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Mathlib.CategoryTheory.Limits.MonoCoprod

Categories where inclusions into coproducts are monomorphisms #

If C is a category, the class MonoCoprod C expresses that left inclusions A ⟶ A ⨿ B are monomorphisms when HasCoproduct A B is satisfied. If it is so, it is shown that right inclusions are also monomorphisms.

More generally, we deduce that when suitable coproducts exists, then if X : I → C and ι : J → I is an injective map, then the canonical morphism ∐ (X ∘ ι) ⟶ ∐ X is a monomorphism. It also follows that for any i : I, Sigma.ι X i : X i ⟶ ∐ X is a monomorphism.

TODO: define distributive categories, and show that they satisfy MonoCoprod, see

This condition expresses that inclusion morphisms into coproducts are monomorphisms.

Instances

    Given a family of objects X : I₁ ⊕ I₂ → C, a cofan of X, and two colimit cofans of X ∘ Sum.inl and X ∘ Sum.inr, this is a cofan for c₁.pt and c₂.pt whose point is c.pt.

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      The binary cofan binaryCofanSum c c₁ c₂ hc₁ hc₂ is colimit.

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