A nonzero object whose nonzero endomorphisms are all isomorphisms and whose every
nonzero monomorphism from another object admits a nonzero "lift" X ⟶ Y is simple.
A semisimple ring whose multiplication is commutative admits a complete orthogonal
family of primitive idempotents e i, and each corner ring e i · R · e i containing
a nonzero element is a field (so the element has an inverse).
Helper: if ι, π split an idempotent e : X ⟶ X whose corner π · g · ι has
a two-sided inverse sinv (in the corner sense), then g : fi ⟶ fi is an isomorphism.
If End(𝟙_ C) is semisimple, then there is a complete orthogonal family of
primitive idempotents e i such that for any objects f i that split each e i,
every nonzero endomorphism of f i is an isomorphism.
Each summand f i arising from splitting a primitive idempotent of End(𝟙) is
a simple object, given the endomorphism-isomorphism and lifting hypotheses.
Any morphism between summands f i ⟶ f j corresponding to distinct primitive
orthogonal idempotents of End(𝟙) is zero, using commutativity of End(𝟙).
Strengthening of idempotent_summands_simple: the summands f i are not only
simple but also pairwise non-isomorphic.
A complete orthogonal family of idempotents in End(𝟙_ C) gives rise to a
biproduct decomposition 𝟙_ C ≅ ⨁ f i whose summands split each idempotent.
Instances For
Under semisimplicity of End(𝟙_ C) and a lifting axiom, the unit object decomposes
as a biproduct of pairwise non-isomorphic simple objects. This is the categorical form
of Corollary 1.15.2 and Theorem 1.15.8 (ii).
A nonzero object in a preadditive category with binary biproducts is indecomposable whenever every nonzero endomorphism is an isomorphism.
Corollary 1.15.2: the unit object decomposes as a biproduct of pairwise
non-isomorphic indecomposable objects, assuming End(𝟙_ C) is semisimple.
Predicate asserting that the unit object 𝟙_ C decomposes as a finite biproduct
of pairwise non-isomorphic simple objects — the content of Theorem 1.15.8 (ii).
Instances For
If X is a retract of the unit object via (ι, π), then every nonzero monomorphism
m : Y ⟶ X admits a nonzero "lift" X ⟶ Y. This supplies the hlift hypothesis needed
to establish simplicity for unit summands.
Theorem 1.15.8 (ii): in an abelian rigid monoidally biexact category with Artinian
unit object and Artinian endomorphism ring of 𝟙, the unit decomposes as a finite
biproduct of pairwise non-isomorphic simple objects.