Open Systems for the Working Mathematician
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The ACT Pipeline from Abstraction to Application
Addressing hypercomplexity: The ACT pipeline from abstraction to application David I. Spivak (Joint with Brendan Fong, Paolo Perrone, Evan Patterson) Department of Mathematics Massachusetts Institute of Technology 0 / 25 Introduction Outline 1 Introduction Why? How? What? Plan of the talk 2 Some new applications 3 Catlab.jl 4 Conclusion 0 / 25 Introduction Why? What's wrong? As problem complexity increases, good organization becomes crucial. (Courtesy of Spencer Breiner, NIST) 1 / 25 Introduction Why? The problem is hypercomplexity The world is becoming more complex. We need to deal with more and more diverse sorts of interaction. Rather than closed dynamical systems, everything is open. The state of one system affects those of systems in its environment. Examples are everywhere. Designing anything requires input from more diverse stakeholders. Software is developed by larger teams, touching a common codebase. Scientists need to use experimental data obtained by their peers. (But the data structure and assumptions don't simply line up.) We need to handle this new complexity; traditional methods are faltering. 2 / 25 Introduction How? How might we deal with hypercomplexity? Organize the problem spaces effectively, as objects in their own right. Find commonalities in the various problems we work on. Consider the ways we build complex problems out of simple ones. Think of problem spaces and solutions as mathematical entities. With that in hand, how do you solve a hypercomplex problem? Migrating subproblems to groups who can solve them. Take the returned solutions and fit them together. Ensure in advance that these solutions will compose. Do this according the mathematical problem-space structure. -
Diagrammatics in Categorification and Compositionality
Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 ABSTRACT Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 Copyright c 2019 by Dmitry Vagner All rights reserved Abstract In the present work, I explore the theme of diagrammatics and their capacity to shed insight on two trends|categorification and compositionality|in and around contemporary category theory. The work begins with an introduction of these meta- phenomena in the context of elementary sets and maps. Towards generalizing their study to more complicated domains, we provide a self-contained treatment|from a pedagogically novel perspective that introduces almost all concepts via diagrammatic language|of the categorical machinery with which we may express the broader no- tions found in the sequel. The work then branches into two seemingly unrelated disciplines: dynamical systems and knot theory. In particular, the former research defines what it means to compose dynamical systems in a manner analogous to how one composes simple maps. The latter work concerns the categorification of the slN link invariant. In particular, we use a virtual filtration to give a more diagrammatic reconstruction of Khovanov-Rozansky homology via a smooth TQFT. -
Duality and Traces for Indexed Monoidal Categories
DUALITY AND TRACES FOR INDEXED MONOIDAL CATEGORIES KATE PONTO AND MICHAEL SHULMAN Abstract. By the Lefschetz fixed point theorem, if an endomorphism of a topological space is fixed-point-free, then its Lefschetz number vanishes. This necessary condition is not usually sufficient, however; for that we need a re- finement of the Lefschetz number called the Reidemeister trace. Abstractly, the Lefschetz number is a trace in a symmetric monoidal cate- gory, while the Reidemeister trace is a trace in a bicategory. In this paper, we show that for any symmetric monoidal category with an associated indexed symmetric monoidal category, there is an associated bicategory which produces refinements of trace analogous to the Reidemeister trace. This bicategory also produces a new notion of trace for parametrized spaces with dualizable fibers, which refines the obvious “fiberwise" traces by incorporating the action of the fundamental group of the base space. Our abstract framework lays the foundation for generalizations of these ideas to other contexts. Contents 1. Introduction2 2. Indexed monoidal categories5 3. Indexed coproducts8 4. Symmetric monoidal traces 12 5. Shadows from indexed monoidal categories 15 6. Fiberwise duality 22 7. Base change objects 27 8. Total duality 29 9. String diagrams for objects 39 10. String diagrams for morphisms 42 11. Proofs for fiberwise duality and trace 49 12. Proofs for total duality and trace 63 References 65 Date: Version of October 31, 2011. Both authors were supported by National Science Foundation postdoctoral fellowships during the writing of this paper. This version of this paper contains diagrams that use dots and dashes to distinguish different features, and is intended for printing on a black and white printer. -
Category Theory Cheat Sheet
Musa Al-hassy https://github.com/alhassy/CatsCheatSheet July 5, 2019 Even when morphisms are functions, the objects need not be sets: Sometimes the objects . are operations —with an appropriate definition of typing for the functions. The categories Reference Sheet for Elementary Category Theory of F -algebras are an example of this. “Gluing” Morphisms Together Categories Traditional function application is replaced by the more generic concept of functional A category C consists of a collection of “objects” Obj C, a collection of “(homo)morphisms” composition suggested by morphism-arrow chaining: HomC(a; b) for any a; b : Obj C —also denoted “a !C b”—, an operation Id associating Whenever we have two morphisms such that the target type of one of them, say g : B A a morphism Ida : Hom(a; a) to each object a, and a dependently-typed “composition” is the same as the source type of the other, say f : C B then “f after g”, their composite operation _ ◦ _ : 8fABC : Objg ! Hom(B; C) ! Hom(A; B) ! Hom(A; C) that is morphism, f ◦ g : C A can be defined. It “glues” f and g together, “sequentially”: required to be associative with Id as identity. It is convenient to define a pair of operations src; tgt from morphisms to objects as follows: f g f◦g C − B − A ) C − A composition-Type f : X !C Y ≡ src f = X ^ tgt f = Y src; tgt-Definition Composition is the basis for gluing morphisms together to build more complex morphisms. Instead of HomC we can instead assume primitive a ternary relation _ : _ !C _ and However, not every two morphisms can be glued together by composition. -
The Joy of String Diagrams Pierre-Louis Curien
The joy of string diagrams Pierre-Louis Curien To cite this version: Pierre-Louis Curien. The joy of string diagrams. Computer Science Logic, Sep 2008, Bertinoro, Italy. pp.15-22. hal-00697115 HAL Id: hal-00697115 https://hal.archives-ouvertes.fr/hal-00697115 Submitted on 14 May 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Joy of String Diagrams Pierre-Louis Curien Preuves, Programmes et Syst`emes,CNRS and University Paris 7 May 14, 2012 Abstract In the past recent years, I have been using string diagrams to teach basic category theory (adjunctions, Kan extensions, but also limits and Yoneda embedding). Us- ing graphical notations is undoubtedly joyful, and brings us close to other graphical syntaxes of circuits, interaction nets, etc... It saves us from laborious verifications of naturality, which is built-in in string diagrams. On the other hand, the language of string diagrams is more demanding in terms of typing: one may need to introduce explicit coercions for equalities of functors, or for distinguishing a morphism from a point in the corresponding internal homset. So that in some sense, string diagrams look more like a language ”`ala Church", while the usual mathematics of, say, Mac Lane's "Categories for the working mathematician" are more ”`ala Curry". -
SHEAVES, TOPOSES, LANGUAGES Acceleration Based on A
Chapter 7 Logic of behavior: Sheaves, toposes, and internal languages 7.1 How can we prove our machine is safe? Imagine you are trying to design a system of interacting components. You wouldn’t be doing this if you didn’t have a goal in mind: you want the system to do something, to behave in a certain way. In other words, you want to restrict its possibilities to a smaller set: you want the car to remain on the road, you want the temperature to remain in a particular range, you want the bridge to be safe for trucks to pass. Out of all the possibilities, your system should only permit some. Since your system is made of components that interact in specified ways, the possible behavior of the whole—in any environment—is determined by the possible behaviors of each of its components in their local environments, together with the precise way in which they interact.1 In this chapter, we will discuss a logic wherein one can describe general types of behavior that occur over time, and prove properties of a larger-scale system from the properties and interaction patterns of its components. For example, suppose we want an autonomous vehicle to maintain a distance of some safe R from other objects. To do so, several components must interact: a 2 sensor that approximates the real distance by an internal variable S0, a controller that uses S0 to decide what action A to take, and a motor that moves the vehicle with an 1 The well-known concept of emergence is not about possibilities, it is about prediction. -
UNIVERSITY of CALIFORNIA RIVERSIDE the Grothendieck
UNIVERSITY OF CALIFORNIA RIVERSIDE The Grothendieck Construction in Categorical Network Theory A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics by Joseph Patrick Moeller December 2020 Dissertation Committee: Dr. John C. Baez, Chairperson Dr. Wee Liang Gan Dr. Carl Mautner Copyright by Joseph Patrick Moeller 2020 The Dissertation of Joseph Patrick Moeller is approved: Committee Chairperson University of California, Riverside Acknowledgments First of all, I owe all of my achievements to my wife, Paola. I couldn't have gotten here without my parents: Daniel, Andrea, Tonie, Maria, and Luis, or my siblings: Danielle, Anthony, Samantha, David, and Luis. I would like to thank my advisor, John Baez, for his support, dedication, and his unique and brilliant style of advising. I could not have become the researcher I am under another's instruction. I would also like to thank Christina Vasilakopoulou, whose kindness, energy, and expertise cultivated a deeper appreciation of category theory in me. My expe- rience was also greatly enriched by my academic siblings: Daniel Cicala, Kenny Courser, Brandon Coya, Jason Erbele, Jade Master, Franciscus Rebro, and Christian Williams, and by my cohort: Justin Davis, Ethan Kowalenko, Derek Lowenberg, Michel Manrique, and Michael Pierce. I would like to thank the UCR math department. Professors from whom I learned a ton of algebra, topology, and category theory include Julie Bergner, Vyjayanthi Chari, Wee-Liang Gan, Jos´eGonzalez, Jacob Greenstein, Carl Mautner, Reinhard Schultz, and Steffano Vidussi. Special thanks goes to the department chair Yat-Sun Poon, as well as Margarita Roman, Randy Morgan, and James Marberry, and many others who keep the whole thing together. -
Categorification and (Virtual) Knots
Categorification and (virtual) knots Daniel Tubbenhauer If you really want to understand something - (try to) categorify it! 13.02.2013 = = Daniel Tubbenhauer Categorification and (virtual) knots 13.02.2013 1 / 38 1 Categorification What is categorification? Two examples The ladder of categories 2 What we want to categorify Virtual knots and links The virtual Jones polynomial The virtual sln polynomial 3 The categorification The algebraic perspective The categorical perspective More to do! Daniel Tubbenhauer Categorification and (virtual) knots 13.02.2013 2 / 38 What is categorification? Categorification is a scary word, but it refers to a very simple idea and is a huge business nowadays. If I had to explain the idea in one sentence, then I would choose Some facts can be best explained using a categorical language. Do you need more details? Categorification can be easily explained by two basic examples - the categorification of the natural numbers through the category of finite sets FinSet and the categorification of the Betti numbers through homology groups. Let us take a look on these two examples in more detail. Daniel Tubbenhauer Categorification and (virtual) knots 13.02.2013 3 / 38 Finite Combinatorics and counting Let us consider the category FinSet - objects are finite sets and morphisms are maps between these sets. The set of isomorphism classes of its objects are the natural numbers N with 0. This process is the inverse of categorification, called decategorification- the spirit should always be that decategorification should be simple while categorification could be hard. We note the following observations. Daniel Tubbenhauer Categorification and (virtual) knots 13.02.2013 4 / 38 Finite Combinatorics and counting Much information is lost, i.e. -
Mathematics and the Brain: a Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness
entropy Review Mathematics and the Brain: A Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness 1,2,3, , 4,5,6,7, 8, Georg Northoff * y, Naotsugu Tsuchiya y and Hayato Saigo y 1 Mental Health Centre, Zhejiang University School of Medicine, Hangzhou 310058, China 2 Institute of Mental Health Research, University of Ottawa, Ottawa, ON K1Z 7K4 Canada 3 Centre for Cognition and Brain Disorders, Hangzhou Normal University, Hangzhou 310036, China 4 School of Psychological Sciences, Faculty of Medicine, Nursing and Health Sciences, Monash University, Melbourne, Victoria 3800, Australia; [email protected] 5 Turner Institute for Brain and Mental Health, Monash University, Melbourne, Victoria 3800, Australia 6 Advanced Telecommunication Research, Soraku-gun, Kyoto 619-0288, Japan 7 Center for Information and Neural Networks (CiNet), National Institute of Information and Communications Technology (NICT), Suita, Osaka 565-0871, Japan 8 Nagahama Institute of Bio-Science and Technology, Nagahama 526-0829, Japan; [email protected] * Correspondence: georg.northoff@theroyal.ca All authors contributed equally to the paper as it was a conjoint and equally distributed work between all y three authors. Received: 18 July 2019; Accepted: 9 October 2019; Published: 17 December 2019 Abstract: Consciousness is a central issue in neuroscience, however, we still lack a formal framework that can address the nature of the relationship between consciousness and its physical substrates. In this review, we provide a novel mathematical framework of category theory (CT), in which we can define and study the sameness between different domains of phenomena such as consciousness and its neural substrates. CT was designed and developed to deal with the relationships between various domains of phenomena. -
Categorification of the Müller-Wichards System Performance Estimation Model: Model Symmetries, Invariants, and Closed Forms
Article Categorification of the Müller-Wichards System Performance Estimation Model: Model Symmetries, Invariants, and Closed Forms Allen D. Parks 1,* and David J. Marchette 2 1 Electromagnetic and Sensor Systems Department, Naval Surface Warfare Center Dahlgren Division, Dahlgren, VA 22448, USA 2 Strategic and Computing Systems Department, Naval Surface Warfare Center Dahlgren Division, Dahlgren, VA 22448, USA; [email protected] * Correspondence: [email protected] Received: 25 October 2018; Accepted: 11 December 2018; Published: 24 January 2019 Abstract: The Müller-Wichards model (MW) is an algebraic method that quantitatively estimates the performance of sequential and/or parallel computer applications. Because of category theory’s expressive power and mathematical precision, a category theoretic reformulation of MW, i.e., CMW, is presented in this paper. The CMW is effectively numerically equivalent to MW and can be used to estimate the performance of any system that can be represented as numerical sequences of arithmetic, data movement, and delay processes. The CMW fundamental symmetry group is introduced and CMW’s category theoretic formalism is used to facilitate the identification of associated model invariants. The formalism also yields a natural approach to dividing systems into subsystems in a manner that preserves performance. Closed form models are developed and studied statistically, and special case closed form models are used to abstractly quantify the effect of parallelization upon processing time vs. loading, as well as to establish a system performance stationary action principle. Keywords: system performance modelling; categorification; performance functor; symmetries; invariants; effect of parallelization; system performance stationary action principle 1. Introduction In the late 1980s, D. -
Category Theory Using String Diagrams
Category Theory Using String Diagrams Dan Marsden November 11, 2014 Abstract In [Fokkinga, 1992a,b] and [Fokkinga and Meertens, 1994] a calcula- tional approach to category theory is developed. The scheme has many merits, but sacrifices useful type information in the move to an equational style of reasoning. By contrast, traditional proofs by diagram pasting re- tain the vital type information, but poorly express the reasoning and development of categorical proofs. In order to combine the strengths of these two perspectives, we propose the use of string diagrams, common folklore in the category theory community, allowing us to retain the type information whilst pursuing a calculational form of proof. These graphi- cal representations provide a topological perspective on categorical proofs, and silently handle functoriality and naturality conditions that require awkward bookkeeping in more traditional notation. Our approach is to proceed primarily by example, systematically ap- plying graphical techniques to many aspects of category theory. We de- velop string diagrammatic formulations of many common notions, includ- ing adjunctions, monads, Kan extensions, limits and colimits. We describe representable functors graphically, and exploit these as a uniform source of graphical calculation rules for many category theoretic concepts. These graphical tools are then used to explicitly prove many standard results in our proposed diagrammatic style. 1 Introduction arXiv:1401.7220v2 [math.CT] 9 Nov 2014 This work develops in some detail many aspects of basic category theory, with the aim of demonstrating the combined effectiveness of two key concepts: 1. The calculational reasoning approach to mathematics. 2. The use of string diagrams in category theory. -
Semantics for a Lambda Calculus for String Diagrams
Semantics for a Lambda Calculus for String Diagrams Bert Lindenhovius Michael Mislove Department of Computer Science Department of Computer Science Tulane University Tulane University Vladimir Zamdzhiev Université de Lorraine, CNRS, Inria, LORIA Linear/non-linear (LNL) models, as described by Benton, soundly model a LNL term calculus and LNL logic closely related to intuitionistic linear logic. Every such model induces a canonical en- richment that we show soundly models a LNL lambda calculus for string diagrams, introduced by Rios and Selinger (with primary application in quantum computing). Our abstract treatment of this language leads to simpler concrete models compared to those presented so far. We also extend the language with general recursion and prove soundness. Finally, we present an adequacy result for the diagram-free fragment of the language which corresponds to a modified version of Benton and Wadler’s adjoint calculus with recursion. In keeping with the purpose of the special issue, we also describe the influence of Samson Abram- sky’s research on these results, and on the overall project of which this is a part. 1 Dedication We are pleased to contribute to this volume honoring SAMSON ABRAMSKY’s many contributions to logic and its applications in theoretical computer science. This contribution is an extended version of the paper [28], which concerns the semantics of high-level functional quantum programming languages. This research is part of a project on the same general theme. The project itself would not exist if it weren’t for Samson’s support, guidance and participation. And, as we document in the final section of this contribution, Samson’s research has had a direct impact on this work, and also points the way for further results along the general line we are pursuing.