theory transformation functionone imports Functors begin
(* guess the third axiom is implied by the fifth *) locale natural_transformation = two_cats + fixes F and G and u assumes"Functor F : AA ⟶ BB" andFunctor G : AA ⟶ BB" and "u : ob AA → ar BB" and "u ∈ extensional (ob AA)" and "\forallA∈ Hom(FA) (G<^esub> A" and "∀Ob <>B∈f∈a(u A) = (u B) ∙BBa^esub
abbreviation
nt_syn (‹
"u : F ==> AA 🚫o
(* is this doing what I think its doing? *) locale endoNT = natural_transformation + one_cat
theorem (in endoNT) id_restrict_natural "(λA∈ Id A) : (id_func AA) ==> (id_func AA) in Func(AA,AA)" proof (intro natural_transformation.intro natural_transformation_axioms.intro
two_catsballI show by (rule funcsetI) auto show"(\lambdaA\inOb. IdA) ∈ by (rule restrict_extensional) A assume A: "A ∈ hence"Id A ∈ thus "by(rule funcsetIauto
usingimpdef) fix B and f
ssume<n Ob" and "f ∈ Ob" hence "f ∈ Hom A A" .. using A by (simp_all add: hom_def) thus "(id_func AA)<a> f ∙ (λA∈Ob. Id A) A
= (λA∈Ob. Id A) B ∙ (id_func AA)<a> f" by (simp add: id_func_def) qed (auto intro: id_func_functor, unfold_
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