Safe Haskell	None
Language	GHC2024

Proarrow.Category.Monoidal.Hypergraph

Contents

Orphan instances

Description

Hypergraph categories: compact closed categories where every object carries a Frobenius structure (a compatible Monoid and Comonoid), giving n-to-m spiders, cups and caps. This is the setting for string diagrams with arbitrary fan-in/fan-out such as Proarrow.Category.Instance.ZX.

Synopsis

class (CommutativeMonoid a, CocommutativeComonoid a) => Frobenius (a :: k)
spider :: forall {k} (n :: Nat) (m :: Nat) (a :: k). (Frobenius a, SNatI n, SNatI m) => NFold n a ~> NFold m a
spiderS :: forall {k} (n :: Nat) (m :: Nat) (a :: k). (Frobenius a, SNatI n, SNatI m) => NFoldS n a ~> NFoldS m a
cup :: forall {k} (a :: k). Frobenius a => (Unit :: k) ~> (a ** a)
cupS :: forall {a1} (a2 :: a1). Frobenius a2 => ('[] :: [a1]) ~> '[a2, a2]
cap :: forall {k} (a :: k). Frobenius a => (a ** a) ~> (Unit :: k)
capS :: forall {k} (a :: k). Frobenius a => '[a, a] ~> ('[] :: [k])
class (Supplies Frobenius k, CompactClosed k) => Hypergraph k
dualHG :: forall {k} (a :: k) (b :: k). Hypergraph k => (a ~> b) -> b ~> a
linDistHG :: forall {k} (a :: k) (b :: k) (c :: k). (Hypergraph k, Ob a, Ob b) => ((a ** b) ~> c) -> a ~> (b ** c)
linDistInvHG :: forall {k} (a :: k) (b :: k) (c :: k). (Hypergraph k, Ob b, Ob c) => (a ~> (b ** c)) -> (a ** b) ~> c
traceHG :: forall {k} (u :: k) (x :: k) (y :: k). (Hypergraph k, Ob x, Ob y, Ob u) => ((u ** x) ~> (u ** y)) -> x ~> y
type ExpHG (a :: k) (b :: k) = a ** b
curryHG :: forall {k} (a :: k) (b :: k) (c :: k). (Hypergraph k, Ob a, Ob b) => ((a ** b) ~> c) -> a ~> ExpHG b c
applyHG :: forall {k} (b :: k) (c :: k). (Hypergraph k, Ob b, Ob c) => (ExpHG b c ** b) ~> c
type FrobeniusStructures = '[Monoidal, SymMonoidal, Supplies Monoid, Supplies Comonoid]

Documentation

class (CommutativeMonoid a, CocommutativeComonoid a) => Frobenius (a :: k) Source Github #

A special commutative Frobenius algebra: a commutative monoid and cocommutative comonoid satisfying speciality (mappend . comult = id) and the Frobenius law. A Hypergraph category supplies this structure at every object, and with it the spider from n-fold a to m-fold a is the unique connected map (commutativity/cocommutativity make fanIn/fanOut independent of wiring order). The bare notion of a Frobenius monoid needs neither (co)commutativity, but the library only ever uses the special commutative one.

Instances

Instances details

KnownNat a => Frobenius (a :: Nat) Source Github #
Instance details Defined in Proarrow.Category.Instance.ZX
SNatI a => Frobenius ('FR a :: FINREL) Source Github #
Instance details Defined in Proarrow.Category.Instance.FinRel
Ob as => Frobenius ('D as :: DOT) Source Github #	The points are spiders: on each wire, merging and copying are the special commutative Frobenius structure.
Instance details Defined in Proarrow.Tools.Diagrams.Dot
Ob as => Frobenius ('S as :: SVG) Source Github #
Instance details Defined in Proarrow.Tools.Diagrams.Svg
(HasPushouts k, HasCoproducts k, Ob a) => Frobenius ('CS a :: COSPAN k) Source Github #
Instance details Defined in Proarrow.Category.Instance.Cospan
(Num a, IsNat n) => Frobenius ('M n :: MatK a) Source Github #
Instance details Defined in Proarrow.Category.Instance.Mat
(HasPullbacks k, HasProducts k, Ob a) => Frobenius ('SP a :: SPAN k) Source Github #
Instance details Defined in Proarrow.Category.Instance.Span
(CommutativeMonoid a, CocommutativeComonoid a) => Frobenius (a :: FREE cs p) Source Github #	In the free category the supply generators (see `Supplies Monoid`/`Supplies Comonoid` in Proarrow.Monoid) are compatible by fiat, so monoid + comonoid is already `Frobenius`. With both supplies in `cs`, `Supplies Frobenius` and `Hypergraph` are derived, with no structure of their own. Superclasses are taken directly as the context to keep dictionary construction acyclic. Bundling them into an `All`-style constraint here builds a dictionary that references itself through the quantified `Supplies` constraint, looping at runtime.
Instance details Defined in Proarrow.Category.Monoidal.Hypergraph

spider :: forall {k} (n :: Nat) (m :: Nat) (a :: k). (Frobenius a, SNatI n, SNatI m) => NFold n a ~> NFold m a Source Github #

spiderS :: forall {k} (n :: Nat) (m :: Nat) (a :: k). (Frobenius a, SNatI n, SNatI m) => NFoldS n a ~> NFoldS m a Source Github #

cup :: forall {k} (a :: k). Frobenius a => (Unit :: k) ~> (a ** a) Source Github #

cupS :: forall {a1} (a2 :: a1). Frobenius a2 => ('[] :: [a1]) ~> '[a2, a2] Source Github #

cap :: forall {k} (a :: k). Frobenius a => (a ** a) ~> (Unit :: k) Source Github #

capS :: forall {k} (a :: k). Frobenius a => '[a, a] ~> ('[] :: [k]) Source Github #

class (Supplies Frobenius k, CompactClosed k) => Hypergraph k Source Github #

A hypergraph category has a special frobenius algebra for every object, and the frobenius algebra of any tensor product X ⊗ Y is induced in the canonical way from those of X and Y.

Instances

Instances details

Hypergraph Nat Source Github #
Instance details Defined in Proarrow.Category.Instance.ZX
Hypergraph FINREL Source Github #
Instance details Defined in Proarrow.Category.Instance.FinRel
Hypergraph DOT Source Github #	The points make every object a special commutative Frobenius object, so a diagram's wires can be bent: each object is its own dual, with cups and caps drawn as a copy or merge point next to a unit or counit point.
Instance details Defined in Proarrow.Tools.Diagrams.Dot
Hypergraph SVG Source Github #
Instance details Defined in Proarrow.Tools.Diagrams.Svg
(HasPushouts k, HasCoproducts k) => Hypergraph (COSPAN k) Source Github #
Instance details Defined in Proarrow.Category.Instance.Cospan
Num a => Hypergraph (MatK a) Source Github #
Instance details Defined in Proarrow.Category.Instance.Mat
(HasPullbacks k, HasProducts k) => Hypergraph (SPAN k) Source Github #
Instance details Defined in Proarrow.Category.Instance.Span
(Supplies Frobenius (FREE cs p), CompactClosed (FREE cs p)) => Hypergraph (FREE cs p) Source Github #
Instance details Defined in Proarrow.Category.Monoidal.Hypergraph

dualHG :: forall {k} (a :: k) (b :: k). Hypergraph k => (a ~> b) -> b ~> a Source Github #

A hypergraph category is self-dual compact closed.

linDistHG :: forall {k} (a :: k) (b :: k) (c :: k). (Hypergraph k, Ob a, Ob b) => ((a ** b) ~> c) -> a ~> (b ** c) Source Github #

linDistInvHG :: forall {k} (a :: k) (b :: k) (c :: k). (Hypergraph k, Ob b, Ob c) => (a ~> (b ** c)) -> (a ** b) ~> c Source Github #

traceHG :: forall {k} (u :: k) (x :: k) (y :: k). (Hypergraph k, Ob x, Ob y, Ob u) => ((u ** x) ~> (u ** y)) -> x ~> y Source Github #

A hypergraph category has a trace.

type ExpHG (a :: k) (b :: k) = a ** b Source Github #

A hypergraph category is monoidal closed.

curryHG :: forall {k} (a :: k) (b :: k) (c :: k). (Hypergraph k, Ob a, Ob b) => ((a ** b) ~> c) -> a ~> ExpHG b c Source Github #

applyHG :: forall {k} (b :: k) (c :: k). (Hypergraph k, Ob b, Ob c) => (ExpHG b c ** b) ~> c Source Github #

type FrobeniusStructures = '[Monoidal, SymMonoidal, Supplies Monoid, Supplies Comonoid] Source Github #

The structures the laws of a category supplying special commutative Frobenius algebras are stated for: the monoids and comonoids, together.

Orphan instances

Laws FrobeniusStructures Source Github #	The supplied monoids and comonoids are special and satisfy the Frobenius law. Their monoid and comonoid laws, and their commutativity, are separate instances, in Proarrow.Monoid.
Instance details Methods laws :: [Law FrobeniusStructures] Source Github #