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Mathlib.CategoryTheory.Bicategory.LocallyDiscrete

Locally discrete bicategories #

A category C can be promoted to a strict bicategory LocallyDiscrete C. The objects and the 1-morphisms in LocallyDiscrete C are the same as the objects and the morphisms, respectively, in C, and the 2-morphisms in LocallyDiscrete C are the equalities between 1-morphisms. In other words, the category consisting of the 1-morphisms between each pair of objects X and Y in LocallyDiscrete C is defined as the discrete category associated with the type X ⟶ Y.

A wrapper for promoting any category to a bicategory, with the only 2-morphisms being equalities.

  • as : C

    A wrapper for promoting any category to a bicategory, with the only 2-morphisms being equalities.

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    LocallyDiscrete C is equivalent to the original type C.

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      Extract the equation from a 2-morphism in a locally discrete 2-category.

      The locally discrete bicategory on a category is a bicategory in which the objects and the 1-morphisms are the same as those in the underlying category, and the 2-morphisms are the equalities between 1-morphisms.

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      A locally discrete bicategory is strict.

      @[simp]
      theorem CategoryTheory.PrelaxFunctor.map₂_eqToHom {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : PrelaxFunctor B C) {a b : B} {f g : a b} (h : f = g) :
      @[reducible, inline]

      A bicategory is locally discrete if the categories of 1-morphisms are discrete.

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        def Quiver.Hom.toLoc {C : Type u} [CategoryTheory.CategoryStruct.{v, u} C] {a b : C} (f : a b) :
        { as := a } { as := b }

        The 1-morphism in LocallyDiscrete C associated to a given morphism f : a ⟶ b in C

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          @[simp]