Seven Sketches in Compositionality: An Invitation to Applied Category Theory

Generative effects: Orders and adjunctions
More than the sum of their parts
A first look at generative effects
Ordering systems
What is order?
Review of sets, relations, and functions
Preorders
Monotone maps
Meets and joins
Definition and basic examples
Back to observations and generative effects
Galois connections
Definition and examples of Galois connections
Back to partitions
Basic theory of Galois connections
Closure operators
Level shifting
Summary and further reading
Resources: monoidal preorders and enrichment
Getting from a to b
Symmetric monoidal preorders
Definition and first examples
Introducing wiring diagrams
Applied examples
Abstract examples
Monoidal monotone maps
Enrichment
V-categories
Preorders as Bool-categories
Lawvere metric spaces
V-variations on preorders and metric spaces
Constructions on V-categories
Changing the base of enrichment
Enriched functors
Product V-categories
Computing presented V-categories with matrix mult.
Monoidal closed preorders
Quantales
Matrix multiplication in a quantale
Summary and further reading
Databases: Categories, functors, and (co)limits
What is a database?
Categories
Free categories
Presenting categories via path equations
Preorders and free categories: two ends of a spectrum
Important categories in mathematics
Isomorphisms in a category
Functors, natural transformations, and databases
Sets and functions as databases
Functors
Database instances as Set-valued functors
Natural transformations
The category of instances on a schema
Adjunctions and data migration
Pulling back data along a functor
Adjunctions
Left and right pushforward functors, and
Single set summaries of databases
Bonus: An introduction to limits and colimits
Terminal objects and products
Limits
Finite limits in Set
A brief note on colimits
Summary and further reading
Co-design: profunctors and monoidal categories
Can we build it?
Enriched profunctors
Feasibility relationships as Bool-profunctors
V-profunctors
Back to co-design diagrams
Categories of profunctors
Composing profunctors
The categories V-Prof and Feas
Fun profunctor facts: companions, conjoints, collages
Categorification
The basic idea of categorification
A reflection on wiring diagrams
Monoidal categories
Categories enriched in a symmetric monoidal category
Profunctors form a compact closed category
Compact closed categories
Feas as a compact closed category
Summary and further reading
Signal flow graphs: Props, presentations, & proofs
Comparing systems as interacting signal processors
Props and presentations
Props: definition and first examples
The prop of port graphs
Free constructions and universal properties
The free prop on a signature
Props via presentations
Simplified signal flow graphs
Rigs
The iconography of signal flow graphs
The prop of matrices over a rig
Turning signal flow graphs into matrices
The idea of functorial semantics
Graphical linear algebra
A presentation of Mat(R)
Aside: monoid objects in a monoidal category
Signal flow graphs: feedback and more
Summary and further reading
Circuits: hypergraph categories and operads
The ubiquity of network languages
Colimits and connection
Initial objects
Coproducts
Pushouts
Finite colimits
Cospans
Hypergraph categories
Frobenius monoids
Wiring diagrams for hypergraph categories
Definition of hypergraph category
Decorated cospans
Symmetric monoidal functors
Decorated cospans
Electric circuits
Operads and their algebras
Operads design wiring diagrams
Operads from symmetric monoidal categories
The operad for hypergraph props
Summary and further reading
Logic of behavior: Sheaves, toposes, languages
How can we prove our machine is safe?
The category Set as an exemplar topos
Set-like properties enjoyed by any topos
The subobject classifier
Logic in the topos Set
Sheaves
Presheaves
Topological spaces
Sheaves on topological spaces
Toposes
The subobject classifier in a sheaf topos
Logic in a sheaf topos
Predicates
Quantification
Modalities
Type theories and semantics
A topos of behavior types
The interval domain
Sheaves on I R
Safety proofs in temporal logic
Summary and further reading
Exercise solutions
Solutions for Chapter 1
Solutions for Chapter 2
Solutions for Chapter 3
Solutions for Chapter 4
Solutions for Chapter 5
Solutions for Chapter 6
Solutions for Chapter 7
Bibliography
Index