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Try doing some basic maths questions in the Lean Theorem Prover. Functions, real numbers, equivalence relations and groups. Click on README.md and then on "Open in CoCalc with one click".

Project: Xena
Views: 18536
License: APACHE
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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: Patrick Massot, Mario Carneiro
-/
import topology.category.Top.basic
import category_theory.adjunction.basic

universe u

open category_theory
open Top

namespace Top

/-- Equipping a type with the discrete topology is left adjoint to the forgetful functor `Top ⥤ Type`. -/
def adj₁ : discrete ⊣ forget Top :=
{ hom_equiv := λ X Y,
  { to_fun := λ f, f,
    inv_fun := λ f, ⟨f, continuous_bot⟩,
    left_inv := by tidy,
    right_inv := by tidy },
  unit := { app := λ X, id },
  counit := { app := λ X, ⟨id, continuous_bot⟩ } }

/-- Equipping a type with the trivial topology is right adjoint to the forgetful functor `Top ⥤ Type`. -/
def adj₂ : forget Top ⊣ trivial :=
{ hom_equiv := λ X Y,
  { to_fun := λ f, ⟨f, continuous_top⟩,
    inv_fun := λ f, f,
    left_inv := by tidy,
    right_inv := by tidy },
  unit := { app := λ X, ⟨id, continuous_top⟩ },
  counit := { app := λ X, id } }

end Top