44 lines
1.6 KiB
Text
44 lines
1.6 KiB
Text
/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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colimit_functor ⊣ Δ ⊣ limit_functor
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-/
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import .colimits ..functor.adjoint
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open eq functor category is_trunc prod nat_trans
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namespace category
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definition limit_functor_adjoint [constructor] (D I : Precategory) [H : has_limits_of_shape D I] :
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constant_diagram D I ⊣ limit_functor D I :=
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adjoint.mk'
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begin
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fapply natural_iso.MK,
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{ intro dF η, induction dF with d F, esimp at *,
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fapply hom_limit,
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{ exact natural_map η},
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{ intro i j f, exact !naturality ⬝ !id_right}},
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{ esimp, intro dF dF' fθ, induction dF with d F, induction dF' with d' F',
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induction fθ with f θ, esimp at *, apply eq_of_homotopy, intro η,
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apply eq_hom_limit, intro i,
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rewrite [assoc, limit_hom_limit_commute,
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-assoc, assoc (limit_morphism F i), hom_limit_commute]},
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{ esimp, intro dF f, induction dF with d F, esimp at *,
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refine !limit_nat_trans ∘n constant_nat_trans I f},
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{ esimp, intro dF, induction dF with d F, esimp, apply eq_of_homotopy, intro η,
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apply nat_trans_eq, intro i, esimp, apply hom_limit_commute},
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{ esimp, intro dF, induction dF with d F, esimp, apply eq_of_homotopy, intro f,
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symmetry, apply eq_hom_limit, intro i, reflexivity}
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end
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/-
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definition adjoint_colimit_functor [constructor] (D I : Precategory)
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[H : has_colimits_of_shape D I] : colimit_functor D I ⊣ constant_diagram D I :=
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have H : colimit_functor D I ⊣ (constant_diagram Dᵒᵖ Iᵒᵖ)ᵒᵖ', from _,
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_
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-/
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end category
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