Finite categories

A category is finite in this sense if it has finitely many objects, and finitely many morphisms.

Implementation

Prior to #14046, FinCategory required a DecidableEq instance on the object and morphism types. This does not seem to have had any practical payoff (i.e. making some definition constructive) so we have removed these requirements to avoid having to supply instances or delay with non-defeq conflicts between instances.

instance CategoryTheory.discreteFintype {α : Type u_1} [Fintype α] : Fintype (CategoryTheory.Discrete α)

Equations

• CategoryTheory.discreteFintype = Fintype.ofEquiv α CategoryTheory.discreteEquiv.symm

instance CategoryTheory.discreteHomFintype {α : Type u_1} (X : CategoryTheory.Discrete α) (Y : CategoryTheory.Discrete α) : Fintype (X ⟶ Y)

Equations

• CategoryTheory.discreteHomFintype X Y = ULift.fintype (PLift (X.as = Y.as))

class CategoryTheory.FinCategory (J : Type v) [CategoryTheory.SmallCategory J] : Type v

A category with a Fintype of objects, and a Fintype for each morphism space.

• fintypeObj : Fintype J
• fintypeHom : (j j' : J) → Fintype (j ⟶ j')

instance CategoryTheory.finCategoryDiscreteOfFintype (J : Type v) [Fintype J] : CategoryTheory.FinCategory (CategoryTheory.Discrete J)

Equations

• CategoryTheory.finCategoryDiscreteOfFintype J = { fintypeObj := inferInstance, fintypeHom := inferInstance }

instance CategoryTheory.finCategoryOpposite {J : Type v} [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] : CategoryTheory.FinCategory Jᵒᵖ

The opposite of a finite category is finite.

Equations

• One or more equations did not get rendered due to their size.

instance CategoryTheory.finCategoryUlift {J : Type v} [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] : CategoryTheory.FinCategory (CategoryTheory.ULiftHom (ULift.{w, v} J))

Applying ULift to morphisms and objects of a category preserves finiteness.

Equations

• CategoryTheory.finCategoryUlift = { fintypeObj := ULift.fintype J, fintypeHom := fun (x x_1 : CategoryTheory.ULiftHom (ULift.{w, v} J)) => ULift.fintype (x.objDown ⟶ x_1.objDown) }