[Merged by Bors] - feat: existence of a limit in a concrete category implies smallness #11625
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…11625) In this PR, it is shown that if a functor `G : J ⥤ C` to a concrete category has a limit and that `forget C` is corepresentable, then `G ⋙ forget C).sections` is small. As the corepresentability property holds in many concrete categories (e.g. groups, abelian groups) and that we already know since #11420 that limits exist under the smallness assumption in such categories, then this lemma may be used in future PR in order to show that usual forgetful functors preserve all limits (regardless of universe assumptions). This shall be convenient in the development of sheaves of modules. In this PR, universes assumptions have also been generalized in the file `Limits.Yoneda`. In order to do this, a small refactor of the file `Limits.Types` was necessary. This introduces bijections like `compCoyonedaSectionsEquiv (F : J ⥤ C) (X : C) : (F ⋙ coyoneda.obj (op X)).sections ≃ ((const J).obj X ⟶ F)` with general universe parameters. In order to reduce imports in `Limits.Yoneda`, part of the file `Limits.Types` was moved to a new file `Limits.TypesFiltered`. Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr>
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…11625) In this PR, it is shown that if a functor `G : J ⥤ C` to a concrete category has a limit and that `forget C` is corepresentable, then `G ⋙ forget C).sections` is small. As the corepresentability property holds in many concrete categories (e.g. groups, abelian groups) and that we already know since #11420 that limits exist under the smallness assumption in such categories, then this lemma may be used in future PR in order to show that usual forgetful functors preserve all limits (regardless of universe assumptions). This shall be convenient in the development of sheaves of modules. In this PR, universes assumptions have also been generalized in the file `Limits.Yoneda`. In order to do this, a small refactor of the file `Limits.Types` was necessary. This introduces bijections like `compCoyonedaSectionsEquiv (F : J ⥤ C) (X : C) : (F ⋙ coyoneda.obj (op X)).sections ≃ ((const J).obj X ⟶ F)` with general universe parameters. In order to reduce imports in `Limits.Yoneda`, part of the file `Limits.Types` was moved to a new file `Limits.TypesFiltered`. Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr>
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…11625) In this PR, it is shown that if a functor `G : J ⥤ C` to a concrete category has a limit and that `forget C` is corepresentable, then `G ⋙ forget C).sections` is small. As the corepresentability property holds in many concrete categories (e.g. groups, abelian groups) and that we already know since #11420 that limits exist under the smallness assumption in such categories, then this lemma may be used in future PR in order to show that usual forgetful functors preserve all limits (regardless of universe assumptions). This shall be convenient in the development of sheaves of modules. In this PR, universes assumptions have also been generalized in the file `Limits.Yoneda`. In order to do this, a small refactor of the file `Limits.Types` was necessary. This introduces bijections like `compCoyonedaSectionsEquiv (F : J ⥤ C) (X : C) : (F ⋙ coyoneda.obj (op X)).sections ≃ ((const J).obj X ⟶ F)` with general universe parameters. In order to reduce imports in `Limits.Yoneda`, part of the file `Limits.Types` was moved to a new file `Limits.TypesFiltered`. Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr>
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…11625) In this PR, it is shown that if a functor `G : J ⥤ C` to a concrete category has a limit and that `forget C` is corepresentable, then `G ⋙ forget C).sections` is small. As the corepresentability property holds in many concrete categories (e.g. groups, abelian groups) and that we already know since #11420 that limits exist under the smallness assumption in such categories, then this lemma may be used in future PR in order to show that usual forgetful functors preserve all limits (regardless of universe assumptions). This shall be convenient in the development of sheaves of modules. In this PR, universes assumptions have also been generalized in the file `Limits.Yoneda`. In order to do this, a small refactor of the file `Limits.Types` was necessary. This introduces bijections like `compCoyonedaSectionsEquiv (F : J ⥤ C) (X : C) : (F ⋙ coyoneda.obj (op X)).sections ≃ ((const J).obj X ⟶ F)` with general universe parameters. In order to reduce imports in `Limits.Yoneda`, part of the file `Limits.Types` was moved to a new file `Limits.TypesFiltered`. Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr>
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…11625) In this PR, it is shown that if a functor `G : J ⥤ C` to a concrete category has a limit and that `forget C` is corepresentable, then `G ⋙ forget C).sections` is small. As the corepresentability property holds in many concrete categories (e.g. groups, abelian groups) and that we already know since #11420 that limits exist under the smallness assumption in such categories, then this lemma may be used in future PR in order to show that usual forgetful functors preserve all limits (regardless of universe assumptions). This shall be convenient in the development of sheaves of modules. In this PR, universes assumptions have also been generalized in the file `Limits.Yoneda`. In order to do this, a small refactor of the file `Limits.Types` was necessary. This introduces bijections like `compCoyonedaSectionsEquiv (F : J ⥤ C) (X : C) : (F ⋙ coyoneda.obj (op X)).sections ≃ ((const J).obj X ⟶ F)` with general universe parameters. In order to reduce imports in `Limits.Yoneda`, part of the file `Limits.Types` was moved to a new file `Limits.TypesFiltered`. Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr>
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In this PR, it is shown that if a functor
G : J ⥤ Cto a concrete category has a limit and thatforget Cis corepresentable, thenG ⋙ forget C).sectionsis small. As the corepresentability property holds in many concrete categories (e.g. groups, abelian groups) and that we already know since #11420 that limits exist under the smallness assumption in such categories, then this lemma may be used in future PR in order to show that usual forgetful functors preserve all limits (regardless of universe assumptions). This shall be convenient in the development of sheaves of modules.In this PR, universes assumptions have also been generalized in the file
Limits.Yoneda. In order to do this, a small refactor of the fileLimits.Typeswas necessary. This introduces bijections likecompCoyonedaSectionsEquiv (F : J ⥤ C) (X : C) : (F ⋙ coyoneda.obj (op X)).sections ≃ ((const J).obj X ⟶ F)with general universe parameters. In order to reduce imports inLimits.Yoneda, part of the fileLimits.Typeswas moved to a new fileLimits.TypesFiltered.