This course is based on the textbook:
Fong, B. and D. I. Spivak. An Invitation to Applied Category Theory: Seven Sketches in Compositionality. Cambridge University Press, 2019. ISBN: 9781108482295.
An online version is freely available on Cornell University’s arXiv.org e-Print archive site as well as from OCW as a single file: An Invitation to Applied Category Theory: Seven Sketches in Compositionality (PDF - 2.6MB).
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Session 1 Chapter 1: Generative Effects: Orders and Adjunctions Part 1 by Dr. David I. Spivak |
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Session 2 Chapter 1: Generative Effects: Orders and Adjunctions Part 2 by Dr. Brendan Fong |
Internal links in the chapter file below are non-functional. For working links, open the file for the full textbook: An Invitation to Applied Category Theory: Seven Sketches in Compositionality (PDF - 2.6MB).
Chapter 1: Generative Effects: Orders and Adjunctions (PDF)
1.1 More than the sum of their parts
1.1.1 A first look at generative effects
1.1.2 Ordering systems
1.2 What is order?
1.2.1 Review of sets, relations, and functions
1.2.2 Preorders
1.2.3 Monotone maps
1.3 Meets and joins
1.3.1 Definition and basic examples
1.3.2 Back to observations and generative effects
1.4 Galois connections
1.4.1 Definition and examples of Galois connections
1.4.2 Back to partitions
1.4.3 Basic theory of Galois connections
1.4.4 Closure operators
1.4.5 Level shifting
1.5 Summary and further reading
Problem set 1 (PDF) due at the beginning of Session 6.
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Session 3 Chapter 2: Resources: Monoidal Preorders and Enrichment Part 1 by Dr. David I. Spivak |
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Session 4 Chapter 2: Resources: Monoidal Preorders and Enrichment Part 2 by Dr. Brendan Fong |
Internal links in the chapter file below are non-functional. For working links, open the file for the full textbook: An Invitation to Applied Category Theory: Seven Sketches in Compositionality (PDF - 2.6MB).
Chapter 2: Resources: Monoidal Preorders and Enrichment (PDF)
2.1 Getting from \(a\) to \(b\)
2.2 Symmetric monoidal preorders
2.2.1 Definition and first examples
2.2.2 Introducing wiring diagrams
2.2.3 Applied examples
2.2.4 Abstract examples
2.2.5 Monoidal monotone maps
2.3 Enrichment
2.3.1 \(\mathcal{V}\)-categories
2.3.2 Preorders as Bool-categories
2.3.3 Lawvere metric spaces
2.3.4 \(\mathcal{V}\)-variations on preorders and metric spaces
2.4 Constructions on \(\mathcal{V}\)-categories
2.4.1 Changing the base of enrichment
2.4.2 Enriched functors
2.4.3 Product \(\mathcal{V}\)-categories
2.5 Computing presented \(\mathcal{V}\)-categories with matrix multiplication
2.5.1 Monoidal closed preorders
2.5.2 Quantales
2.5.3 Matrix multiplication in a quantale
2.6 Summary and further reading
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Session 5 Chapter 3: Databases: Categories, Functors, and (Co)Limits Part 1 by Dr. David I. Spivak |
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Session 6 Chapter 3: Databases: Categories, Functors, and (Co)Limits Part 2 by Dr. Brendan Fong |
Internal links in the chapter file below are non-functional. For working links, open the file for the full textbook: An Invitation to Applied Category Theory: Seven Sketches in Compositionality (PDF - 2.6MB).
Chapter 3: Databases: Categories, Functors, and (Co)Limits (PDF)
3.1 What is a database?
3.2 Categories
3.2.1 Free categories
3.2.2 Presenting categories via path equations
3.2.3 Preorders and free categories: two ends of a spectrum
3.2.4 Important categories in mathematics
3.2.5 Isomorphisms in a category
3.3 Functors, natural transformations, and databases
3.3.1 Sets and functions as databases
3.3.2 Functors
3.3.3 Database instances as Set-valued functors
3.3.4 Natural transformations
3.3.5 The category of instances on a schema
3.4 Adjunctions and data migration
3.4.1 Pulling back data along a functor
3.4.2 Adjunctions
3.4.3 Left and right pushforward functors, \(\Sigma\) and \(\Pi\)
3.4.4 Single set summaries of databases
3.5 Bonus: An introduction to limits and colimits
3.5.1 Terminal objects and products
3.5.2 Limits
3.5.3 Finite limits in Set
3.5.4 A brief note on colimits
3.6 Summary and further reading
Problem set 2 (PDF) due at the beginning of Session 10.
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Session 7 Chapter 4: Co-design: Profunctors and Monoidal Categories Part 1 by Dr. David I. Spivak |
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Session 8 Chapter 4: Co-design: Profunctors and Monoidal Categories Part 2 by Dr. Brendan Fong |
Internal links in the chapter file below are non-functional. For working links, open the file for the full textbook: An Invitation to Applied Category Theory: Seven Sketches in Compositionality (PDF - 2.6MB).
Chapter 4: Co-design: Profunctors and Monoidal Categories (PDF)
4.1 Can we build it?
4.2 Enriched profunctors
4.2.1 Feasibility relationships as Bool-profunctors
4.2.2 \(\mathcal{V}\)-profunctors
4.2.3 Back to co-design diagrams
4.3 Categories of profunctors
4.3.1 Composing profunctors
4.3.2 The categories \(\mathcal{V}\)-Prof and Feas
4.3.3 Fun profunctor facts: companions, conjoints, collages
4.4 Categorification
4.4.1 The basic idea of categorification
4.4.2 A reflection on wiring diagrams
4.4.3 Monoidal categories
4.4.4 Categories enriched in a symmetric monoidal category
4.5 Profunctors form a compact closed category
4.5.1 Compact closed categories
4.5.2 Feas as a compact closed category
4.6 Summary and further reading
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Session 9 Chapter 5: Signal Flow Graphs: Props, Presentations, and Proofs Part 1 by Dr. David I. Spivak |
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Session 10 Chapter 5: Signal Flow Graphs: Props, Presentations, and Proofs Part 2 by Dr. Brendan Fong |
Internal links in the chapter file below are non-functional. For working links, open the file for the full textbook: An Invitation to Applied Category Theory: Seven Sketches in Compositionality (PDF - 2.6MB).
Chapter 5: Signal Flow Graphs: Props, Presentations, and Proofs (PDF)
5.1 Comparing systems as interacting signal processors
5.2 Props and presentations
5.2.1 Props: definition and first examples
5.2.2 The prop of port graphs
5.2.3 Free constructions and universal properties
5.2.4 The free prop on a signature
5.2.5 Props via presentations
5.3 Simplified signal flow graphs
5.3.1 Rigs
5.3.2 The iconography of signal flow graphs
5.3.3 The prop of matrices over a rig
5.3.4 Turning signal flow graphs into matrices
5.3.5 The idea of functorial semantics
5.4 Graphical linear algebra
5.4.1 A presentation of Mat(\(R\))
5.4.2 Aside: monoid objects in a monoidal category
5.4.3 Signal flow graphs: feedback and more
5.5 Summary and further reading
Problem set 3 (PDF) due at the beginning of Session 14.
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Session 11 Chapter 6: Circuits: Hypergraph Categories and Operads Part 1 by Dr. David I. Spivak |
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Session 12 Chapter 6: Circuits: Hypergraph Categories and Operads Part 2 by Dr. Brendan Fong |
Internal links in the chapter file below are non-functional. For working links, open the file for the full textbook: An Invitation to Applied Category Theory: Seven Sketches in Compositionality (PDF - 2.6MB).
Chapter 6: Circuits: Hypergraph Categories and Operads (PDF)
6.1 The ubiquity of network languages
6.2 Colimits and connection
6.2.1 Initial objects
6.2.2 Coproducts
6.2.3 Pushouts
6.2.4 Finite colimits
6.2.5 Cospans
6.3 Hypergraph categories
6.3.1 Frobenius monoids
6.3.2 Wiring diagrams for hypergraph categories
6.3.3 Definition of hypergraph category
6.4 Decorated cospans
6.4.1 Symmetric monoidal functors
6.4.2 Decorated cospans
6.4.3 Electric circuits
6.5 Operads and their algebras
6.5.1 Operads design wiring diagrams
6.5.2 Operads from symmetric monoidal categories
6.5.3 The operad for hypergraph props
6.6 Summary and further reading
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Session 13 Chapter 7: Logic of Behavior: Sheaves, Toposes, Languages Part 1 by Dr. David I. Spivak |
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Session 14 Chapter 7: Logic of Behavior: Sheaves, Toposes, Languages Part 2 by Dr. Brendan Fong |
Internal links in the chapter file below are non-functional. For working links, open the file for the full textbook: An Invitation to Applied Category Theory: Seven Sketches in Compositionality (PDF - 2.6MB).
Chapter 7: Logic of Behavior: Sheaves, Toposes, Languages (PDF)
7.1 How can we prove our machine is safe?
7.2 The category Set as an exemplar topos
7.2.1 Set-like properties enjoyed by any topos
7.2.2 The subobject classifier
7.2.3 Logic in the topos Set
7.3 Sheaves
7.3.1 Presheaves
7.3.2 Topological spaces
7.3.3 Sheaves on topological spaces
7.4 Toposes
7.4.1 The subobject classifier \(\Omega\) in a sheaf topos
7.4.2 Logic in a sheaf topos
7.4.3 Predicates
7.4.4 Quantification
7.4.5 Modalities
7.4.6 Type theories and semantics
7.5 A topos of behavior types
7.5.1 The interval domain
7.5.2 Sheaves on \(\mathbb{IR}\)
7.5.3 Safety proofs in temporal logic
7.6 Summary and further reading
