Constructions related to weakly initial objects #

This file gives constructions related to weakly initial objects, namely:

These are primarily useful to show the General Adjoint Functor Theorem.

theorem CategoryTheory.has_weakly_initial_of_weakly_initial_set_and_hasProducts {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {ι : Type v} {B : ιC} (hB : ∀ (A : C), ∃ (i : ι), Nonempty (B i A)) :
∃ (T : C), ∀ (X : C), Nonempty (T X)

If C has (small) products and a small weakly initial set of objects, then it has a weakly initial object.

If C has (small) wide equalizers and a weakly initial object, then it has an initial object.

The initial object is constructed as the wide equalizer of all endomorphisms on the given weakly initial object.