Theorems · Theorem · category theory
CategoryTheory.simple_iff_subobject_isSimpleOrder
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasZeroObject C] (X : C),
CategoryTheory.Simple X ↔ IsSimpleOrder (CategoryTheory.Subobject X)X is simple iff it has subobject lattice {⊥, ⊤}.
- Defined in
- Mathlib.CategoryTheory.Simple
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CategoryTheory.Categorystatement and proof · cited by 32,673
- CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
- CategoryTheory.Limits.HasZeroObjectstatement and proof · cited by 1,298
- CategoryTheory.Subobjectstatement and proof · cited by 385
- IsSimpleOrderstatement and proof · cited by 54
- CategoryTheory.Simplestatement and proof · cited by 39
- CategoryTheory.simple_of_isSimpleOrder_subobjectproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- simple_iff_isSimpleModuleproof · cited by 1
- CategoryTheory.subobject_simple_iff_isAtomproof · cited by 1