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Seven Sketches in Compositionality: An Invitation to Applied Category Theory

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Seven Sketches in Compositionality: An Invitation to Applied Category Theory

Category theory is a relatively new branch of mathematics that has transformed much of pure math research. The technical advance is that category theory provides a framework in which to organize formal systems and by which to translate between them, allowing one to transfer knowledge from one field to another. But this same organizational framework also has many compelling examples outside of pure math. In this text, we will give seven sketches on real-world applications of category theory.

Applied MathematicsMathematics
Author
Brendan Fong & David I. Spivak
Publisher
Massachusetts Institute of Technology
Source
LibreTexts
License
CC BY-NC-SA
Language
EN
Category
English
Read online Download PDF (open licence)
Contents· 55 sections
  1. 01: Generative Effects - Orders and Adjunctions
  2. 1.01: What is Order
  3. 1.02: Meets and Joins
  4. 1.03: Galois Connections
  5. 1.04: Summary and further reading
  6. 1.05: More than the sum of their parts
  7. 02: Resources - Monoidal Preorders and Enrichment
  8. 2.01: Getting from a to b
  9. 2.02: Symmetric Monoidal Preorders
  10. 2.03: Enrichment
  11. 2.04: Constructions on V-categories
  12. 2.05: Computing presented V-categories with matrix mult
  13. 2.06: Summary and further reading
  14. 03: Databases- Categories functors and (co)limits
  15. 3.01: What is a Database
  16. 3.02: Categories
  17. 3.03: Functors natural transformations and databases
  18. 3.04: Adjunctions and Data Migration
  19. 3.05: Bonus- An introduction to limits and colimits
  20. 3.06: Summary and further reading
  21. 04: Co-design - Profunctors and Monoidal Categories
  22. 4.01: Can we build it
  23. 4.02: Enriched Profunctors
  24. 4.03: Categories of Profunctors
  25. 4.04: Categorification
  26. 4.05: Profunctors form a Compact Closed Category
  27. 4.06: Summary and further reading
  28. 05: Signal flow graphs- Props presentations and proofs
  29. 5.01: Comparing systems as interacting signal processors
  30. 5.02: Props and Presentations
  31. 5.03: Simplified Signal Flow Graphs
  32. 5.04: Graphical linear algebra
  33. 5.05: Summary and further reading
  34. 06: Circuits - Hypergraph Categories and Operads
  35. 6.01: The ubiquity of network languages
  36. 6.02: Colimits and Connection
  37. 6.03: Hyper Graph Categories
  38. 6.04: Decorated cospans
  39. 6.05: Operads and their algebras
  40. 6.06: Summary and further reading
  41. 07: Logic of Behavior - Sheaves Toposes Languages
  42. 7.01: How can we prove our machine is safe
  43. 7.02: The Category Set as an Exemplar Topos
  44. 7.03: Sheaves
  45. 7.04: Toposes
  46. 7.05: A topos of behavior types
  47. 7.06: Summary and further reading
  48. 08: Exercise solutions
  49. 8.01: Solutions for Chapter 1
  50. 8.02: Solutions for Chapter 2
  51. 8.03: Solutions for Chapter 3
  52. 8.04: Solutions for Chapter 4
  53. 8.05: Solutions for Chapter 5
  54. 8.06: Solutions for Chapter6
  55. 8.07: Solutions for Chapter 7