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category_theory.idempotents.biproducts

Biproducts in the idempotent completion of a preadditive category #

In this file, we define an instance expressing that if C is an additive category, then karoubi C is also an additive category.

We also obtain that for all P : karoubi C where C is a preadditive category C, there is a canonical isomorphism P ⊞ P.complement ≅ (to_karoubi C).obj P.X in the category karoubi C where P.complement is the formal direct factor of P.X corresponding to the idempotent endomorphism 𝟙 P.X - P.p.

The bicone used in order to obtain the existence of the biproduct of a functor J ⥤ karoubi C when the category C is additive.

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P.complement is the formal direct factor of P.X given by the idempotent endomorphism 𝟙 P.X - P.p

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Instances for category_theory.idempotents.karoubi.complement

A formal direct factor P : karoubi C of an object P.X : C in a preadditive category is actually a direct factor of the image (to_karoubi C).obj P.X of P.X in the category karoubi C

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