding them appropriately.

Our example has some other properties, too! Each element c \in k defines a morphism from V to itself, namely scalar multiplication by c:

x \mapsto c x

We draw this as follows:

These morphisms are compatible with the ones so far:

Moreover, all the ‘rig operations’ in k—that is, addition, multiplication, 0 and 1, but not subtraction or division—can be recovered from what we have so far:

We summarize this by saying our vector space V is a bicommutative bimonoid ‘over k‘.

More generally, suppose we have a bicommutative bimonoid A in a symmetric monoidal category. Let \mathrm{End}(A) be the set of bicommutative bimonoid homomorphisms from A to itself. This is actually a rig: there’s a way to add these homomorphisms, and also a way to ‘multiply’ them (namely, compose them).

Suppose R is any commutative rig. Then we say A is a bicommutative bimonoid over R if it’s equipped with a rig homomorphism

\Phi : R \to \mathrm{End}(A)

This is a way of summarizing the diagrams I just showed you! You see, each c \in R gives a morphism from A to itself, which we write as

The fact that this is a bicommutative bimonoid endomorphism says precisely this:

And the fact that \Phi is a rig homomorphism says precisely this:

So sometimes the right word is worth a dozen pictures!

What Jason and I showed is that for any field k, the \mathrm{FinVect}_k is the free symmetric monoidal category on a bicommutative bimonoid over k. This means that the above rules, which are rules for manipulating signal flow diagrams, completely characterize the world of linear algebra!

Bonchi, Sobociński and Zanasi used ‘PROPs’ to prove a similar result where the field is replaced by a sufficiently nice commutative ring. And Wadlsey and Woods used PROPS to generalize even further to the case of an arbitrary commutative rig!

But what are PROPs?


A PROP is a particularly tractable sort of symmetric monoidal category: a strict symmetric monoidal category where the objects are natural numbers and the tensor product of objects is given by ordinary addition. The symmetric monoidal category \mathrm{FinVect}_k is equivalent to the PROP \mathrm{Mat}(k), where a morphism f : m \to n is an n \times m matrix with entries in k, composition of morphisms is given by matrix multiplication, and the tensor product of morphisms is the direct sum of matrices.

We can define a similar PROP \mathrm{Mat}(R) whenever R is a commutative rig, and Wadsley and Woods gave an elegant description of the ‘algebras’ of \mathrm{Mat}(R). Suppose C is a PROP and D is a strict symmetric monoidal category. Then the category of algebras of C in D is the category of strict symmetric monoidal functors F : C \to D and natural transformations between these.

If for every choice of D the category of algebras of C in D is equivalent to the category of algebraic structures of some kind in D, we say C is the PROP for structures of that kind. This explains the theorem Wadsley and Woods proved:

Theorem. Whenever R is a commutative rig, \mathrm{Mat}(R) is the PROP for bicommutative bimonoids over R.

The fact that an algebra of \mathrm{Mat}(R) is a bicommutative bimonoid is equivalent to all this stuff:

The fact that \Phi(c) is a bimonoid homomorphism for all c \in R is equivalent to this stuff:

And the fact that \Phi is a rig homomorphism is equivalent to this stuff:

This is a great result because it includes some nice new examples.

First, the commutative rig of natural numbers gives a PROP \mathrm{Mat}. This is equivalent to the symmetric monoidal category \mathrm{FinSpan}, where morphisms are isomorphism classes of spans of finite sets, with disjoint union as the tensor product. Steve Lack had already shown that \mathrm{FinSpan} is the PROP for bicommutative bimonoids. But this also follows from the result of Wadsley and Woods, since every bicommutative bimonoid V is automatically equipped with a unique rig homomorphism

\Phi : \mathbb{N} \to \mathrm{End}(V)

Second, the commutative rig of booleans

\mathbb{B} = \{F,T\}

with ‘or’ as addition and ‘and’ as multiplication gives a PROP \mathrm{Mat}(\mathbb{B}). This is equivalent to the symmetric monoidal category \mathrm{FinRel} where morphisms are relations between finite sets, with disjoint union as the tensor product. Samuel Mimram had already shown that this is the PROP for special bicommutative bimonoids, meaning those where comultiplication followed by multiplication is the identity:

But again, this follows from the general result of Wadsley and Woods!

Finally, taking the commutative ring of integers \mathbb{Z}, Wadsley and Woods showed that \mathrm{Mat}(\mathbb{Z}) is the PROP for bicommutative Hopf monoids. The key here is that scalar multiplication by -1 obeys the axioms for an antipode—the extra morphism that makes a bimonoid into a Hopf monoid. Here are those axioms:

More generally, whenever R is a commutative ring, the presence of -1 \in R guarantees that a bimonoid over R is automatically a Hopf monoid over R. So, when R is a commutative ring, Wadsley and Woods’ result implies that \mathrm{Mat}(R) is the PROP for Hopf monoids over R.

Earlier, in their paper on ‘interacting Hopf algebras’, Bonchi, Sobociński and Zanasi had given an elegant and very different proof that \mathrm{Mat}(R) is the PROP for Hopf monoids over R whenever R is a principal ideal domain. The advantage of their argument is that they build up the PROP for Hopf monoids over R from smaller pieces, using some ideas developed by Steve Lack. But the new argument by Wadsley and Woods has its own charm.

In short, we’re getting the diagrammatics of linear algebra worked out very nicely, providing a solid mathematical foundation for signal flow diagrams in control theory!

Resource Theories

12 May, 2015

by Brendan Fong

Hugo Nava-Kopp and I have a new paper on resource theories:

• Brendan Fong and Hugo Nava-Kopp, Additive monotones for resource theories of parallel-combinable processes with discarding.

A mathematical theory of resources is Tobias Fritz’s current big project. He’s already explained how ordered monoids can be viewed as theories of resource convertibility in a three part series on this blog.

Ordered monoids are great, and quite familiar in network theory: for example, a Petri net can be viewed as a presentation for an ordered commutative monoid. But this work started in symmetric monoidal categories, together with my (Oxford) supervisor Bob Coecke and Rob Spekkens.

The main idea is this: think of the objects of your symmetric monoidal category as resources, and your morphisms as ways to convert one resource into another. The monoidal product or ‘tensor product’ in your category allows you to talk about collections of your resources. So, for example, in the resource theory of chemical reactions, our objects are molecules like oxygen O2, hydrogen H2, and water H2O, and morphisms things like the electrolysis of water:

2H2O → O2 + 2H2

This is a categorification of the ordered commutative monoid of resource convertibility: we now keep track of how we convert resources into one another, instead of just whether we can convert them.

Categorically, I find the other direction easier to state: being a category, the resource theory is enriched over \mathrm{Set}, while a poset is enriched over the poset of truth values or ‘booleans’ \mathbb{B} = \{0,1\}. If we ‘partially decategorify’ by changing the base of enrichment along the functor \mathrm{Set} \to \mathbb{B} that maps the empty set to 0 and any nonempty set to 1, we obtain the ordered monoid corresponding to the resource theory.

But the research programme didn’t start at resource theories either. The starting point was ‘partitioned process theories’.

Here’s an example that guided the definitions. Suppose we have a bunch of labs with interacting quantum systems, separated in space. With enough cooperation and funding, they can do big joint operations on their systems, like create entangled pairs between two locations. For ‘free’, however, they’re limited to classical communication between the locations, although they can do the full range of quantum operations on their local system. So you’ve got a symmetric monoidal category with objects quantum systems and morphisms quantum operations, together with a wide (all-object-including) symmetric monoidal subcategory that contains the morphisms you can do with local quantum operations and classical communication (known as LOCC operations).

This general structure: a symmetric monoidal category (or SMC for short) with a wide symmetric monoidal subcategory, is called a partitioned process theory. We call the morphisms in the SMC processes, and those in the subSMC free processes.

There are a number of methods for building a resource theory (i.e. an SMC) from a partitioned process theory. The unifying idea though, is that your new SMC has the processes f,g as objects, and morphisms f \to g ways of using the free processes to build g from f.

But we don’t have to go to fancy sounding quantum situations to find examples of partitioned process theories. Instead, just look at any SMC in which each object is equipped with an algebraic structure. Then the morphisms defining this structure can be taken as our ‘free’ processes.

For example, in a multigraph category every object has the structure of a ‘special commutative Frobenius algebra’. That’s a bit of a mouthful, but John defined it a while back, and examples include categories where morphisms are electrical circuits, and categories where morphisms are signal flow diagrams.

So these categories give partitioned process theories! This idea of partitioning the morphisms into ‘free’ ones and ‘costly’ ones is reminiscent of what I was saying earlier about the operad of wiring diagrams about it being useful to separate behavioural structure from interconnection structure.

This suggests that we can also view the free processes as generating some sort of operad, that describes the ways we allow ourselves to use free processes to turn processes into other processes. If we really want to roll a big machine out to play with this stuff, framed bicategories may also be interesting; Spivak is already using them to get at questions about operads. But that’s all conjecture, and a bit of a digression.

To get back to the point, this was all just to say that if you find yourself with a bunch of resistors, and you ask ‘what can I build?’, then you’re after the resource theory apparatus.

You can read more about this stuff here:

• Bob Coecke, Tobias Fritz and Rob W. Spekkens, A mathematical theory of resources.

• Tobias Fritz, The mathematical structure of theories of resource convertibility I.

Cospans, Wiring Diagrams, and the Behavioral Approach

5 May, 2015

joint with Brendan Fong

We’re getting ready for the Turin workshop on the
Categorical Foundations of Network Theory. So, we’re trying to get our thoughts in order.

Last time we talked about understanding types of networks as categories of decorated cospans. Earlier, David Spivak told us about understanding networks as algebras of an operad. Both these frameworks work at capturing notions of modularity and interconnection. Are they then related? How?

In this post we want to discuss some similarities between decorated cospan categories and algebras for Spivak’s operad of wiring diagrams. The main idea is that the two approaches are ‘essentially’ equivalent, but that compared to decorated cospans, Spivak’s operad formalism puts greater emphasis on the distinction between the ‘duplication’ and ‘deletion’ morphisms and other morphisms in our category.

The precise details are still to be worked out—jump in and help us!


We begin with a bit about operads in general. Recall that an operad is similar to a category, except that instead of a set \mathrm{hom}(x,y) of morphisms from any object x to any object y, you have a set \mathrm{hom}(x_1,\dots,x_n;y) of operations from any finite list of objects x_1,...,x_n to any object y . If we have an operation f \in \mathrm{hom}(x_1,\dots,x_n;y), we can call x_1, \dots, x_n the inputs of f and call y the output of f .

We can compose operations in an operad. To understand how, it’s easiest to use pictures. We draw an operation in \mathrm{hom}(x_1,\dots,x_n;y) as a little box with n wires coming in and one wire coming out:

The input wires should be labelled with the objects x_1, \dots, x_n and the output wire with the object y, but I haven’t done this.

We are allowed to compose these operations as follows:

as long as the outputs of the operations g_1,\dots,g_n match the inputs of the operation f . The result is a new operation which we call f \circ (g_1,\dots,g_n) .

We demand that there be unary operations 1_x \in \mathrm{hom}(x;x) serving as identities for composition, and we impose an associative law that makes a composite of composites like this well-defined:

So far this is the definition of a operad without permutations. In a full-fledged permutative operad, we can also permute the inputs of an operation f and get a new operation:

which we call f \sigma if \sigma is the the permutation of the inputs. We demand that (f \sigma) \sigma' = f (\sigma \sigma') . And finally, we demand that permutations act in a way that is compatible with composition. For example:

Here we see that (f \sigma) \circ (g_1, \dots, g_n) is equal to some obvious other thing.

Finally, there is a law saying

f \circ (g_1 \sigma_1, \dots, g_n \sigma_n) = (f \circ (g_1 , \dots, g_n)) \sigma

for some choice of \sigma that you can cook up from the permuations \sigma_i in an obvious way. We leave it as an exercise to work out the details. By the way, one well-known book on operads accidentally omits this law, so here’s a rather more lengthy exercise: read this book, see which theorems require this law, and correct their proofs!

Operads are similar to symmetric monoidal categories. The idea is that in a symmetric monoidal category you can just form the tensor product x_1 \otimes \dots \otimes x_n and talk about the set of morphisms x_1 \otimes \cdots \otimes \cdots x_n \to y . Indeed any symmetric monoidal category gives an operad in this way: just define \mathrm{hom}(x_1,...,x_n;y) to be \mathrm{hom}(x_1 \otimes \cdots \otimes x_n, y) . If we do this with Set, which is a symmetric monoidal category using the usual cartesian product of sets, we get an operad called \mathrm{Set}.

An algebra for an operad O is an operad homomorphism O \to \mathrm{Set}. We haven’t said what an operad homomorphism is, but you can probably figure it out yourself. The point is this: an algebra for O turns the abstract operations in O into actual operations on sets!

Finally, we should warn you that operads come in several flavors, and we’ve been talking about ‘typed permutative operads’. ‘Typed’ means that there’s more than one object; ‘permutative’ means that we have the ability to permute the input wires. When people say ‘operad’, they often mean an untyped permutative operad. For that, just specialize down to the case where there’s only one object x.

You can see a fully precise definition of untyped permutative operads here:

Operad theory, Wikipedia.

along with the definition of an untyped operad without permutations.

The operad of wiring diagrams

Spivak’s favorite operad is the operad of wiring diagrams. The operad of wiring diagrams WD is an operad version of \mathrm{Cospan}(\mathrm{FinSet}), constructed in the vein suggested above: the objects are finite sets, and an operation from a list of sets X_1,...,X_n to a set Y is a cospan

X_1+ \cdots +X_n \rightarrow S \leftarrow Y

Spivak draws such a thing as a big circle with n small circles cut out from the interior:

The outside of the big circle has a set Y of terminals marked on it, and each small circle has a set X_i of terminals marked on it. Then in the interior of this shape there are wires connecting these terminals. This what he calls a wiring diagrams.

You compose these wiring diagrams by pasting other wiring diagrams into each of the small circles.

The relationship with our Frobenius monoid diagrams is pretty simple: we draw our ‘wiring diagrams’ X \to Y in a square, with the X terminals on the left and Y terminals on the right. To get a Spivak-approved wiring diagram, glue the top and bottom edges of this square together, then flatten the cylinder you get down into an annulus, with the X-side on the inside and Y-side on the outside. If X = X_1+X_2 you can imagine gluing opposite edges of the inside circle together to divide it into two small circles accordingly, and so on.

Relational algebras of type A

Algebras for wiring diagrams tell you what components you have available to wire together with your diagrams. An algebra for the operad of wiring diagrams is an operad homomorphism

WD \to \mathrm{Set}

What does this look like? Just like a functor for categories, it assigns to each natural number a set, and each wiring diagram a function.

In work related to decorated cospans (such as our paper on circuits or John and Jason’s work on signal flow diagrams), our semantics usually is constructed from a field of values—not a physicist’s ‘field’, bt an algebraist’s sort of ‘field’, where you can add, multiply, subtract and divide. For example, we like being able to assign a real number like a velocity, or potential, or current to a variable. This gives us vector spaces and a bunch of nice linear-algebraic structures.

Spivak works more generally: he’s interested in the structure when you just have a set of values. While this means we can’t do some of the nice things we could do with a field, it also means this framework can do things like talk about logic gates, where the variables are boolean ones, or number theoretic questions, where you’re interested in the natural numbers.

So to discuss semantics we pick a set A of values, such as the real numbers or natural numbers or booleans or colors. We imagine then associating elements of this set to each wire in a wiring diagram. More technically, the algebra

\mathrm{Rel}A: WD \to \mathrm{Set}

then maps each finite set X to the power set \mathcal{P}(A^X) of the set A^X of functions X \to A .

On the morphisms (the wiring diagrams themselves), this functor behaves as follows. Note that a function (X \to A) can be thought of as an ‘X-vector’ (a_1,...,a_x) of ‘A-coordinates’. A wiring diagram X \to Y is just a cospan

X \to N  \leftarrow Y

in \mathrm{FinSet}, so it can be thought of as some compares

X \to N

followed by some copies

N \to Y

Thus, given a wiring diagram X \to Y, we can consider a partial function that maps an X-vector to the Y-vector by doing these compares, and if it passes them does the copies and returns the resulting Y-vector, but if not returns ‘undefined’. We can then define a map \mathcal{P}(A^X) \to \mathcal{P}(A^Y) which takes a set of X-vectors to its image under this partial function.

This semantics is called the relational WD-algebra of type A. We can think of it as being like the ‘light operations’ fragment of the signal flow calculus. By ‘light operations’, we mean the operations of duplication and deletion, which form a cocommutative comonoid:

and their time-reversed versions, ‘coduplication’ and ‘codeletion’, which form a commutatative monoid:

These fit together to form a Frobenius monoid, meaning that these equations hold:

And it’s actually extra-special, meaning that these equations hold:

(If you don’t understand these hieroglyphics, please reread our post about categories in control theory, and ask some questions!)

Note that we can’t do the ‘dark operations’, because we only have a set A of values, not a field, and the dark operations involve addition and zero!

Operads and the behavioral approach

In formulating Frobenius monoids this way, Spivak achieves something that we’ve been working hard to find ways to achieve: a separation of the behavioral structure from the interconnection structure.

What do I mean by this? In his ‘behavioral approach‘, Willems makes the point that for all their elaborate and elegant formulation, in the end physical laws just come down to dividing the set of what might a priori be possible (the ‘universum’) into the set of things that actually are possible (the ‘behavior’), and the set of things that aren’t). Here the universum is the set A^X: a priori, on each of the wires in X, we might associate any value of A . For example, to the two wires at the ends of a resistor, we might a priori associate any pair of currents. But physical law, here Kirchhoff’s current law, says otherwise: the currents must be equal and opposite. So the ‘behavior’ is the subset (i,-i) of the universum \mathbb{R}^2.

So you can say that to each object X in the operad of wiring diagrams the relational algebra of type A associates the set \mathcal{P}(A^X) of possible behaviors—the universum is A^X . (\mathcal{P}(A^X) forms some sort of meta-universum, where you can discuss physical laws about physical laws, commonly called ‘principles’.)

The second key aspect of the behavioral approach is that the behaviors of larger systems can be constructed from the behaviors of its subsystems, if we understand the so-called ‘interconnection structure’ well enough. This is a key principle in engineering: we build big, complicated systems from a much smaller set of components, whether it be electronics from resistors and inductors, or mechanical devices from wheels and rods and ropes, or houses from Lego bricks. The various interconnection structures here are the wiring diagrams, and our relational algebras say they act by what Willems calls ‘variable sharing’.

This division between behavior and interconnection motivates the decorated cospan construction (where the decorations are the ‘components’, the cospans the ‘interconnection’) and also the multigraph categories discussed by Aleks Kissinger (where morphisms are the ‘components’, and the Frobenius monoid operations are the ‘interconnection’):

• Aleks Kissinger, Finite matrices are complete for (dagger-)multigraph categories.

So it’s good to have this additional way of thinking about things in our repertoire: operads describe ‘interconnection’, their algebras ‘behaviors’.

The separation Spivak achieves, however, seems to me to come at the cost of neat ways to talk about individual components, and perhaps this can be seen as the essential difference between the two approaches. By including our components as morphisms, we can talk more carefully about them and additional structure individual components have. On the other hand, by lumping all the components into the objects, Spivak can talk more carefully about how the interconnection structure acts on all behaviors at once.

Other operads of wiring diagrams

One advantage of the operad approach is that you can easily tweak your operad to talk about different sorts of network structure. Sometimes you can make similar adjustments with decorated cospans too, such as working over the category of typed finite sets, rather than just finite sets, to discuss networks in which wires have types, and only wires of the same types can be connected together. A physical example is a model of a hydroelectric power plant, where you don’t want to connect a water pipe with an electrical cable! This is also a common technique in computer science, where you don’t want to try to multiply two strings of text, or try to interpret a telephone number as a truth value.

But some modifications are harder to do with decorated cospans. In some other papers, Spivak employs a more restricted operad of wiring diagrams, in which joining wires and terminating wires is not allowed, among other things. He uses this to formalise graphical languages for certain types of discrete-time processes, open dynamical systems, including mode-dependent ones.

For more detail, read these:

• Brendan Fong, Decorated cospans.

• David Spivak, The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits.

Decorated Cospans

1 May, 2015

Last time I talked about a new paper I wrote with Brendan Fong. It’s about electrical circuits made of ‘passive’ components, like resistors, inductors and capacitors. We showed these circuits are morphisms in a category. Moreover, there’s a functor sending each circuit to its ‘external behavior’: what it does, as seen by someone who can only measure voltages and currents at the terminals.

Our paper uses a formalism that Brendan developed here:

• Brendan Fong, Decorated cospans.

The idea here is we may want to take something like a graph with edges labelled by positive numbers:

and say that some of its nodes are ‘inputs’, while others are ‘outputs’:

This lets us treat our labelled graph as a ‘morphism’ from the set X to the set Y.

The point is that we can compose such morphisms. For example, suppose we have another one of these things, going from Y to Z:

Since the points of Y are sitting in both things:

we can glue them together and get a thing going from X to Z:

That’s how we compose these morphisms.

Note how we’re specifying some nodes of our original thing as inputs and outputs:

We’re using maps from two sets X and Y to the set N of nodes of our graph. And a bit surprisingly, we’re not demanding that these maps be one-to-one. That turns out to be useful—and in general, when doing math, it’s dumb to make your definitions forbid certain possibilities unless you really need to.

So, our thing is really a cospan of finite sets—that is, a diagram of finite sets and functions like this:

together some extra structure on the set N. This extra structure is what Brendan calls a decoration, and it makes the cospan into a ‘decorated cospan’. In this particular example, a decoration on N is a way of making it into the nodes of a graph with edges labelled by positive numbers. But one could consider many other kinds of decorations: this idea is very general.

To formalize the idea of ‘a kind of decoration’, Brendan uses a functor

F: \mathrm{FinSet} \to \mathrm{Set}

sending each finite set N to a set of F(N). This set F(N) is the set of decorations of the given kind that we can put on N.

So, for any such functor F, a decorated cospan of finite sets is a cospan of finite sets:

together with an element of F(N).

But in fact, Brendan goes further. He’s not content to use a functor

F: \mathrm{FinSet} \to \mathrm{Set}

to decorate his cospans.

First, there’s no need to limit ourselves to cospans of finite sets: we can replace \mathrm{FinSet} with some other category! If C is any category with finite colimits, there’s a category \mathrm{Cospan}(C) with:

• objects of C as its objects,
• isomorphism classes of cospans between these as morphisms.

Second, there’s no need to limit ourselves to decorations that are elements of a set: we can replace \mathrm{Set} with some other category! If D is any symmetric monoidal category, we can define an element of an object d \in D to be a morphism

f: I \to d

where I is the unit for the tensor product in D.

So, Brendan defines decorated cospans at this high level of generality, and shows that under some mild conditions we can compose them, just as in the pictures we saw earlier.

Here’s one of the theorems Brendan proves:

Theorem. Suppose C is a category with finite colimits, and make C into a symmetric monoidal category with its coproduct as the tensor product. Suppose D is a symmetric monoidal category, and suppose F: C \to D is a lax symmetric monoidal functor. Define an F-decorated cospan to be a cospan

in C together with an element of F(N). Then there is a category with

• objects of C as its objects,
• isomorphism classes of F-decorated cospans as its morphisms.

This is called the F-decorated cospan category, FCospan. This category becomes symmetric monoidal in a natural way. It is then a dagger compact category.

(You may not know all this jargon, but ‘lax symmetric monoidal’, for example, talks about how we can take decorations on two things and get a decoration on their disjoint union, or ‘coproduct’. We need to be able to do this—as should be obvious from the pictures I drew. Also, a ‘dagger compact category’ is the kind of category whose morphisms can be drawn as networks.)

Brendan also explains how to get functors between decorated cospan categories. We need this in our paper on electrical circuits, because we consider several categories where morphisms is a circuit, or something that captures some aspect of a circuit. Most of these categories are decorated cospan categories. We want to get functors between them. And often we can just use Brendan’s general results to get the job done! No fuss, no muss: all the hard work has been done ahead of time.

I expect to use this technology a lot in my work on network theory.


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