Search Results for "category-theory-for-the-sciences"

Category Theory for the Sciences

Category Theory for the Sciences

  • Author: David I. Spivak
  • Publisher: MIT Press
  • ISBN: 0262028131
  • Category: Computers
  • Page: 486
  • View: 9744
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An introduction to category theory as a rigorous, flexible, and coherent modeling language that can be used across the sciences.

Handbook of Research Methods in Complexity Science

Handbook of Research Methods in Complexity Science

Theory and Applications

  • Author: Eve Mitleton-Kelly,Alexandros Paraskevas,Christopher Day
  • Publisher: Edward Elgar Publishing
  • ISBN: 1785364421
  • Category:
  • Page: N.A
  • View: 6887
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This comprehensive Handbook is aimed at both academic researchers and practitioners in the field of complexity science. The book’s 26 chapters, specially written by leading experts, provide in-depth coverage of research methods based on the sciences of complexity. The research methods presented are illustratively applied to practical cases and are readily accessible to researchers and decision makers alike.

Operads of Wiring Diagrams

Operads of Wiring Diagrams

  • Author: Donald Yau
  • Publisher: Springer
  • ISBN: 3319950010
  • Category: Mathematics
  • Page: 308
  • View: 4130
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Wiring diagrams form a kind of graphical language that describes operations or processes with multiple inputs and outputs, and shows how such operations are wired together to form a larger and more complex operation. This monograph presents a comprehensive study of the combinatorial structure of the various operads of wiring diagrams, their algebras, and the relationships between these operads. The book proves finite presentation theorems for operads of wiring diagrams as well as their algebras. These theorems describe the operad in terms of just a few operadic generators and a small number of generating relations. The author further explores recent trends in the application of operad theory to wiring diagrams and related structures, including finite presentations for the propagator algebra, the algebra of discrete systems, the algebra of open dynamical systems, and the relational algebra. A partial verification of David Spivak’s conjecture regarding the quotient-freeness of the relational algebra is also provided. In the final part, the author constructs operad maps between the various operads of wiring diagrams and identifies their images. Assuming only basic knowledge of algebra, combinatorics, and set theory, this book is aimed at advanced undergraduate and graduate students as well as researchers working in operad theory and its applications. Numerous illustrations, examples, and practice exercises are included, making this a self-contained volume suitable for self-study.

Category Theory and Computer Science

Category Theory and Computer Science

Paris, France, September 3-6, 1991. Proceedings

  • Author: David H. Pitt,Pierre-Louis Curien,Samson Abramsky,Andrew Pitts,Axel Poigne,David E. Rydeheard
  • Publisher: Springer Science & Business Media
  • ISBN: 9783540544951
  • Category: Mathematics
  • Page: 304
  • View: 7515
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The papers in this volume were presented at the fourth biennial Summer Conference on Category Theory and Computer Science, held in Paris, September3-6, 1991. Category theory continues to be an important tool in foundationalstudies in computer science. It has been widely applied by logicians to get concise interpretations of many logical concepts. Links between logic and computer science have been developed now for over twenty years, notably via the Curry-Howard isomorphism which identifies programs with proofs and types with propositions. The triangle category theory - logic - programming presents a rich world of interconnections. Topics covered in this volume include the following. Type theory: stratification of types and propositions can be discussed in a categorical setting. Domain theory: synthetic domain theory develops domain theory internally in the constructive universe of the effective topos. Linear logic: the reconstruction of logic based on propositions as resources leads to alternatives to traditional syntaxes. The proceedings of the previous three category theory conferences appear as Lecture Notes in Computer Science Volumes 240, 283 and 389.

Handbook of Categorical Algebra: Volume 1, Basic Category Theory

Handbook of Categorical Algebra: Volume 1, Basic Category Theory

  • Author: Francis Borceux
  • Publisher: Cambridge University Press
  • ISBN: 9780521441780
  • Category: Mathematics
  • Page: 345
  • View: 8128
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First of a 3-volume work giving a detailed account of what should be known by all working in, or using category theory. Volume 1 covers basic concepts.

Relative category theory and geometric morphisms

Relative category theory and geometric morphisms

a logical approach

  • Author: Jonathan Chapman,Frederick Rowbottom
  • Publisher: Oxford University Press, USA
  • ISBN: N.A
  • Category: Mathematics
  • Page: 263
  • View: 6550
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Topos theory provides an important setting and language for much of mathematical logic and set theory. It is well known that a typed language can be given for a topos to be regarded as a category of sets. This enables a fruitful interplay between category theory and set theory. However, one stumbling block to a logical approach to topos theory has been the treatment of geometric morphisms. This book presents a convenient and natural solution to this problem by developing the notion of a frame relative to an elementary topos. The authors show how this technique enables a logical approach to be taken to topics such as category theory relative to a topos and the relative Giraud theorem. The work is self-contained except that the authors presuppose a familiarity with basic category theory and topos theory. Logicians, set and category theorists, and computer scientist working in the field will find this work essential reading.

A New Foundation for Representation in Cognitive and Brain Science

A New Foundation for Representation in Cognitive and Brain Science

Category Theory and the Hippocampus

  • Author: Jaime Gómez-Ramirez
  • Publisher: Springer Science & Business Media
  • ISBN: 9400777388
  • Category: Medical
  • Page: 193
  • View: 1558
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The purpose of the book is to advance in the understanding of brain function by defining a general framework for representation based on category theory. The idea is to bring this mathematical formalism into the domain of neural representation of physical spaces, setting the basis for a theory of mental representation, able to relate empirical findings, uniting them into a sound theoretical corpus. The innovative approach presented in the book provides a horizon of interdisciplinary collaboration that aims to set up a common agenda that synthesizes mathematical formalization and empirical procedures in a systemic way. Category theory has been successfully applied to qualitative analysis, mainly in theoretical computer science to deal with programming language semantics. Nevertheless, the potential of category theoretic tools for quantitative analysis of networks has not been tackled so far. Statistical methods to investigate graph structure typically rely on network parameters. Category theory can be seen as an abstraction of graph theory. Thus, new categorical properties can be added into network analysis and graph theoretic constructs can be accordingly extended in more fundamental basis. By generalizing networks using category theory we can address questions and elaborate answers in a more fundamental way without waiving graph theoretic tools. The vital issue is to establish a new framework for quantitative analysis of networks using the theory of categories, in which computational neuroscientists and network theorists may tackle in more efficient ways the dynamics of brain cognitive networks. The intended audience of the book is researchers who wish to explore the validity of mathematical principles in the understanding of cognitive systems. All the actors in cognitive science: philosophers, engineers, neurobiologists, cognitive psychologists, computer scientists etc. are akin to discover along its pages new unforeseen connections through the development of concepts and formal theories described in the book. Practitioners of both pure and applied mathematics e.g., network theorists, will be delighted with the mapping of abstract mathematical concepts in the terra incognita of cognition.

Category Theory for Computing Science

Category Theory for Computing Science

  • Author: Michael Barr
  • Publisher: N.A
  • ISBN: N.A
  • Category: Mathematics
  • Page: 325
  • View: 9697
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Antipositivist Theories of the Sciences

Antipositivist Theories of the Sciences

Critical Rationalism, Critical Theory and Scientific Realism

  • Author: N. Stockman
  • Publisher: Springer Science & Business Media
  • ISBN: 9401576785
  • Category: Science
  • Page: 284
  • View: 7384
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The sciences are too important to be left exclusively to scientists, and indeed they have not been. The structure of scientific knowledge, the role of the sciences in society, the appropriate social contexts for the pursuit of scientific inquiry, have long been matters for reflection and debate about the sciences carried on both within academe and outside it. Even within the universities this reflection has not been the property of any single discipline. Philosophy might have been first in the field, but history and the social sciences have also entered the fray. For the latter, new problems came to the fore, since reflection on the sciences is, in the case of the social sciences, necessarily also reflection on themselves as sciences. Reflection on the natural sciences and self-reflection by the social sciences came to be dominated in the 1960s by the term 'positivism'. At the time when this word had been invented, the sciences were flourishing; their social and material environment had become increasingly favourable to scientific progress, and the sciences were pointing the way to an optimistic future. In the later twentieth century, however, 'positivism' came to be a word used more frequently by those less sure of nineteenth century certainties. In both sociology and philosophy, 'positivism' was now something to be rejected, and, symbolizing the collapse of an earlier consensus, it became itself the shibboleth of a new dissensus, as different groups of reflective thinkers, in rejecting 'positivism', rejected something different, and often rejected each other.

Probability Theory and Applications

Probability Theory and Applications

Essays to the Memory of József Mogyoródi

  • Author: Janos Galambos,Imre Kátai
  • Publisher: Springer Science & Business Media
  • ISBN: 9780792319221
  • Category: Mathematics
  • Page: 350
  • View: 5191
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"Et moi, ... , si j'avait su comment en revenir, je One service mathematics bas rendered the human race. It bas put common sense back n'y serais point all~.' where it belongs, on the topmost shelf next to lu1esVeme the dusty canister labelled 'discarded nonsense'~ Eric T. Bell 1be series is divergent; therefore we may be able to do something with it O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlineari­ ties abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sci­ ences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One ser­ vice topology has rendered mathematical physics ... '; 'One service logic has rendered computer science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d 'etre of this series.