arXiv subjects

David I. Spivak

Publications and source records attributed to David I. Spivak.

At least 55 records · Page 3Linked to original sources

arXiv2020

Behavioral Mereology (Proofs and Properties)

Mereology is the study of parts and the relationships that hold between them. We introduce a behavioral approach to mereology, in which systems and their parts are known only by the types of behavior they can exhibit. Our discussion is formally topos-theoretic, and agnostic to the topos, providing maximal generality; however, by using only its internal logic we can hide the details and readers may assume a completely elementary set-theoretic discussion. We consider the relationship between various parts of a whole in terms of how behavioral constraints are passed between them, and give an inter-modal logic that generalizes the usual alethic modalities in the setting of symmetric accessibility.

math.LO↗

arXiv2020

A Compositional Sheaf-Theoretic Framework for Event-Based Systems (Extended Version)

A compositional sheaf-theoretic framework for the modeling of complex event-based systems is presented. We show that event-based systems are machines, with inputs and outputs, and that they can be composed with machines of different types, all within a unified, sheaf-theoretic formalism. We take robotic systems as an exemplar of complex systems and rigorously describe actuators, sensors, and algorithms using this framework.

eess.SY↗

arXiv2020

Poly: An abundant categorical setting for mode-dependent dynamics

Dynamical systems---by which we mean machines that take time-varying input, change their state, and produce output---can be wired together to form more complex systems. Previous work has shown how to allow collections of machines to reconfigure their wiring diagram dynamically, based on their collective state. This notion was called "mode dependence", and while the framework was compositional (forming an operad of re-wiring diagrams and algebra of mode-dependent dynamical systems on it), the formulation itself was more "creative" than it was natural. In this paper we show that the theory of mode-dependent dynamical systems can be more naturally recast within the category Poly of polynomial functors. This category is almost superlatively abundant in its structure: for example, it has \emph{four} interacting monoidal structures $(+,\times,\otimes,\circ)$, two of which ($\times,\otimes$) are monoidal closed, and the comonoids for $\circ$ are precisely categories in the usual sense. We discuss how the various structures in Poly show up in the theory of dynamical systems. We also show that the usual coalgebraic formalism for dynamical systems takes place within Poly. Indeed one can see coalgebras as special dynamical systems---ones that do not record their history---formally analogous to contractible groupoids as special categories.

math.CT↗

arXiv2020

Dirichlet Functors are Contravariant Polynomial Functors

Polynomial functors are sums of covariant representable functors from the category of sets to itself. They have a robust theory with many applications -- from operads and opetopes to combinatorial species. In this paper, we define a contravariant analogue of polynomial functors: Dirichlet functors. We develop the basic theory of Dirichlet functors, and relate them to their covariant analogues.

math.CT↗

arXiv2020

A Categorical Semantics for Guarded Petri Nets

We build on the correspondence between Petri nets and free symmetric strict monoidal categories already investigated in the literature, and present a categorical semantics for Petri nets with guards. This comes in two flavors: Deterministic and with side-effects. Using the Grothendieck construction, we show how the guard semantics can be internalized in the net itself.

math.CT↗

arXiv2019

Decomposition-space slices are toposes

We show that the category of decomposition spaces and CULF maps is locally a topos. Precisely, the slice category over any decomposition space D is a presheaf topos, namely decomp/D=Psh(tw D).

math.CT↗

arXiv2019

Backprop as Functor: A compositional perspective on supervised learning

A supervised learning algorithm searches over a set of functions $A \to B$ parametrised by a space $P$ to find the best approximation to some ideal function $f\colon A \to B$. It does this by taking examples $(a,f(a)) \in A\times B$, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent---with respect to a fixed step size and an error function satisfying a certain property---defines a monoidal functor from a category of parametrised functions to this category of update rules. This provides a structural perspective on backpropagation, as well as a broad generalisation of neural networks.

math.CT↗

arXiv2019

Categorical Data Integration for Computational Science

Categorical Query Language is an open-source query and data integration scripting language that can be applied to common challenges in the field of computational science. We discuss how the structure-preserving nature of CQL data migrations protect those who publicly share data from the misinterpretation of their data. Likewise, this feature of CQL migrations allows those who draw from public data sources to be sure only data which meets their specification will actually be transferred. We argue some open problems in the field of data sharing in computational science are addressable by working within this paradigm of functorial data migration. We demonstrate these tools by integrating data from the Open Quantum Materials Database with some alternative materials databases.

cs.DB↗

arXiv2019

Dynamical Systems and Sheaves

A categorical framework for modeling and analyzing systems in a broad sense is proposed. These systems should be thought of as `machines' with inputs and outputs, carrying some sort of signal that occurs through some notion of time. Special cases include continuous and discrete dynamical systems (e.g. Moore machines). Additionally, morphisms between the different types of systems allow their translation in a common framework. A central goal is to understand the systems that result from arbitrary interconnection of component subsystems, possibly of different types, as well as establish conditions that ensure totality and determinism compositionally. The fundamental categorical tools used here include lax monoidal functors, which provide a language of compositionality, as well as sheaf theory, which flexibly captures the crucial notion of time.

math.CT↗

arXiv2018

Evaluating the Pixel Array Method as Applied to Partial Differential Equations

The Pixel Array (PA) Method, originally introduced by Spivak et. al., is a fast method for solving nonlinear or linear systems. One of its distinguishing features is that it presents all solutions within a bounding box, represented by a plot whose axes are the values of "exposed variables." Here we develop a set-theoretic variant of the PA method, named the Pixel Array Solution-Set (PASS) method, that gives PA access to "hidden variables" whose values are not displayed on plot axes. We evaluate the effectiveness of PASS at numerically finding steady states for several partial differential equations. We discretize several one-dimensional solved reaction-diffusion equations, such as the Fisher equation and the Benjamin-Bona-Mohany equation, using finite differences. Then, we run PASS on each equation, and determine whether it successfully finds all boundary conditions for which a numerical steady state might exist. Then we verify whether the steady states found by PASS are correct. Finally, we discuss the benefits and weaknesses of PASS.

math.AP↗

arXiv2018

Abstraction, Composition and Contracts: A Sheaf Theoretic Approach

Complex systems of systems (SoS) are characterized by multiple interconnected subsystems. Typically, each subsystem is designed and analyzed using methodologies and formalisms that are specific to the particular subsystem model of computation considered --- Petri nets, continuous time ODEs, nondeterministic automata, to name a few. When interconnecting subsystems, a designer needs to choose, based on the specific subsystems models, a common abstraction framework to analyze the composition. In this paper we introduce a new framework for abstraction, composition and analysis of SoS that builds on results and methods developed in sheaf theory, category theory and topos theory. In particular, we will be modeling behaviors of systems using sheaves, leverage category theoretic methods to define wiring diagrams and formalize composition and, by establishing a connection with topos theory, define a formal (intuitionistic/constructive) logic with a sound sheaf semantics

cs.LO↗

arXiv2017

Temporal Type Theory: A topos-theoretic approach to systems and behavior

This book introduces a temporal type theory, the first of its kind as far as we know. It is based on a standard core, and as such it can be formalized in a proof assistant such as Coq or Lean by adding a number of axioms. Well-known temporal logics---such as Linear and Metric Temporal Logic (LTL and MTL)---embed within the logic of temporal type theory. The types in this theory represent "behavior types". The language is rich enough to allow one to define arbitrary hybrid dynamical systems, which are mixtures of continuous dynamics---e.g. as described by a differential equation---and discrete jumps. In particular, the derivative of a continuous real-valued function is internally defined. We construct a semantics for the temporal type theory in the topos of sheaves on a translation-invariant quotient of the standard interval domain. In fact, domain theory plays a recurring role in both the semantics and the type theory.

math.CT↗

arXiv2017

Pixel matrices: An elementary technique for solving nonlinear systems

A new technique for approximating the entire solution set for a nonlinear system of relations (nonlinear equations, inequalities, etc. involving algebraic, smooth, or even continuous functions) is presented. The technique is to first plot each function as a pixel matrix, and to then perform a sequence of basic matrix operations, as dictated by how variables are shared by the relations in the system. The result is a pixel matrix graphing the approximated simultaneous solution set for the system.

math.NA↗

arXiv2017

An algebra of open continuous time dynamical systems and networks

Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a series of papers David Spivak with collaborators formalized these kinds of structures (systems of systems) as algebras over presentable colored operads. It is also very useful to consider maps between dynamical systems. This amounts to viewing dynamical systems as objects in an appropriate category. This is the point taken by DeVille and Lerman in the study of dynamics on networks. The goal of this paper is to describe an algebraic structure that encompasses both approaches to systems of systems. To this end we replace the monoidal category of wiring diagrams by a monoidal double category whose objects are surjective submersions. This allows us, on one hand, build new large open systems out of collections of smaller open subsystems and on the other keep track of maps between open systems. As a special case we recover the results of DeVille and Lerman on fibrations of networks of manifolds.

math.DS↗

arXiv2017

Pixel Arrays: A fast and elementary method for solving nonlinear systems

We present a new method, called the pixel array method, for approximating all solutions in a bounding box for an arbitrary nonlinear system of relations. In contrast with other solvers, our approach requires that the user must specify which variables are to be exposed, and which are to be left latent. The entire solution set is then obtained---in terms of these exposed variables---by performing a series of array multiplications on the $n_i$-dimensional plots of the individual relations $R_i$. This procedure introduces no false negatives and is much faster than Newton-based solvers. The key is the unexposed variables, which Newton methods can make no use of. In fact, we found that with even a single unexposed variable our method was more than 10x faster than Julia's NLsolve. Due to its relative simplicity, the pixel array method is also applicable to a broader class of systems than Newton-based solvers are. The purpose of this article is to give an account of this new method.

math.NA↗

arXiv2016

String diagrams for traced and compact categories are oriented 1-cobordisms

We give an alternate conception of string diagrams as labeled 1-dimensional oriented cobordisms, the operad of which we denote by Cob/O, where O is the set of string labels. The axioms of traced (symmetric monoidal) categories are fully encoded by Cob/O in the sense that there is an equivalence between (Cob/O)-algebras, for varying O, and traced categories with varying object set. The same holds for compact (closed) categories, the difference being in terms of variance in O. As a consequence of our main theorem, we give a characterization of the 2-category of traced categories solely in terms of those of monoidal and compact categories, without any reference to the usual structures or axioms of traced categories. In an appendix we offer a complete proof of the well-known relationship between the 2-category of monoidal categories with strong monoidal functors and the 2-category of monoidal categories whose object set is free with strict functors; similarly for traced and compact categories.

math.CT↗

arXiv2016

Homotopy composition of cospans

It is well known that the category of finite sets and cospans, composed by pushout, contains the universal {\em special} commutative Frobenius algebra. In this note we observe that the same construction yields also general commutative Frobenius algebras, if just the pushouts are changed to homotopy pushouts.

math.CT↗

arXiv2016

The steady states of coupled dynamical systems compose according to matrix arithmetic

Open dynamical systems are mathematical models of machines that take input, change their internal state, and produce output. For example, one may model anything from neurons to robots in this way. Several open dynamical systems can be arranged in series, in parallel, and with feedback to form a new dynamical system---this is called compositionality---and the process can be repeated in a fractal-like manner to form more complex systems of systems. One issue is that as larger systems are created, their state space grows exponentially. In this paper a technique for calculating the steady states of an interconnected system of systems, in terms of the steady states of its component dynamical systems, is provided. These are organized into "steady state matrices" which generalize bifurcation diagrams. It is shown that the compositionality structure of dynamical systems fits with the familiar monoidal structure for the steady state matrices, where serial, parallel, and feedback composition of matrices correspond to multiplication, Kronecker product, and partial trace operations. The steady state matrices of dynamical systems respect this compositionality structure, exponentially reducing the complexity involved in studying the steady states of composite dynamical systems.

math.DS↗