Cartesian Lenses Between Polynomial Functors

A Lens P Q between polynomial functors carries a forward map on positions (toFunA : P.A → Q.A) and, for each position a : P.A, a backward map on directions (toFunB a : Q.B (toFunA a) → P.B a). We say the lens is cartesian when every backward map toFunB a is a bijection.

In the language of the fibration Poly → Set that sends a polynomial functor to its set of positions, cartesian lenses are exactly the cartesian morphisms: the type-square underlying each fiber is a pullback. A cartesian lens is a fiberwise isomorphism over an arbitrary forward map on positions.

The two classes in the orthogonal factorization system on Poly are captured separately:

• a vertical lens has bijective toFunA (it changes position coordinates without losing or adding positions); and
• a cartesian lens has each toFunB a a bijection (each direction fiber is rigid over an arbitrary position map).

The canonical vertical leg in the factorization is identity on positions, but identity is a chosen representative of the more general bijective-on-positions class. See PFunctor.Lens.IsVertical in Lens/Factorization.lean.

A PFunctor.Lens.Equiv (a categorical isomorphism in the lens category) is the intersection of the two: toFunA is a bijection AND every toFunB a is a bijection. Thus every Equiv is cartesian, but a cartesian lens whose position map is not a bijection (e.g. Lens.inl : Lens P (P + Q)) is not an Equiv.

The downstream consumer of this file is LawfulSubSpec in VCVio/OracleComp/Coercions/SubSpec.lean: a LawfulSubSpec is by definition the predicate "the underlying Lens spec.toPFunctor superSpec.toPFunctor is cartesian", which is the exact condition needed to pull the uniform distribution on superSpec.Range back through a sub-spec embedding without distortion.

def PFunctor.Lens.IsCartesian {P : PFunctor.{uA₁, uB₁}} {Q : PFunctor.{uA₂, uB₂}} (l : P.Lens Q) : Prop

A lens l : Lens P Q is cartesian when every backward fiber l.toFunB a : Q.B (l.toFunA a) → P.B a is a bijection. Equivalently, l is a cartesian morphism in the fibration Poly → Set of polynomial functors over their position sets.

@[simp]

theorem PFunctor.Lens.IsCartesian.id (P : PFunctor.{uA, uB}) : (Lens.id P).IsCartesian

theorem PFunctor.Lens.IsCartesian.comp {P : PFunctor.{uA₁, uB₁}} {Q : PFunctor.{uA₂, uB₂}} {R : PFunctor.{uA₃, uB₃}} {l₁ : Q.Lens R} {l₂ : P.Lens Q} (h₁ : l₁.IsCartesian) (h₂ : l₂.IsCartesian) : (l₁ ∘ₗ l₂).IsCartesian

Cartesianness of canonical lens constructors

Most "embedding-shaped" lenses produced by the lens algebra are cartesian. We collect the witnesses here so that downstream classes (notably LawfulSubSpec) can be discharged uniformly.

theorem PFunctor.Lens.IsCartesian.inl {P : PFunctor.{uA₁, uB}} {Q : PFunctor.{uA₂, uB}} : Lens.inl.IsCartesian

theorem PFunctor.Lens.IsCartesian.inr {P : PFunctor.{uA₁, uB}} {Q : PFunctor.{uA₂, uB}} : Lens.inr.IsCartesian

theorem PFunctor.Lens.IsCartesian.sumMap {P : PFunctor.{uA₁, uB₁}} {Q : PFunctor.{uA₂, uB₁}} {R : PFunctor.{uA₃, uB₃}} {W : PFunctor.{uA₄, uB₃}} {l₁ : P.Lens R} {l₂ : Q.Lens W} (h₁ : l₁.IsCartesian) (h₂ : l₂.IsCartesian) : (l₁ ⊎ₗ l₂).IsCartesian

theorem PFunctor.Lens.IsCartesian.sumPair {P : PFunctor.{uA₁, uB}} {Q : PFunctor.{uA₂, uB}} {R : PFunctor.{uA₃, uB₃}} {l₁ : P.Lens R} {l₂ : Q.Lens R} (h₁ : l₁.IsCartesian) (h₂ : l₂.IsCartesian) : [l₁,l₂]ₗ.IsCartesian

theorem PFunctor.Lens.IsCartesian.sigmaMap {I : Type v} {F : I → PFunctor.{uA₁, uB₁}} {G : I → PFunctor.{uA₂, uB₂}} {l : (i : I) → (F i).Lens (G i)} (hl : ∀ (i : I), (l i).IsCartesian) : (Lens.sigmaMap l).IsCartesian

theorem PFunctor.Lens.IsCartesian.sigmaExists {I : Type v} {F : I → PFunctor.{uA₁, uB₁}} {R : PFunctor.{uA₂, uB₂}} {l : (i : I) → (F i).Lens R} (hl : ∀ (i : I), (l i).IsCartesian) : (Lens.sigmaExists l).IsCartesian

Sigma injection

The natural inclusion of the j-th summand into a sigma: the unique lens whose forward action picks the j-th tag and whose backward action is the identity on directions. This is the Lens whose corresponding MonadLift underlies OracleSpec.subSpec_sigma.

def PFunctor.Lens.sigmaInj {I : Type v} {F : I → PFunctor.{uA, uB}} (j : I) : (F j).Lens (sigma F)

Lens injection Lens (F j) (sigma F) for a fixed index j : I.

theorem PFunctor.Lens.IsCartesian.sigmaInj {I : Type v} {F : I → PFunctor.{uA, uB}} (j : I) : (Lens.sigmaInj j).IsCartesian