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feat(category_theory/closed): cartesian closed category with zero obj…
…ect is trivial (#4924)
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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 | ||
-/ | ||
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import category_theory.closed.cartesian | ||
import category_theory.limits.shapes.zero | ||
import category_theory.punit | ||
import category_theory.conj | ||
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/-! | ||
# A cartesian closed category with zero object is trivial | ||
A cartesian closed category with zero object is trivial: it is equivalent to the category with one | ||
object and one morphism. | ||
## References | ||
* https://mathoverflow.net/a/136480 | ||
-/ | ||
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universes v u | ||
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noncomputable theory | ||
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namespace category_theory | ||
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open category limits | ||
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variables {C : Type u} [category.{v} C] | ||
variables [has_finite_products C] [cartesian_closed C] | ||
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/-- | ||
If a cartesian closed category has an initial object which is isomorphic to the terminal object, | ||
then each homset has exactly one element. | ||
-/ | ||
def unique_homset_of_initial_iso_terminal [has_initial C] (i : ⊥_ C ≅ ⊤_ C) (X Y : C) : | ||
unique (X ⟶ Y) := | ||
equiv.unique $ | ||
calc (X ⟶ Y) ≃ (X ⨯ ⊤_ C ⟶ Y) : iso.hom_congr (prod.right_unitor _).symm (iso.refl _) | ||
... ≃ (X ⨯ ⊥_ C ⟶ Y) : iso.hom_congr (prod.map_iso (iso.refl _) i.symm) (iso.refl _) | ||
... ≃ (⊥_ C ⟶ Y ^^ X) : (exp.adjunction _).hom_equiv _ _ | ||
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local attribute [instance] has_zero_object.has_zero | ||
local attribute [instance] has_zero_object.unique_to has_zero_object.unique_from | ||
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/-- If a cartesian closed category has a zero object, each homset has exactly one element. -/ | ||
def unique_homset_of_zero [has_zero_object C] (X Y : C) : | ||
unique (X ⟶ Y) := | ||
begin | ||
haveI : has_initial C := has_zero_object.has_initial, | ||
apply unique_homset_of_initial_iso_terminal _ X Y, | ||
refine ⟨default _, default (⊤_ C ⟶ 0) ≫ default _, _, _⟩; | ||
simp, | ||
end | ||
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local attribute [instance] unique_homset_of_zero | ||
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/-- | ||
A cartesian closed category with a zero object is equivalent to the category with one object and | ||
one morphism. | ||
-/ | ||
def equiv_punit [has_zero_object C] : C ≌ discrete punit := | ||
equivalence.mk | ||
(functor.star C) | ||
(functor.from_punit 0) | ||
(nat_iso.of_components | ||
(λ X, { hom := default (X ⟶ 0), | ||
inv := default (0 ⟶ X) }) | ||
(λ X Y f, dec_trivial)) | ||
(functor.punit_ext _ _) | ||
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end category_theory |