counitCoalgHom — Mathlib

English

The counit of a coalgebra is realized as a CoalgHom from A to R, i.e., a coalgebra morphism ε: A → R with the standard counit properties.

Русский

Коуниц коалгебры реализуется как коалгоморфизм ε: A → R, обладающий стандартными свойствами кушюбий.

LaTeX

$$$\varepsilon: A \to R \quad\text{is a coalgebra morphism}$$$

Lean4

/-- The counit of a coalgebra as a `CoalgHom`. -/
noncomputable def counitCoalgHom : A →ₗc[R] R :=
  { counit with
    counit_comp := by ext; simp
    map_comp_comul := by
      ext
      simp only [LinearMap.coe_comp, Function.comp_apply, CommSemiring.comul_apply, ← LinearMap.lTensor_comp_rTensor,
        rTensor_counit_comul, LinearMap.lTensor_tmul] }

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