Theorems · Definition · category theory

CategoryTheory.StructuredArrow.subobjectEquiv

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {D : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        {S : D} →
          {T : CategoryTheory.Functor C D} →
            [CategoryTheory.Limits.HasFiniteLimits C] →
              [CategoryTheory.Limits.PreservesFiniteLimits T] →
                (A : CategoryTheory.StructuredArrow S T) →
                  CategoryTheory.Subobject A ≃o
                    { P // ∃ q, CategoryTheory.CategoryStruct.comp q (T.map P.arrow) = A.hom }

If A : S → T.obj B is a structured arrow for S : D and T : C ⥤ D, then we can explicitly describe the subobjects of A as the subobjects P of B in C for which A.hom factors through the image of P under T.

Defined in

Mathlib.CategoryTheory.Subobject.Comma

Cited by

0 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasFiniteLimitsCategoryTheory.Limits.PreservesFiniteLimits