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Mathlib.CategoryTheory.Category.KleisliCat

The Kleisli construction on the Type category #

Define the Kleisli category for (control) monads. CategoryTheory/Monad/Kleisli defines the general version for a monad on C, and demonstrates the equivalence between the two.

TODO #

Generalise this to work with CategoryTheory.Monad

def CategoryTheory.KleisliCat :
(Type u → Type v)Type (u + 1)

The Kleisli category on the (type-)monad m. Note that the monad is not assumed to be lawful yet.

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    Construct an object of the Kleisli category from a type.

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      • One or more equations did not get rendered due to their size.
      theorem CategoryTheory.KleisliCat.comp_def {m : Type u_1 → Type u_2} [Monad m] (α : CategoryTheory.KleisliCat m) (β : CategoryTheory.KleisliCat m) (γ : CategoryTheory.KleisliCat m) (xs : α β) (ys : β γ) (a : α) :
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      • CategoryTheory.instInhabitedMkId = { default := let_fun this := default; this }