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