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Category theory for structural characterization

  • Jochen Mau
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BIOKYBERNETIKA
This chapter is in the book BIOKYBERNETIKA

Abstract

Category theory, in mathematics a part of homological algebra, is abstract theory to study and compare mathematical systems by relations between elements only. For its application outside of mathematics, it is important to notice that it considers a complex system’s structure of effectuation in which its dynamics are confined to evolve. Structural equivalence becomes prerequisite when dynamics of systems are compared. Here, the whole human-body system (HBS) serves for motivation. Concepts of category theory are explained for application to a triangular structure that arises by functional decomposition of HBS, in structurally analogous human-machine systems (HMS) and with a system of axiomatic functional effectuation (AFE), when focus is laid on the interaction of power, tools, and operational mind to generate physical activity. Some previous definitions of relations in these systems need revision to fit into category theory context. The roles of Yoneda embedding and the Yoneda lemma are emphasized as they jointly permit characterization of one object by all the relations it has with all other objects in the category, which sounds like a mathematical paradigm of an artificial intelligence.

Abstract

Category theory, in mathematics a part of homological algebra, is abstract theory to study and compare mathematical systems by relations between elements only. For its application outside of mathematics, it is important to notice that it considers a complex system’s structure of effectuation in which its dynamics are confined to evolve. Structural equivalence becomes prerequisite when dynamics of systems are compared. Here, the whole human-body system (HBS) serves for motivation. Concepts of category theory are explained for application to a triangular structure that arises by functional decomposition of HBS, in structurally analogous human-machine systems (HMS) and with a system of axiomatic functional effectuation (AFE), when focus is laid on the interaction of power, tools, and operational mind to generate physical activity. Some previous definitions of relations in these systems need revision to fit into category theory context. The roles of Yoneda embedding and the Yoneda lemma are emphasized as they jointly permit characterization of one object by all the relations it has with all other objects in the category, which sounds like a mathematical paradigm of an artificial intelligence.

Chapters in this book

  1. Frontmatter I
  2. Prologue I VII
  3. Prologue II XI
  4. Prologue III XIII
  5. Preface XVII
  6. Overview XIX
  7. Contents XXXIII
  8. Part I: Theories
  9. Part I-A: Overarching theory
  10. Introduction 1
  11. Universal axioms in classical Chinese philosophy 5
  12. Category theory for structural characterization 15
  13. Axiomatic bipolar dynamics and their control 45
  14. Part I-B: Systems theories
  15. Introduction 75
  16. Stochastic formalization of agent-oriented systems 79
  17. Simplification of high-dimensional multitempo dynamic models 109
  18. Ideas of symmetry as a biophysical basis of system biomedicine 123
  19. Disorder of multiscale control 149
  20. Part II: Person’s life-sphere
  21. Part II-A: Person’s biosphere
  22. Introduction 185
  23. Mutations as activators of biological evolutionary processes at population levels 189
  24. Immunometabolism of T-cells in COVID-19 209
  25. Part II-A.2: Body’s vital functions
  26. Introduction 245
  27. Structural modeling of vascular networks 249
  28. Mathematical modeling of AI application for the diagnosis of blood flow disorders 283
  29. Modeling of glucose and insulin regulation within the framework of a self-consistent model of the cardiovascular system 303
  30. Hemodynamics in residual myocardial ischemia 319
  31. The quasi-one-dimensional model of the lymph flow in the human lymphatic system 335
  32. An integrate-and-fire mechanism for modeling rhythmicity in the neuroendocrine system 365
  33. Kinetic network modeling of the neuroendocrine hypothalamic-pituitary-adrenal axis dynamics with particular attention on the role of alcohol as a digestif 377
  34. Inflammation and immune response in atherosclerosis 393
  35. Part II-A.3: Body’s motor functions
  36. Introduction 423
  37. A magnetic resonance spectroscopy approach to quantitatively measure GABA and phosphorus level changes in the primary motor cortex elicited by transcranial direct current stimulation 427
  38. Part II-A.4: Body’s operational functions
  39. Introduction 441
  40. The fermionic mind hypothesis–a category-theoretic verification of consciousness 445
  41. Cross-task cognitive workload measurement based on the sample selection of the EEG data 459
  42. Part II-B: Person’s eco-sphere exposures
  43. Introduction 475
  44. The spread of SARS-CoV-2 in Russia and the evolution of the properties of the pathogen 479
  45. Agent-based modeling of epidemic spread via kinetic Monte Carlo method 491
  46. Control of SARS-nCoV outbreaks in China 2020 513
  47. Part II-B.2: Civilization
  48. Introduction 531
  49. Pesticide exposure: Toward holistic environmental modeling 535
  50. Part II-C: Person’s sociosphere exposures
  51. Introduction 559
  52. Evolution of the health system in Shanghai, China, 2016–2020 563
  53. Part III: Technologies
  54. Introduction 577
  55. Design-process automation using functional process blocks 581
  56. Slow/fast dynamic models with applications to engineering problems 601
  57. Part III-B: Information sciences
  58. Introduction 613
  59. Numerical modeling of medical ultrasound using the grid-characteristic method 617
  60. The direct and the inverse magnetic encephalography problem 635
  61. Part III-C: Data-analytic sciences
  62. Introduction 653
  63. Assessing the bioequivalence of two different drugs with the same active ingredient 655
  64. Estimation of adjusted relative risks in log-binomial regression using the Bekhit–Schöpe–Wagenpfeil algorithm 665
  65. Part IV: Clinical medicine
  66. Introduction 679
  67. Finding optimal two-stage combined treatment protocols for a blood cancer model 681
  68. Unraveling the mysteries: Mathematical perspectives on traditional Chinese medicine meridians 697
  69. Epilogue 721
  70. Index 723
https://doi.org/10.1515/9783111341996-003

Chapters in this book

  1. Frontmatter I
  2. Prologue I VII
  3. Prologue II XI
  4. Prologue III XIII
  5. Preface XVII
  6. Overview XIX
  7. Contents XXXIII
  8. Part I: Theories
  9. Part I-A: Overarching theory
  10. Introduction 1
  11. Universal axioms in classical Chinese philosophy 5
  12. Category theory for structural characterization 15
  13. Axiomatic bipolar dynamics and their control 45
  14. Part I-B: Systems theories
  15. Introduction 75
  16. Stochastic formalization of agent-oriented systems 79
  17. Simplification of high-dimensional multitempo dynamic models 109
  18. Ideas of symmetry as a biophysical basis of system biomedicine 123
  19. Disorder of multiscale control 149
  20. Part II: Person’s life-sphere
  21. Part II-A: Person’s biosphere
  22. Introduction 185
  23. Mutations as activators of biological evolutionary processes at population levels 189
  24. Immunometabolism of T-cells in COVID-19 209
  25. Part II-A.2: Body’s vital functions
  26. Introduction 245
  27. Structural modeling of vascular networks 249
  28. Mathematical modeling of AI application for the diagnosis of blood flow disorders 283
  29. Modeling of glucose and insulin regulation within the framework of a self-consistent model of the cardiovascular system 303
  30. Hemodynamics in residual myocardial ischemia 319
  31. The quasi-one-dimensional model of the lymph flow in the human lymphatic system 335
  32. An integrate-and-fire mechanism for modeling rhythmicity in the neuroendocrine system 365
  33. Kinetic network modeling of the neuroendocrine hypothalamic-pituitary-adrenal axis dynamics with particular attention on the role of alcohol as a digestif 377
  34. Inflammation and immune response in atherosclerosis 393
  35. Part II-A.3: Body’s motor functions
  36. Introduction 423
  37. A magnetic resonance spectroscopy approach to quantitatively measure GABA and phosphorus level changes in the primary motor cortex elicited by transcranial direct current stimulation 427
  38. Part II-A.4: Body’s operational functions
  39. Introduction 441
  40. The fermionic mind hypothesis–a category-theoretic verification of consciousness 445
  41. Cross-task cognitive workload measurement based on the sample selection of the EEG data 459
  42. Part II-B: Person’s eco-sphere exposures
  43. Introduction 475
  44. The spread of SARS-CoV-2 in Russia and the evolution of the properties of the pathogen 479
  45. Agent-based modeling of epidemic spread via kinetic Monte Carlo method 491
  46. Control of SARS-nCoV outbreaks in China 2020 513
  47. Part II-B.2: Civilization
  48. Introduction 531
  49. Pesticide exposure: Toward holistic environmental modeling 535
  50. Part II-C: Person’s sociosphere exposures
  51. Introduction 559
  52. Evolution of the health system in Shanghai, China, 2016–2020 563
  53. Part III: Technologies
  54. Introduction 577
  55. Design-process automation using functional process blocks 581
  56. Slow/fast dynamic models with applications to engineering problems 601
  57. Part III-B: Information sciences
  58. Introduction 613
  59. Numerical modeling of medical ultrasound using the grid-characteristic method 617
  60. The direct and the inverse magnetic encephalography problem 635
  61. Part III-C: Data-analytic sciences
  62. Introduction 653
  63. Assessing the bioequivalence of two different drugs with the same active ingredient 655
  64. Estimation of adjusted relative risks in log-binomial regression using the Bekhit–Schöpe–Wagenpfeil algorithm 665
  65. Part IV: Clinical medicine
  66. Introduction 679
  67. Finding optimal two-stage combined treatment protocols for a blood cancer model 681
  68. Unraveling the mysteries: Mathematical perspectives on traditional Chinese medicine meridians 697
  69. Epilogue 721
  70. Index 723
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