# mathlibdocumentation

order.category.Preorder

# Category of preorders #

This defines Preorder, the category of preorders with monotone maps.

def Preorder  :
Type (u_1+1)

The category of preorders.

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Instances for Preorder
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def Preorder.of (α : Type u_1) [preorder α] :

Construct a bundled Preorder from the underlying type and typeclass.

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@[simp]
theorem Preorder.coe_of (α : Type u_1) [preorder α] :
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def Preorder.preorder (α : Preorder) :
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def Preorder.iso.mk {α β : Preorder} (e : α ≃o β) :
α β

Constructs an equivalence between preorders from an order isomorphism between them.

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theorem Preorder.iso.mk_inv {α β : Preorder} (e : α ≃o β) :
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theorem Preorder.iso.mk_hom {α β : Preorder} (e : α ≃o β) :

order_dual as a functor.

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theorem Preorder.dual_map (X Y : Preorder) (ᾰ : X →o Y) :
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theorem Preorder.dual_obj (X : Preorder) :
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The equivalence between Preorder and itself induced by order_dual both ways.

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theorem Preorder_to_Cat_obj (X : Preorder) :

The embedding of Preorder into Cat.

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Instances for Preorder_to_Cat
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theorem Preorder_to_Cat_map (X Y : Preorder) (f : X Y) :
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