Theorems · Theorem · category theory

CategoryTheory.PositiveCategory.copy_comp_natural

∀ {C : Type u} {inst : CategoryTheory.Category.{v, u} C} {inst_1 : CategoryTheory.MonoidalCategory C}
  [self : CategoryTheory.PositiveCategory C] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z)
  [h : CategoryTheory.Deterministic (CategoryTheory.CategoryStruct.comp f g)],
  CategoryTheory.CategoryStruct.comp f
      (CategoryTheory.CategoryStruct.comp CategoryTheory.ComonObj.comul
        (CategoryTheory.MonoidalCategoryStruct.tensorHom g (CategoryTheory.CategoryStruct.id Y))) =
    CategoryTheory.CategoryStruct.comp CategoryTheory.ComonObj.comul
      (CategoryTheory.MonoidalCategoryStruct.tensorHom (CategoryTheory.CategoryStruct.comp f g) f)

Given morphisms f : X ⟶ Y and g : Y ⟶ Z, if their composition is deterministic, then process f, copy and then process g equals copy and process f and g independently.

Defined in

Mathlib.CategoryTheory.MarkovCategory.Positive

Cited by

0 results in Mathlib

Foundations

Depth 5 from the axioms · uses no axioms

Assumes

CategoryTheory.PositiveCategoryCategoryTheory.Deterministic