Simplicity of the tensor unit

In the setting of Deligne's theorem — an abelian ℂ-linear rigid monoidal category whose unit endomorphisms are exactly the scalars — the tensor unit is a simple object. This is Proposition 1.17 of Deligne–Milne, Tannakian categories.

The argument follows Deligne–Milne. Given a nonzero subobject i : U ⟶ 𝟙, write V for the cokernel of i and D : 𝟙 ⟶ 𝟙 ⊗ Uᘁ for the mate of i under the duality adjunction, and set W := ker D. Exactness of the tensor product in each variable (rigidity provides two-sided adjoints to whiskering) yields V ⊗ U = 0, U ⊗ V = 0 and W ⊗ U = 0, from which the composite W ⟶ 𝟙 ⟶ V is an isomorphism. The resulting splitting of 𝟙 ↠ V produces an idempotent unit endomorphism, and End (𝟙) = ℂ has no idempotents besides 0 and 1: the value 1 would force i = 0, so the idempotent vanishes, V = 0, and i is an isomorphism.

Scalars on the unit

Consequences of HasScalarUnit alone: the identity of the unit is nonzero, and the only idempotent endomorphisms of the unit are 0 and the identity.

theorem RS.id_unit_ne_zero {A : Type u} [CategoryTheory.Category.{v, u} A] [CategoryTheory.Preadditive A] [CategoryTheory.Linear ℂ A] [CategoryTheory.MonoidalCategory A] (hu : HasScalarUnit A) : CategoryTheory.CategoryStruct.id (CategoryTheory.MonoidalCategoryStruct.tensorUnit A) ≠ 0

Under HasScalarUnit, the identity of the unit is nonzero.

theorem RS.HasScalarUnit.idempotent_eq_zero_or_id {A : Type u} [CategoryTheory.Category.{v, u} A] [CategoryTheory.Preadditive A] [CategoryTheory.Linear ℂ A] [CategoryTheory.MonoidalCategory A] (hu : HasScalarUnit A) {e : CategoryTheory.MonoidalCategoryStruct.tensorUnit A ⟶ CategoryTheory.MonoidalCategoryStruct.tensorUnit A} (he : CategoryTheory.CategoryStruct.comp e e = e) : e = 0 ∨ e = CategoryTheory.CategoryStruct.id (CategoryTheory.MonoidalCategoryStruct.tensorUnit A)

Under HasScalarUnit, the only idempotent endomorphisms of the unit are 0 and the identity: composition of scalar endomorphisms is multiplication in ℂ, and a field has no other idempotents.

Exactness of whiskering

In a rigid category each whiskering functor has adjoints on both sides, so it preserves monomorphisms, epimorphisms and kernels.

The mate of a subobject of the unit

For a monomorphism i : U ⟶ 𝟙 the mate D : 𝟙 ⟶ 𝟙 ⊗ Uᘁ under the duality adjunction detects which maps into the unit are annihilated by tensoring with U: if x ≫ D = 0 then x ▷ U = 0.

Vanishing tensor products

For a subobject i : U ⟶ 𝟙 with cokernel V, both V ⊗ U and U ⊗ V vanish: the composite of the whiskered epimorphism and the whiskered monomorphism through them is i ≫ π = 0. The kernel W of the mate also satisfies W ⊗ U = 0, by construction of the mate.

The kernel of the mate maps onto the cokernel

Whiskering the mate by the cokernel V kills it, because the mate factors through i and V ⊗ U = 0. Since whiskering preserves kernels, V ◁ kernel.ι is then the kernel of a zero map, hence an isomorphism V ⊗ W ≅ V ⊗ 𝟙.

Subobjects of U killed by ⊗ U

A subobject T of U with T ⊗ U = 0 is zero: T ⊗ V vanishes because U ⊗ V does, and whiskering the exact sequence U ⟶ 𝟙 ⟶ V by T exhibits T ≅ T ⊗ 𝟙 as an extension of the two vanishing ends.

The theorem

theorem RS.simple_unit_of_hasScalarUnit {A : Type u} [CategoryTheory.Category.{v, u} A] [CategoryTheory.Abelian A] [CategoryTheory.Linear ℂ A] [CategoryTheory.MonoidalCategory A] [CategoryTheory.MonoidalPreadditive A] [CategoryTheory.RigidCategory A] (hu : HasScalarUnit A) : CategoryTheory.Simple (CategoryTheory.MonoidalCategoryStruct.tensorUnit A)

The tensor unit is simple in the setting of Deligne's theorem: in an abelian ℂ-linear rigid monoidal category whose unit endomorphisms are exactly the scalars, 𝟙_ A is a simple object. This is Proposition 1.17 of Deligne–Milne, Tannakian categories.