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Classically, the splitting principle says how to pull back a vector bundle in such a way that it splits into line bundles and the pullback map induces an injection on $K$-theory. Here we categorify the splitting principle and generalize it to the context of 2-rigs. A 2-rig is a kind of categorified "ring without negatives", such as a category of vector bundles with $\oplus$ as addition and $\otimes$ as multiplication. Technically, we define a 2-rig to be a Cauchy complete $k$-linear symmetric monoidal category where $k$ has characteristic zero. We conjecture that for any suitably finite-dimensional object $r$ of a 2-rig $\mathsf{R}$, there is a 2-rig map $E \colon \mathsf{R} \to \mathsf{R'}$ such that $E(r)$ splits as a direct sum of finitely many "subline objects" and $E$ has various good properties: it is faithful, conservative, essentially injective, and the induced map of Grothendieck rings $K(E) \colon K(\mathsf{R}) \to K(\mathsf{R'})$ is injective. We prove this conjecture for the free 2-rig on one object, namely the category of Schur functors, whose Grothendieck ring is the free $\lambda$-ring on one generator, also known as the ring of symmetric functions. We use this task as an excuse to develop the representation theory of affine categories - that is, categories enriched in affine schemes - using the theory of 2-rigs., Comment: 47 pages

This short book is an elementary course on entropy, leading up to a calculation of the entropy of hydrogen gas at standard temperature and pressure. Topics covered include information, Shannon entropy and Gibbs entropy, the principle of maximum entropy, the Boltzmann distribution, temperature and coolness, the relation between entropy, expected energy and temperature, the equipartition theorem, the partition function, the relation between expected energy, free energy and entropy, the entropy of a classical harmonic oscillator, the entropy of a classical particle in a box, and the entropy of a classical ideal gas., Comment: 129 pages

A "Littlewood polynomial" is a polynomial whose coefficients are all 1 or -1. The set of all complex roots of all Littlewood polynomials exhibits many complicated, beautiful and fascinating patterns. Some fractal regions of this set closely resemble "dragon sets" formed by iterated function systems. A heuristic argument for this is known, but no precise theorem along these lines has been proved. We invite the reader to try.

The icosidodecahedron has 30 vertices, one at the center of each edge of a regular icosahedron -- or equivalently, one at the center of each edge of a regular dodecahedron. It is a beautiful, highly symmetrical shape. But it is just a projection down to 3d space of a more symmetrical 6-dimensional polytope with 60 vertices. It is also a slice of a more symmetrical 4d polytope with 120 vertices, which in turn is the projection down to 4d space of an even more symmetrical 8-dimensional polytope with 240 vertices: the E8 root polytope. Here we explain all these constructions, and their connection to the quaternions and icosians., Comment: 4 pages, 4 figures

During what Vietnamese call the American War, Alexander Grothendieck spent three weeks teaching mathematics in and near Hanoi. Ho\`ang Xu\^an S\'inh took notes on his lectures and later did her thesis work with him by correspondence. In her thesis she developed the theory of "Gr-categories", which are monoidal categories in which all objects and morphisms have inverses. Now often called "2-groups", these structures allow the study of symmetries that themselves have symmetries. After a brief account of how Ho\`ang Xu\^an S\'inh wrote her thesis, we explain some of its main results, and its context in the history of mathematics., Comment: 22 pages with xypic figures

Underlying the Riemann Hypothesis there is a question whose full answer still eludes us: what do the zeros of the Riemann zeta function really mean? As a step toward answering this, Andr\'e Weil proposed a series of conjectures that include a simplified version of the Riemann Hypothesis in which the meaning of the zeros becomes somewhat easier to understand. Grothendieck and others worked for decades to prove Weil's conjectures, inventing a large chunk of modern algebraic geometry in the process. This quest, still in part unfulfilled, led Grothendieck to dream of "motives": mysterious building blocks that could explain the zeros (and poles) of Weil's analogue of the Riemann zeta function. This exposition by a complete amateur tries to sketch some of these ideas in ways that other amateurs can enjoy., Comment: 13 pages, 6 figures

These are the third 50 issues of This Week's Finds of Mathematical Physics, from April 9, 1997 to June 18, 2000. These issues discuss quantum gravity, topological quantum field theory, n-categories and other topics in mathematics and physics. They also contain a series of expository mini-articles on homotopy theory. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird; you can get a better-looking version at https://math.ucr.edu/home/baez/twf_latex/), Comment: 348 pages

Young diagrams are ubiquitous in combinatorics and representation theory. Here we explain these diagrams, focusing on how they are used to classify representations of the symmetric groups $S_n$ and various "classical groups": famous groups of matrices such as the general linear group $\mathrm{GL}(n,\mathbb{C})$ consisting of all invertible $n \times n$ complex matrices, the special linear group $\mathrm{SL}(n,\mathbb{C})$ consisting of all $n \times n$ complex matrices with determinant 1, the group $\mathrm{U}(n)$ consisting of all unitary $n \times n$ matrices, and the special unitary group $\mathrm{SU}(n)$ consisting of all unitary $n \times n$ matrices with determinant 1. We also discuss representations of the full linear monoid consisting of all linear transformations of $\mathbb{C}^n$. These notes, based on the column This Week's Finds in Mathematical Physics, are made to accompany a series of lecture videos., Comment: 19 pages with TikZ figures and one png figure

Mathematicians love dualities. After a brief explanation of dualities, with examples, we turn to one of the purest and most beautiful: Isbell duality. For any category $\mathsf{C}$, this gives an adjunction between the category of presheaves on $\mathsf{C}$, namely the functor category $[\mathsf{C}^{\text{op}}, \mathsf{Set}]$, and the opposite of the category of copresheaves on $\mathsf{C}$, namely $[\mathsf{C}, \mathsf{Set}]^{\text{op}}$., Comment: 3 pages

Stock and flow diagrams are already an important tool in epidemiology, but category theory lets us go further and treat these diagrams as mathematical entities in their own right. In this chapter we use communicable disease models created with our software, StockFlow.jl, to explain the benefits of the categorical approach. We first explain the category of stock-flow diagrams and note the clear separation between the syntax of these diagrams and their semantics, demonstrating three examples of semantics already implemented in the software: ODEs, causal loop diagrams, and system structure diagrams. We then turn to two methods for building large stock-flow diagrams from smaller ones in a modular fashion: composition and stratification. Finally, we introduce the open-source ModelCollab software for diagram-based collaborative modeling. The graphical user interface of this web-based software lets modelers take advantage of the ideas discussed here without any knowledge of their categorical foundations.

The Kuramoto-Sivashinsky equation was introduced as a simple 1-dimensional model of instabilities in flames, but it turned out to mathematically fascinating in its own right. One reason is that this equation is a simple model of Galilean-invariant chaos with an arrow of time. Starting from random initial conditions, manifestly time-asymmetric stripe-like patterns emerge. As we move forward in time, it appears that these stripes are born and merge, but do not die or split. We pose a precise conjecture to this effect, which requires a precise definition of 'stripes'., Comment: 3 pages, 2 figures

These are the second 50 issues of This Week's Finds of Mathematical Physics, from April 23, 1995 to March 23, 1997. These issues discuss quantum gravity, topological quantum field theory and other topics in mathematics and physics. They also include two expository series: one on ADE classifications and one on categories and higher categories. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird)., Comment: 281 pages

The Gauss-Lucas theorem says that for any complex polynomial $P$, the roots of the derivative $P'$ lie in the convex hull of the roots of $P$. In other words, the roots of $P'$ lie inside the smallest convex subset of the complex plane containing all the roots of $P$. This theorem is not hard to prove, but is there an intuitive explanation? In fact there is, using physics -- or more precisely, electrostatics in 2-dimensional space, Comment: 2 pages, one picture by Greg Egan

We define a thermostatic system to be a convex space of states together with a concave function sending each state to its entropy, which is an extended real number. This definition applies to classical thermodynamics, classical statistical mechanics, quantum statistical mechanics, and also generalized probabilistic theories of the sort studied in quantum foundations. It also allows us to treat a heat bath as a thermostatic system on an equal footing with any other. We construct an operad whose operations are convex relations from a product of convex spaces to a single convex space, and prove that thermostatic systems are algebras of this operad. This gives a general, rigorous formalism for combining thermostatic systems, which captures the fact that such systems maximize entropy subject to whatever constraints are imposed upon them., Comment: 25 pages, 4 figures

Suppose we have $n$ different types of self-replicating entity, with the population $P_i$ of the $i$th type changing at a rate equal to $P_i$ times the fitness $f_i$ of that type. Suppose the fitness $f_i$ is any continuous function of all the populations $P_1, \dots, P_n$. Let $p_i$ be the fraction of replicators that are of the $i$th type. Then $p = (p_1, \dots, p_n)$ is a time-dependent probability distribution, and we prove that its speed as measured by the Fisher information metric equals the variance in fitness. In rough terms, this says that the speed at which information is updated through natural selection equals the variance in fitness. This result can be seen as a modified version of Fisher's fundamental theorem of natural selection. We compare it to Fisher's original result as interpreted by Price, Ewens and Edwards., Comment: 6 pages

It is known that the Grothendieck group of the category of Schur functors is the ring of symmetric functions. This ring has a rich structure, much of which is encapsulated in the fact that it is a "plethory": a monoid in the category of birings with its substitution monoidal structure. We show that similarly the category of Schur functors is a "2-plethory", which descends to give the plethory structure on symmetric functions. Thus, much of the structure of symmetric functions exists at a higher level in the category of Schur functors., Comment: 53 pages

The "Brownian map" is a fundamental object in mathematics, in some sense a 2-dimensional analogue of Brownian motion. Here we briefly explain this object and a bit of its history., Comment: 3 pages, one figure by Thomas Budzinski

One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor $L \colon \mathsf{A} \to \mathsf{X}$, a "structured cospan" is a diagram in $\mathsf{X}$ of the form $L(a) \rightarrow x \leftarrow L(b)$. If $\mathsf{A}$ and $\mathsf{X}$ have finite colimits and $L$ preserves them, it is known that there is a symmetric monoidal double category whose objects are those of $\mathsf{A}$ and whose horizontal 1-cells are structured cospans. Second, given a pseudofunctor $F \colon \mathsf{A} \to \mathbf{Cat}$, a "decorated cospan" is a diagram in $\mathsf{A}$ of the form $a \rightarrow m \leftarrow b$ together with an object of $F(m)$. Generalizing the work of Fong, we show that if $\mathsf{A}$ has finite colimits and $F \colon (\mathsf{A},+) \to (\mathsf{Cat},\times)$ is symmetric lax monoidal, there is a symmetric monoidal double category whose objects are those of $\mathsf{A}$ and whose horizontal 1-cells are decorated cospans. We prove that under certain conditions, these two constructions become isomorphic when we take $\mathsf{X} = \int F$ to be the Grothendieck category of $F$. We illustrate these ideas with applications to electrical circuits, Petri nets, dynamical systems and epidemiological modeling., Comment: 39 pages, version for Compositionality

These are the first 50 issues of This Week's Finds of Mathematical Physics, from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity, topological quantum field theory, knot theory, and applications of $n$-categories to these subjects. However, there are also digressions into Lie algebras, elliptic curves, linear logic and other subjects. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird)., Comment: 241 pages

We present a unified framework for Petri nets and various variants, such as pre-nets and Kock's whole-grain Petri nets. Our framework is based on a less well-studied notion that we call $\Sigma$-nets, which allow finer control over whether tokens are treated using the collective or individual token philosophy. We describe three forms of execution semantics in which pre-nets generate strict monoidal categories, $\Sigma$-nets (including whole-grain Petri nets) generate symmetric strict monoidal categories, and Petri nets generate commutative monoidal categories, all by left adjoint functors. We also construct adjunctions relating these categories of nets to each other, in particular showing that all kinds of net can be embedded in the unifying category of $\Sigma$-nets, in a way that commutes coherently with their execution semantics., Comment: 29 pages

The tenfold way became important in physics around 2010: it implies that there are ten fundamentally different kinds of matter. But it goes back to 1964, when C. T. C. Wall classified real super division algebras. These are finite-dimensional real $\mathbb{Z}/2$-graded algebras where every nonzero homogeneous element is invertible. He found that besides $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$, which give purely even super division algebras, there are seven more. He also showed that these ten algebras are all real or complex Clifford algebras. The eight real ones represent all eight Morita equivalence classes of real Clifford algebras, and the two complex ones do the same for the complex Clifford algebras. The tenfold way thus unites real and complex Bott periodicity. In this expository article we give a quick proof that there are ten real super division algebras, and say a bit about applications of the tenfold way., Comment: 3 pages

System of systems engineering seeks to analyze, design and deploy collections of systems that together can flexibly address an array of complex tasks. In the Complex Adaptive System Composition and Design Environment program, we developed "network operads" as a tool for designing and tasking systems of systems, and applied them to domains including maritime search and rescue. The network operad formalism offers new ways to handle changing levels of abstraction in system-of-system design and tasking., Comment: 2 pages, TikZ figure

We examine the assumptions behind Noether's theorem connecting symmetries and conservation laws. To compare classical and quantum versions of this theorem, we take an algebraic approach. In both classical and quantum mechanics, observables are naturally elements of a Jordan algebra, while generators of one-parameter groups of transformations are naturally elements of a Lie algebra. Noether's theorem holds whenever we can map observables to generators in such a way that each observable generates a one-parameter group that preserves itself. In ordinary complex quantum mechanics this mapping is multiplication by $\sqrt{-1}$. In the more general framework of unital JB-algebras, Alfsen and Shultz call such a mapping a "dynamical correspondence", and show its presence allows us to identify the unital JB-algebra with the self-adjoint part of a complex C*-algebra. However, to prove their result, they impose a second, more obscure, condition on the dynamical correspondence. We show this expresses a relation between quantum and statistical mechanics, closely connected to the principle that "inverse temperature is imaginary time"., Comment: 37 pages

One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce "structured cospans" as a way to study networks with inputs and outputs. Given a functor $L \colon \mathsf{A} \to \mathsf{X}$, a structured cospan is a diagram in $\mathsf{X}$ of the form $L(a) \rightarrow x \leftarrow L(b)$. If $\mathsf{A}$ and $\mathsf{X}$ have finite colimits and $L$ is a left adjoint, we obtain a symmetric monoidal category whose objects are those of $\mathsf{A}$ and whose morphisms are isomorphism classes of structured cospans. This is a hypergraph category. However, it arises from a more fundamental structure: a symmetric monoidal double category where the horizontal 1-cells are structured cospans. We show how structured cospans solve certain problems in the closely related formalism of "decorated cospans", and explain how they work in some examples: electrical circuits, Petri nets, and chemical reaction networks., Comment: 43 pages, TikZ figures

Enriched Lawvere theories are a generalization of Lawvere theories that allow us to describe the operational semantics of formal systems. For example, a graph enriched Lawvere theory describes structures that have a graph of operations of each arity, where the vertices are operations and the edges are rewrites between operations. Enriched theories can be used to equip systems with operational semantics, and maps between enriching categories can serve to translate between different forms of operational and denotational semantics. The Grothendieck construction lets us study all models of all enriched theories in all contexts in a single category. We illustrate these ideas with the SKI-combinator calculus, a variable-free version of the lambda calculus., Comment: In Proceedings ACT 2019, arXiv:2009.06334

Petri networks and network models are two frameworks for the compositional design of systems of interacting entities. Here we show how to combine them using the concept of a "catalyst": an entity that is neither destroyed nor created by any process it engages in. In a Petri net, a place is a catalyst if its in-degree equals its out-degree for every transition. We show how a Petri net with a chosen set of catalysts gives a network model. This network model maps any list of catalysts from the chosen set to the category whose morphisms are all the processes enabled by this list of catalysts. Applying the Grothendieck construction, we obtain a category fibered over the category whose objects are lists of catalysts. This category has as morphisms all processes enabled by some list of catalysts. While this category has a symmetric monoidal structure that describes doing processes in parallel, its fibers also have premonoidal structures that describe doing one process and then another while reusing the catalysts., Comment: 15 pages, TikZ figures

The reachability semantics for Petri nets can be studied using open Petri nets. For us an "open" Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category $\mathsf{Open}(\mathsf{Petri})$, which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category $\mathbb{O}\mathbf{pen}(\mathsf{Petri})$. We describe two forms of semantics for open Petri nets using symmetric monoidal double functors out of $\mathbb{O}\mathbf{pen}(\mathsf{Petri})$. The first, an operational semantics, gives for each open Petri net a category whose morphisms are the processes that this net can carry out. This is done in a compositional way, so that these categories can be computed on smaller subnets and then glued together. The second, a reachability semantics, simply says which markings of the outputs can be reached from a given marking of the inputs., Comment: 30 pages, TikZ figures

Bazhin has analyzed ATP coupling in terms of quasiequilibrium states where fast reactions have reached an approximate steady state while slow reactions have not yet reached equilibrium. After an expository introduction to the relevant aspects of reaction network theory, we review his work and explain the role of emergent conserved quantities in coupling. These are quantities, left unchanged by fast reactions, whose conservation forces exergonic processes such as ATP hydrolysis to drive desired endergonic processes., Comment: 13 pages

The regular icosahedron is connected to many exceptional objects in mathematics. Here we describe two constructions of the $\mathrm{E}_8$ lattice from the icosahedron. One uses a subring of the quaternions called the "icosians", while the other uses du Val's work on the resolution of Kleinian singularities. Together they link the golden ratio, the quaternions, the quintic equation, the 600-cell, and the Poincare homology 3-sphere. We leave it as a challenge to the reader to find the connection between these two constructions., Comment: 9 pages LaTeX

Coarse-graining is a standard method of extracting a simple Markov process from a more complicated one by identifying states. Here we extend coarse-graining to open Markov processes. An "open" Markov process is one where probability can flow in or out of certain states called "inputs" and "outputs". One can build up an ordinary Markov process from smaller open pieces in two basic ways: composition, where we identify the outputs of one open Markov process with the inputs of another, and tensoring, where we set two open Markov processes side by side. In previous work, Fong, Pollard and the first author showed that these constructions make open Markov processes into the morphisms of a symmetric monoidal category. Here we go further by constructing a symmetric monoidal double category where the 2-morphisms are ways of coarse-graining open Markov processes. We also extend the already known "black-boxing" functor from the category of open Markov processes to our double category. Black-boxing sends any open Markov process to the linear relation between input and output data that holds in steady states, including nonequilibrium steady states where there is a nonzero flow of probability through the process. To extend black-boxing to a functor between double categories, we need to prove that black-boxing is compatible with coarse-graining., Comment: 39 pages LaTeX with TikZ figures

Networks can be combined in various ways, such as overlaying one on top of another or setting two side by side. We introduce "network models" to encode these ways of combining networks. Different network models describe different kinds of networks. We show that each network model gives rise to an operad, whose operations are ways of assembling a network of the given kind from smaller parts. Such operads, and their algebras, can serve as tools for designing networks. Technically, a network model is a lax symmetric monoidal functor from the free symmetric monoidal category on some set to $\mathbf{Cat}$, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction., Comment: 46 pages

Generalized span categories provide a framework for formalizing mathematical models of open systems in classical mechanics. We introduce categories $\mathsf{LagSy}$ and $\mathsf{HamSy}$ that respectively provide a categorical framework for the Lagrangian and Hamiltonian descriptions of open classical mechanical systems. The morphisms of $\mathsf{LagSy}$ and $\mathsf{HamSy}$ correspond to such open systems, and composition of morphisms models the construction of systems from subsystems. The Legendre transformation gives rise to a functor from $\mathsf{LagSy}$ to $\mathsf{HamSy}$ that translates from the Lagrangian to the Hamiltonian perspective., Comment: 31 pages

Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermodynamic and chemical components. We can formalize this reasoning using props: that is, strict symmetric monoidal categories where the objects are natural numbers, with the tensor product of objects given by addition. In this approach, each kind of network corresponds to a prop, and each network of this kind is a morphism in that prop. A network with $m$ inputs and $n$ outputs is a morphism from $m$ to $n$, putting networks together in series is composition, and setting them side by side is tensoring. Here we work out the details of this approach for various kinds of electrical circuits, starting with circuits made solely of ideal perfectly conductive wires, then circuits with passive linear components, and then circuits that also have voltage and current sources. Each kind of circuit corresponds to a mathematically natural prop. We describe the "behavior" of these circuits using morphisms between props. In particular, we give a new proof of the black-boxing theorem proved by Fong and the first author; unlike the original proof, this new one easily generalizes to circuits with nonlinear components. We also use a morphism of props to clarify the relation between circuit diagrams and the signal-flow diagrams in control theory. Technically, the key tools are the Rosebrugh-Sabadini-Walters result relating circuits to special commutative Frobenius monoids, the monadic adjunction between props and signatures, and a result saying which symmetric monoidal categories are equivalent to props., Comment: 47 pages LaTeX

Reaction networks, or equivalently Petri nets, are a general framework for describing processes in which entities of various kinds interact and turn into other entities. In chemistry, where the reactions are assigned "rate constants", any reaction network gives rise to a nonlinear dynamical system called its "rate equation". Here we generalize these ideas to "open" reaction networks, which allow entities to flow in and out at certain designated inputs and outputs. We treat open reaction networks as morphisms in a category. Composing two such morphisms connects the outputs of the first to the inputs of the second. We construct a functor sending any open reaction network to its corresponding "open dynamical system". This provides a compositional framework for studying the dynamics of reaction networks. We then turn to statics: that is, steady state solutions of open dynamical systems. We construct a "black-boxing" functor that sends any open dynamical system to the relation that it imposes between input and output variables in steady states. This extends our earlier work on black-boxing for Markov processes., Comment: 44 pages LaTeX

Our assumption that spacetime is a continuum leads to many challenges in mathematical physics. Singularities, divergent integrals and the like threaten many of our favorite theories, from Newtonian gravity to classical electrodynamics, quantum electrodynamics and the Standard Model. In general relativity, singularities are intimately connected to some of the theory's most dramatic successful predictions. We survey these problems and the large amount of work that has gone into dealing with them.

Sunada's work on topological crystallography emphasizes the role of the "maximal abelian cover" of a graph $X$. This is a covering space of $X$ for which the group of deck transformations is the first homology group $H_1(X,\mathbb{Z})$. An embedding of the maximal abelian cover in a vector space can serve as the pattern for a crystal: atoms are located at the vertices, while bonds lie on the edges. We prove that for any connected graph $X$ without bridges, there is a canonical way to embed the maximal abelian cover of $X$ into the vector space $H_1(X,\mathbb{R})$. We call this a "topological crystal". Crystals of graphene and diamond are examples of this construction. We prove that any symmetry of a graph lifts to a symmetry of its topological crystal. We also compute the density of atoms in this topological crystal. We give special attention to the topological crystals coming from Platonic solids. The key technical tools are a way of decomposing the 1-chain coming from a path in $X$ into manageable pieces, and the work of Bacher, de la Harpe and Nagnibeda on integral cycles and integral cuts., Comment: 21 pages LaTeX with .png figures

We construct an operad $\mathrm{Phyl}$ whose operations are the edge-labelled trees used in phylogenetics. This operad is the coproduct of $\mathrm{Com}$, the operad for commutative semigroups, and $[0,\infty)$, the operad with unary operations corresponding to nonnegative real numbers, where composition is addition. We show that there is a homeomorphism between the space of $n$-ary operations of $\mathrm{Phyl}$ and $\mathcal{T}_n\times [0,\infty)^{n+1}$, where $\mathcal{T}_n$ is the space of metric $n$-trees introduced by Billera, Holmes and Vogtmann. Furthermore, we show that the Markov models used to reconstruct phylogenetic trees from genome data give coalgebras of $\mathrm{Phyl}$. These always extend to coalgebras of the larger operad $\mathrm{Com} + [0,\infty]$, since Markov processes on finite sets converge to an equilibrium as time approaches infinity. We show that for any operad $O$, its coproduct with $[0,\infty]$ contains the operad $W(O)$ constucted by Boardman and Vogt. To prove these results, we explicitly describe the coproduct of operads in terms of labelled trees., Comment: 48 pages, 3 figures

In this paper we review various information-theoretic characterizations of the approach to equilibrium in biological systems. The replicator equation, evolutionary game theory, Markov processes and chemical reaction networks all describe the dynamics of a population or probability distribution. Under suitable assumptions, the distribution will approach an equilibrium with the passage of time. Relative entropy - that is, the Kullback--Leibler divergence, or various generalizations of this - provides a quantitative measure of how far from equilibrium the system is. We explain various theorems that give conditions under which relative entropy is nonincreasing. In biochemical applications these results can be seen as versions of the Second Law of Thermodynamics, stating that free energy can never increase with the passage of time. In ecological applications, they make precise the notion that a population gains information from its environment as it approaches equilibrium., Comment: 20 pages

We define the concept of an "open" Markov process, or more precisely, continuous-time Markov chain, which is one where probability can flow in or out of certain states called "inputs" and "outputs". One can build up a Markov process from smaller open pieces. This process is formalized by making open Markov processes into the morphisms of a dagger compact category. We show that the behavior of a detailed balanced open Markov process is determined by a principle of minimum dissipation, closely related to Prigogine's principle of minimum entropy production. Using this fact, we set up a functor mapping open detailed balanced Markov processes to open circuits made of linear resistors. We also describe how to "black box" an open Markov process, obtaining the linear relation between input and output data that holds in any steady state, including nonequilibrium steady states with a nonzero flow of probability through the system. We prove that black boxing gives a symmetric monoidal dagger functor sending open detailed balanced Markov processes to Lagrangian relations between symplectic vector spaces. This allows us to compute the steady state behavior of an open detailed balanced Markov process from the behaviors of smaller pieces from which it is built. We relate this black box functor to a previously constructed black box functor for circuits., Comment: 43 pages, TikZ figures

Passive linear networks are used in a wide variety of engineering applications, but the best studied are electrical circuits made of resistors, inductors and capacitors. We describe a category where a morphism is a circuit of this sort with marked input and output terminals. In this category, composition describes the process of attaching the outputs of one circuit to the inputs of another. We construct a functor, dubbed the "black box functor", that takes a circuit, forgets its internal structure, and remembers only its external behavior. Two circuits have the same external behavior if and only if they impose same relation between currents and potentials at their terminals. The space of these currents and potentials naturally has the structure of a symplectic vector space, and the relation imposed by a circuit is a Lagrangian linear relation. Thus, the black box functor goes from our category of circuits to the category of symplectic vector spaces and Lagrangian linear relations. We prove that this functor is We prove that this functor is symmetric monoidal and indeed a hypergraph functor. We assume the reader is familiar with category theory, but not with circuit theory or symplectic linear algebra., Comment: 54 pages, with TikZ figures

In 1914 Lebesgue defined a "universal covering" to be a convex subset of the plane that contains an isometric copy of any subset of diameter 1. His challenge of finding a universal covering with the least possible area has been addressed by various mathematicians: Pal, Sprague and Hansen have each created a smaller universal covering by removing regions from those known before. However, Hansen's last reduction was microsopic: he claimed to remove an area of $6 \cdot 10^{-18}$, but we show that he actually removed an area of just $8 \cdot 10^{-21}$. In the following, with the help of Greg Egan, we find a new, smaller universal covering with area less than $0.8441153$. This reduces the area of the previous best universal covering by a whopping $2.2 \cdot 10^{-5}$., Comment: 11 pages, 5 jpeg figures, numerical errors corrected

Control theory uses "signal-flow diagrams" to describe processes where real-valued functions of time are added, multiplied by scalars, differentiated and integrated, duplicated and deleted. These diagrams can be seen as string diagrams for the symmetric monoidal category FinVect_k of finite-dimensional vector spaces over the field of rational functions k = R(s), where the variable s acts as differentiation and the monoidal structure is direct sum rather than the usual tensor product of vector spaces. For any field k we give a presentation of FinVect_k in terms of the generators used in signal flow diagrams. A broader class of signal-flow diagrams also includes "caps" and "cups" to model feedback. We show these diagrams can be seen as string diagrams for the symmetric monoidal category FinRel_k, where objects are still finite-dimensional vector spaces but the morphisms are linear relations. We also give a presentation for FinRel_k. The relations say, among other things, that the 1-dimensional vector space k has two special commutative dagger-Frobenius structures, such that the multiplication and unit of either one and the comultiplication and counit of the other fit together to form a bimonoid. This sort of structure, but with tensor product replacing direct sum, is familiar from the "ZX-calculus" obeyed by a finite-dimensional Hilbert space with two mutually unbiased bases., Comment: 42 pages LaTeX

We give a new characterization of relative entropy, also known as the Kullback-Leibler divergence. We use a number of interesting categories related to probability theory. In particular, we consider a category FinStat where an object is a finite set equipped with a probability distribution, while a morphism is a measure-preserving function $f: X \to Y$ together with a stochastic right inverse $s: Y \to X$. The function $f$ can be thought of as a measurement process, while s provides a hypothesis about the state of the measured system given the result of a measurement. Given this data we can define the entropy of the probability distribution on $X$ relative to the "prior" given by pushing the probability distribution on $Y$ forwards along $s$. We say that $s$ is "optimal" if these distributions agree. We show that any convex linear, lower semicontinuous functor from FinStat to the additive monoid $[0,\infty]$ which vanishes when $s$ is optimal must be a scalar multiple of this relative entropy. Our proof is independent of all earlier characterizations, but inspired by the work of Petz., Comment: 32 pages, minor revision