mathlib documentation

category_theory.groupoid

@[class]
structure category_theory.groupoid  :
Type uType (max u (v+1))

A groupoid is a category such that all morphisms are isomorphisms.

Instances
def category_theory.large_groupoid  :
Type (u+1)Type (u+1)

A large_groupoid is a groupoid where the objects live in Type (u+1) while the morphisms live in Type u.

def category_theory.small_groupoid  :
Type uType (u+1)

A small_groupoid is a groupoid where the objects and morphisms live in the same universe.

def category_theory.groupoid.iso_equiv_hom {C : Type u} [category_theory.groupoid C] (X Y : C) :
(X Y) (X Y)

In a groupoid, isomorphisms are equivalent to morphisms.

Equations

A category where every morphism is_iso is a groupoid.

Equations

A category where every morphism has a trunc retraction is computably a groupoid.

Equations