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feat: Additional axioms to TensorSpecies

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jstoobysmith 2024-10-24 14:44:02 +00:00
parent 223a1fcf6c
commit 28e0e4d610
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24
HepLean/Tensors/Tree/Basic.lean
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@ -55,6 +55,30 @@ structure TensorSpecies where
(y : FDiscrete.obj (Discrete.mk (τ c))) :
(contr.app (Discrete.mk c)).hom (x ⊗ₜ[k] y) = (contr.app (Discrete.mk (τ c))).hom
(y ⊗ₜ (FDiscrete.map (Discrete.eqToHom (τ_involution c).symm)).hom x)
/-- Contraction with unit leaves invariant. -/
contr_unit (c : C) (x : FDiscrete.obj (Discrete.mk (c))) :
(λ_ (FDiscrete.obj (Discrete.mk (c)))).hom.hom
(((contr.app (Discrete.mk c)) ▷ (FDiscrete.obj (Discrete.mk (c)))).hom
((α_ _ _ (FDiscrete.obj (Discrete.mk (c)))).inv.hom
(x ⊗ₜ[k] (unit.app (Discrete.mk c)).hom (1 : k)))) = x
/-- The unit is symmetric. -/
unit_symm (c : C) :
((unit.app (Discrete.mk c)).hom (1 : k)) =
((FDiscrete.obj (Discrete.mk (τ (c)))) ◁ (FDiscrete.map (Discrete.eqToHom (τ_involution c)))).hom
((β_ (FDiscrete.obj (Discrete.mk (τ (τ c)))) (FDiscrete.obj (Discrete.mk (τ (c))))).hom.hom
((unit.app (Discrete.mk (τ c))).hom (1 : k)))
/-- On contracting metrics we get back the unit. -/
contr_metric (c : C) :
(β_ (FDiscrete.obj (Discrete.mk c)) (FDiscrete.obj (Discrete.mk (τ c)))).hom.hom
(((FDiscrete.obj (Discrete.mk c)) ◁ (λ_ (FDiscrete.obj (Discrete.mk (τ c)))).hom).hom
(((FDiscrete.obj (Discrete.mk c)) ◁ ((contr.app (Discrete.mk c)) ▷
(FDiscrete.obj (Discrete.mk (τ c))))).hom
(((FDiscrete.obj (Discrete.mk c)) ◁ (α_ (FDiscrete.obj (Discrete.mk (c)))
(FDiscrete.obj (Discrete.mk (τ c))) (FDiscrete.obj (Discrete.mk (τ c)))).inv).hom
((α_ (FDiscrete.obj (Discrete.mk (c))) (FDiscrete.obj (Discrete.mk (c)))
(FDiscrete.obj (Discrete.mk (τ c)) ⊗ FDiscrete.obj (Discrete.mk (τ c)))).hom.hom
((metric.app (Discrete.mk c)).hom (1 : k) ⊗ₜ[k] (metric.app (Discrete.mk (τ c))).hom (1 : k))))))
= (unit.app (Discrete.mk c)).hom (1 : k)
noncomputable section