feat(CategoryTheory/Functor): refactoring Lan #10425
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Co-authored-by: Markus Himmel <markus@himmel-villmar.de>
Co-authored-by: Markus Himmel <markus@himmel-villmar.de>
Co-authored-by: Markus Himmel <markus@himmel-villmar.de>
Co-authored-by: Markus Himmel <markus@himmel-villmar.de>
Co-authored-by: Markus Himmel <markus@himmel-villmar.de>
Co-authored-by: Markus Himmel <markus@himmel-villmar.de>
Co-authored-by: Markus Himmel <markus@himmel-villmar.de>
Co-authored-by: Markus Himmel <markus@himmel-villmar.de>
Co-authored-by: Markus Himmel <markus@himmel-villmar.de>
Co-authored-by: Markus Himmel <markus@himmel-villmar.de>
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Given a functor `F : C ⥤ D`, we define the left Kan extension functor `F.lan : (C ⥤ E) ⥤ (D ⥤ E)` which sends a functor `G : C ⥤ E` to its left Kan extension along `F`. (This is a step towards the refactor of `Lan/Ran` in mathlib using the new API for Kan extensions of functors #10425.)
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Given a functor `F : C ⥤ D`, we define the left Kan extension functor `F.lan : (C ⥤ E) ⥤ (D ⥤ E)` which sends a functor `G : C ⥤ E` to its left Kan extension along `F`. (This is a step towards the refactor of `Lan/Ran` in mathlib using the new API for Kan extensions of functors #10425.)
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Given a functor `F : C ⥤ D`, we define the left Kan extension functor `F.lan : (C ⥤ E) ⥤ (D ⥤ E)` which sends a functor `G : C ⥤ E` to its left Kan extension along `F`. (This is a step towards the refactor of `Lan/Ran` in mathlib using the new API for Kan extensions of functors #10425.)
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This PR refactors the definition of the left Kan extension functor (
Functor.lan : (C ⥤ E) ⥤ (D ⥤ E)) which sends a functor to its left Kan extension along a given functorF : C ⥤ D. This definition used to be a construction using the existence of suitable colimits. The new definition relies on the new general API for left Kan extensions. Then,F.lan.obj Gis now defined asleftKanExtension F G. As a result,(leftKanExtension F G).obj Xis no longer defined as colimit, but when it is known thatGhas a pointwise left Kan extension, it is shown thatleftKanExtension F Gis indeed a pointwise left Kan extension, which means that(leftKanExtension F G).obj Xis the point of a certain colimit cocone. Then, in the downstream applications of this construction we have to use the API for colimit cocones rather that the colimit API.The file
CategoryTheory.Limits.Presheafis also significantly refactored: instead of reconstructing functors by using colimits, we make use of the left Kan extension API instead.This is only a very early draft... It was mostly a test to see how many things would break. I will not proceed with this refactor in the immediate future: first, I will develop a little bit more the left Kan extension API.