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feat(category_theory/limits): preserving pullbacks (#5668)
This touches multiple files but it's essentially the same thing as all my other PRs for preserving limits of special shapes - I can split it up if you'd like but hopefully this is alright?
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src/category_theory/limits/preserves/shapes/pullbacks.lean
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| /- | ||
| Copyright (c) 2020 Bhavik Mehta. All rights reserved. | ||
| Released under Apache 2.0 license as described in the file LICENSE. | ||
| Authors: Bhavik Mehta | ||
| -/ | ||
| import category_theory.limits.shapes.pullbacks | ||
| import category_theory.limits.preserves.basic | ||
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| /-! | ||
| # Preserving pullbacks | ||
| Constructions to relate the notions of preserving pullbacks and reflecting pullbacks to concrete | ||
| pullback cones. | ||
| In particular, we show that `pullback_comparison G f g` is an isomorphism iff `G` preserves | ||
| the pullback of `f` and `g`. | ||
| ## TODO | ||
| * Dualise to pushouts | ||
| * Generalise to wide pullbacks | ||
| -/ | ||
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| noncomputable theory | ||
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| universes v u₁ u₂ | ||
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| open category_theory category_theory.category category_theory.limits | ||
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| variables {C : Type u₁} [category.{v} C] | ||
| variables {D : Type u₂} [category.{v} D] | ||
| variables (G : C ⥤ D) | ||
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| namespace category_theory.limits | ||
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| variables {W X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} {h : W ⟶ X} {k : W ⟶ Y} (comm : h ≫ f = k ≫ g) | ||
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| /-- | ||
| The map of a pullback cone is a limit iff the fork consisting of the mapped morphisms is a limit. | ||
| This essentially lets us commute `pullback_cone.mk` with `functor.map_cone`. | ||
| -/ | ||
| def is_limit_map_cone_pullback_cone_equiv : | ||
| is_limit (G.map_cone (pullback_cone.mk h k comm)) ≃ | ||
| is_limit (pullback_cone.mk (G.map h) (G.map k) (by simp only [← G.map_comp, comm]) | ||
| : pullback_cone (G.map f) (G.map g)) := | ||
| (is_limit.postcompose_hom_equiv (diagram_iso_cospan _) _).symm.trans | ||
| (is_limit.equiv_iso_limit (cones.ext (iso.refl _) (by { rintro (_ | _ | _), tidy }))) | ||
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| /-- The property of preserving pullbacks expressed in terms of binary fans. -/ | ||
| def is_limit_pullback_cone_map_of_is_limit [preserves_limit (cospan f g) G] | ||
| (l : is_limit (pullback_cone.mk h k comm)) : | ||
| is_limit (pullback_cone.mk (G.map h) (G.map k) _) := | ||
| is_limit_map_cone_pullback_cone_equiv G comm (preserves_limit.preserves l) | ||
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| /-- The property of reflecting pullbacks expressed in terms of binary fans. -/ | ||
| def is_limit_of_is_limit_pullback_cone_map [reflects_limit (cospan f g) G] | ||
| (l : is_limit (pullback_cone.mk (G.map h) (G.map k) _)) : | ||
| is_limit (pullback_cone.mk h k comm) := | ||
| reflects_limit.reflects ((is_limit_map_cone_pullback_cone_equiv G comm).symm l) | ||
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| variables (f g) [has_pullback f g] | ||
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| /-- | ||
| If `G` preserves pullbacks and `C` has them, then the pullback cone constructed of the mapped | ||
| morphisms of the pullback cone is a limit. | ||
| -/ | ||
| def is_limit_of_has_pullback_of_preserves_limit | ||
| [preserves_limit (cospan f g) G] : | ||
| is_limit (pullback_cone.mk (G.map pullback.fst) (G.map pullback.snd) _) := | ||
| is_limit_pullback_cone_map_of_is_limit G _ (pullback_is_pullback f g) | ||
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| variables [has_pullback (G.map f) (G.map g)] | ||
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| /-- | ||
| If the pullback comparison map for `G` at `(f,g)` is an isomorphism, then `G` preserves the | ||
| pullback of `(f,g)`. | ||
| -/ | ||
| def preserves_pullback.of_iso_comparison [i : is_iso (pullback_comparison G f g)] : | ||
| preserves_limit (cospan f g) G := | ||
| begin | ||
| apply preserves_limit_of_preserves_limit_cone (pullback_is_pullback f g), | ||
| apply (is_limit_map_cone_pullback_cone_equiv _ _).symm _, | ||
| apply is_limit.of_point_iso (limit.is_limit (cospan (G.map f) (G.map g))), | ||
| apply i, | ||
| end | ||
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| variables [preserves_limit (cospan f g) G] | ||
| /-- | ||
| If `G` preserves the pullback of `(f,g)`, then the pullback comparison map for `G` at `(f,g)` is | ||
| an isomorphism. | ||
| -/ | ||
| def preserves_pullback.iso : | ||
| G.obj (pullback f g) ≅ pullback (G.map f) (G.map g) := | ||
| is_limit.cone_point_unique_up_to_iso | ||
| (is_limit_of_has_pullback_of_preserves_limit G f g) | ||
| (limit.is_limit _) | ||
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| @[simp] | ||
| lemma preserves_pullback.iso_hom : | ||
| (preserves_pullback.iso G f g).hom = pullback_comparison G f g := rfl | ||
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| instance : is_iso (pullback_comparison G f g) := | ||
| begin | ||
| rw ← preserves_pullback.iso_hom, | ||
| apply_instance | ||
| end | ||
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| end category_theory.limits |
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