[Merged by Bors] - feat(category/limits/shapes): fix biproducts #3175
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This is a second attempt at #3102. Previously the definition of a (binary) biproduct in a category with zero morphisms (but not necessarily) preadditive was just wrong. The definition for a "bicone" was just something that was simultaneously a cone and a cocone, with the same cone points. It was a "biproduct bicone" if the cone was a limit cone and the cocone was a colimit cocone. However, this definition was not particularly useful. In particular, there was no way to prove that the two different `map` constructions providing a morphism `W ⊞ X ⟶ Y ⊞ Z` (i.e. by treating the biproducts either as cones, or as cocones) were actually equal. Blech. So, I've added the axioms `inl ≫ fst = 𝟙 P`, `inl ≫ snd = 0`, `inr ≫ fst = 0`, and `inr ≫ snd = 𝟙 Q` to `bicone P Q`. (Note these only require the exist of zero morphisms, not preadditivity.) Now the two map constructions are equal. I've then entirely removed the `has_preadditive_biproduct` typeclass. Instead we have 1. additional theorems providing `total`, when `preadditive C` is available 2. alternative constructors for `has_biproduct` that use `total` rather than `is_limit` and `is_colimit`. This PR also introduces some abbreviations along the lines of `abbreviation has_binary_product (X Y : C) := has_limit (pair X Y)`, just to improve readability. Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
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This is a second attempt at #3102. Previously the definition of a (binary) biproduct in a category with zero morphisms (but not necessarily) preadditive was just wrong. The definition for a "bicone" was just something that was simultaneously a cone and a cocone, with the same cone points. It was a "biproduct bicone" if the cone was a limit cone and the cocone was a colimit cocone. However, this definition was not particularly useful. In particular, there was no way to prove that the two different `map` constructions providing a morphism `W ⊞ X ⟶ Y ⊞ Z` (i.e. by treating the biproducts either as cones, or as cocones) were actually equal. Blech. So, I've added the axioms `inl ≫ fst = 𝟙 P`, `inl ≫ snd = 0`, `inr ≫ fst = 0`, and `inr ≫ snd = 𝟙 Q` to `bicone P Q`. (Note these only require the exist of zero morphisms, not preadditivity.) Now the two map constructions are equal. I've then entirely removed the `has_preadditive_biproduct` typeclass. Instead we have 1. additional theorems providing `total`, when `preadditive C` is available 2. alternative constructors for `has_biproduct` that use `total` rather than `is_limit` and `is_colimit`. This PR also introduces some abbreviations along the lines of `abbreviation has_binary_product (X Y : C) := has_limit (pair X Y)`, just to improve readability. Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
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This is a second attempt at leanprover-community#3102. Previously the definition of a (binary) biproduct in a category with zero morphisms (but not necessarily) preadditive was just wrong. The definition for a "bicone" was just something that was simultaneously a cone and a cocone, with the same cone points. It was a "biproduct bicone" if the cone was a limit cone and the cocone was a colimit cocone. However, this definition was not particularly useful. In particular, there was no way to prove that the two different `map` constructions providing a morphism `W ⊞ X ⟶ Y ⊞ Z` (i.e. by treating the biproducts either as cones, or as cocones) were actually equal. Blech. So, I've added the axioms `inl ≫ fst = 𝟙 P`, `inl ≫ snd = 0`, `inr ≫ fst = 0`, and `inr ≫ snd = 𝟙 Q` to `bicone P Q`. (Note these only require the exist of zero morphisms, not preadditivity.) Now the two map constructions are equal. I've then entirely removed the `has_preadditive_biproduct` typeclass. Instead we have 1. additional theorems providing `total`, when `preadditive C` is available 2. alternative constructors for `has_biproduct` that use `total` rather than `is_limit` and `is_colimit`. This PR also introduces some abbreviations along the lines of `abbreviation has_binary_product (X Y : C) := has_limit (pair X Y)`, just to improve readability. Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
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This is a second attempt at #3102.
Previously the definition of a (binary) biproduct in a category with zero morphisms (but not necessarily) preadditive was just wrong.
The definition for a "bicone" was just something that was simultaneously a cone and a cocone, with the same cone points. It was a "biproduct bicone" if the cone was a limit cone and the cocone was a colimit cocone. However, this definition was not particularly useful. In particular, there was no way to prove that the two different
mapconstructions providing a morphismW ⊞ X ⟶ Y ⊞ Z(i.e. by treating the biproducts either as cones, or as cocones) were actually equal. Blech.So, I've added the axioms
inl ≫ fst = 𝟙 P,inl ≫ snd = 0,inr ≫ fst = 0, andinr ≫ snd = 𝟙 Qtobicone P Q. (Note these only require the exist of zero morphisms, not preadditivity.)Now the two map constructions are equal.
I've then entirely removed the
has_preadditive_biproducttypeclass. Instead we havetotal, whenpreadditive Cis availablehas_biproductthat usetotalrather thanis_limitandis_colimit.This PR also introduces some abbreviations along the lines of
abbreviation has_binary_product (X Y : C) := has_limit (pair X Y), just to improve readability.