cyclesIsoSc'_hom_iCycles — Mathlib

English

The inverse of the cycles-iso interacts with iCycles via composition to recover iCycles.

Русский

Обратное изоморфизма между циклами и sc' действует на iCycles через композицию, возвращая iCycles.

LaTeX

$$(K.cyclesIsoSc' i j k hi hk).inv \circ K.iCycles j = K.iCycles j$$

Lean4

@[reassoc (attr := simp)]
theorem cyclesIsoSc'_hom_iCycles : (K.cyclesIsoSc' i j k hi hk).hom ≫ (K.sc' i j k).iCycles = K.iCycles j :=
  by
  dsimp [cyclesIsoSc']
  simp only [ShortComplex.cyclesMap_i, shortComplexFunctor_obj_X₂, shortComplexFunctor'_obj_X₂, natIsoSc'_hom_app_τ₂,
    comp_id]
  rfl

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