Theorems · Theorem · category theory

CategoryTheory.ModObj.isIso_leftSMul_iff

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C] {M : C}
  [inst_2 : CategoryTheory.MonObj M] {X : C} [inst_3 : CategoryTheory.ModObj M X],
  CategoryTheory.IsIso (CategoryTheory.ModObj.leftSMul M X) ↔ ∀ (Z : C) (x y : Z ⟶ X), ∃! m, m • x = y

The morphism (m, x) ↦ (m • x, x) is an isomorphism if and only if the induced action is pointwise simply transitive.

Defined in

Mathlib.CategoryTheory.Monoidal.Cartesian.Mod

Cited by

0 results in Mathlib

Foundations

Depth 33 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CartesianMonoidalCategoryCategoryTheory.MonObjCategoryTheory.ModObj