abc

The abstract base classes for categories.

These mirror the concrete hierarchy of discopy modules: each class adds the characteristic generator of its categorical structure as an abc.abstractmethod(), e.g. BraidedCategory is a MonoidalCategory with an abstract BraidedCategory.braid().

Software dependencies between modules go top-to-bottom, left-to-right and forgetful functors between categories go the other way.

Each class also declares its discopy.testable.axiom() equations, which every free category inherits along with the structure they axiomatise: Category states the unitality and associativity of composition, the typing of its identities and composites, and the involution and contravariance of its dagger; a ColouredMonoid inherits them as the unitality and associativity of its product, its composition.

Summary

Category	A category is a class with two class variables ob, ar, two attributes dom, cod and two methods id, then.
ColouredMonoid	A coloured monoid is a category whose sequential composition then is given by a monoidal tensor, with the objects C0 (its colours) as the boundaries of its morphisms.
Monoid	A monoid is a coloured monoid with a single, trivial colour.
Nat	Nat is the free monoid on one generator, i.e. the natural numbers with addition as tensor.
MonoidalCategory	A monoidal category is a Category with a method tensor for both its objects and its morphisms.
PRO	A PRO is a MonoidalCategory whose objects are the natural numbers Nat, i.e. the free monoidal category on one generator.
TracedCategory	A traced category is a MonoidalCategory with a method trace for the partial trace of a morphism over some objects.
ResiduatedMonoid	A monoid is residuated when it comes with methods over and under with syntactic sugar << and >>.
BiclosedCategory	A biclosed category is a MonoidalCategory with methods ev and curry for the evaluation and currying of morphisms.
Pregroup	A pregroup is a residuated monoid where the left and right exponentials are given by tensoring with the chosen left and right duals for each object.
RigidCategory	A rigid category is a BiclosedCategory with a Pregroup as object type and methods for cups and caps.
PivotalCategory	A pivotal category is a RigidCategory where the left and right adjoints coincide, hence it is also a TracedCategory.
BraidedCategory	A braided category is a MonoidalCategory with a method braid for the natural isomorphism x @ y -> y @ x.
PROB	A PROB is a BraidedCategory whose objects are the natural numbers Nat, i.e. the free braided category on one generator.
SymmetricCategory	A symmetric category is a BraidedCategory where the braid is its own inverse called swap for the symmetry x @ y -> y @ x.
PROP	A PROP is a SymmetricCategory whose objects are the natural numbers Nat, i.e. the free symmetric category on one generator.
MarkovCategory	A Markov category is a SymmetricCategory with methods copy and merge for the supply of commutative comonoids.
ClosedCategory	A closed category is a symmetric BiclosedCategory.
FeedbackCategory	A feedback category is a MarkovCategory with a delay endofunctor and a feedback operator.
BalancedCategory	A balanced category is a BraidedCategory and a TracedCategory with a method twist for the natural automorphism x -> x.
RibbonCategory	A ribbon category is a PivotalCategory which is also a BalancedCategory, i.e. where diagrams can draw knots and links.
CompactCategory	A compact category is a RibbonCategory which is also a SymmetricCategory, i.e. with cups, caps and swaps and where the twist is the identity.
HypergraphCategory	A hypergraph category is a symmetric category with a supply of spiders, i.e. special commutative Frobenius algebras on each objects.
NamedGeneric	A NamedGeneric is a Generic where the type parameter has a name.