# mathlib3documentation

category_theory.category.Cat

# Category of categories #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This file contains the definition of the category Cat of all categories. In this category objects are categories and morphisms are functors between these categories.

## Implementation notes #

Though Cat is not a concrete category, we use bundled to define its carrier type.

@[nolint]
def category_theory.Cat  :
Type (max (u+1) u (v+1))

Category of categories.

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Construct a bundled Cat from the underlying type and the typeclass.

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Bicategory structure on Cat

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Cat is a strict bicategory.

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Category structure on Cat

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@[simp]
theorem category_theory.Cat.id_map {C : category_theory.Cat} {X Y : C} (f : X Y) :
(𝟙 C).map f = f
@[simp]
theorem category_theory.Cat.comp_obj {C D E : category_theory.Cat} (F : C D) (G : D E) (X : C) :
(F G).obj X = G.obj (F.obj X)
@[simp]
theorem category_theory.Cat.comp_map {C D E : category_theory.Cat} (F : C D) (G : D E) {X Y : C} (f : X Y) :
(F G).map f = G.map (F.map f)

Functor that gets the set of objects of a category. It is not called forget, because it is not a faithful functor.

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Any isomorphism in Cat induces an equivalence of the underlying categories.

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@[simp]
theorem category_theory.Type_to_Cat_map (X Y : Type u) (f : X Y) :

Embedding Type into Cat as discrete categories.

This ought to be modelled as a 2-functor!

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