The exact pairing on the strand object

ExactPairing ⟨1⟩ ⟨1⟩ in the skein category: coevaluation and evaluation are the strand classes, and the zig-zag laws are the snake identities — every structural cast in the categorical formulation lives at equal numeral arities and collapses to the identity class.

theorem RS.bmc_finCongr_refl {R : ℕ} (f : EdgeRankParameter R) {n : ℕ} (h : n = n) : bundleMapClass f (finCongr h) = HomSpace.ofFragment f.val (strandBundle n)

Bundle maps of self-casts are the identity class.

theorem RS.braidClass_eq_bmc {R : ℕ} (f : EdgeRankParameter R) : braidClass f = bundleMapClass f (transposeEquiv 1 1)

The braiding class at one strand is the permutation-fragment braid class.

@[instance_reducible]

noncomputable instance RS.strandExactPairing {R : ℕ} (f : EdgeRankParameter R) : CategoryTheory.ExactPairing { arity := 1 } { arity := 1 }

The exact pairing on the strand object.

Equations

• RS.strandExactPairing f = { coevaluation' := RS.coevClass f, evaluation' := RS.evClass f, coevaluation_evaluation' := ⋯, evaluation_coevaluation' := ⋯ }

theorem RS.strand_ev_symmetry {R : ℕ} (f : EdgeRankParameter R) : CategoryTheory.CategoryStruct.comp (β_ { arity := 1 } { arity := 1 }).hom (ε_ { arity := 1 } { arity := 1 }) = ε_ { arity := 1 } { arity := 1 }

Supersymmetry of the evaluation, categorical form.