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chore(category_theory/category/preorder): split material on galois co…
…nnections (#17339) This is reducing unnecessary imports. Really, however, someone should tackle `order.complete_lattice`, which has unnecessary heavy imports. Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
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/- | ||
Copyright (c) 2017 Scott Morrison. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Stephen Morgan, Scott Morrison, Johannes Hölzl, Reid Barton | ||
-/ | ||
import category_theory.category.preorder | ||
import category_theory.adjunction.basic | ||
import order.galois_connection | ||
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/-! | ||
# Galois connections between preorders are adjunctions. | ||
* `galois_connection.adjunction` is the adjunction associated to a galois connection. | ||
-/ | ||
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universes u v | ||
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section | ||
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variables {X : Type u} {Y : Type v} [preorder X] [preorder Y] | ||
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/-- | ||
A galois connection between preorders induces an adjunction between the associated categories. | ||
-/ | ||
def galois_connection.adjunction {l : X → Y} {u : Y → X} (gc : galois_connection l u) : | ||
gc.monotone_l.functor ⊣ gc.monotone_u.functor := | ||
category_theory.adjunction.mk_of_hom_equiv | ||
{ hom_equiv := λ X Y, ⟨λ f, (gc.le_u f.le).hom, λ f, (gc.l_le f.le).hom, by tidy, by tidy⟩ } | ||
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end | ||
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namespace category_theory | ||
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variables {X : Type u} {Y : Type v} [preorder X] [preorder Y] | ||
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/-- | ||
An adjunction between preorder categories induces a galois connection. | ||
-/ | ||
lemma adjunction.gc {L : X ⥤ Y} {R : Y ⥤ X} (adj : L ⊣ R) : | ||
galois_connection L.obj R.obj := | ||
λ x y, ⟨λ h, ((adj.hom_equiv x y).to_fun h.hom).le, λ h, ((adj.hom_equiv x y).inv_fun h.hom).le⟩ | ||
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end category_theory |
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