# mathlibdocumentation

category_theory.limits.constructions.weakly_initial

# Constructions related to weakly initial objects #

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

• If a category has small products and a small weakly initial set of objects, then it has a weakly initial object.
• If a category has wide equalizers and a weakly initial object, then it has an initial object.

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

theorem category_theory.has_weakly_initial_of_weakly_initial_set_and_has_products {C : Type u} {ι : Type v} {B : ι → C} (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.

theorem category_theory.has_initial_of_weakly_initial_and_has_wide_equalizers {C : Type u} {T : C} (hT : ∀ (X : C), nonempty (T X)) :

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.