Mario RomΓ‘n

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Promonad

Last updated Aug 14, 2023

# Definition

promonad

A promonad $(β„‚,β˜…,{}^{∘})$ over a category $𝕍$ is a profunctor $β„‚ \colon 𝕍 \nrightarrow 𝕍$ together with transformations representing inclusion $({}^{∘}){X,Y} \colon 𝕍(X,Y) \to β„‚(X,Y)$ and multiplication $(β˜…){X,Y} \colon β„‚(X,Y) \times β„‚(Y,Z) \to β„‚(X,Z)$, and such that

Naturality of the multiplication, dinaturality of the multiplication and the functoriality of the unit, are redundant from the definition and can be proved from the rest of the axioms.

Equivalently, promonads are proarrow monoids in the double category of categories, where the dinatural multiplication represents a transformation from the composition of the profunctor $β„‚$ with itself.

# Motivation for promonads

motivation-for-promonads

# Promonad homomorphism

promonad-homomorphism

# Promonad natural transformations

promonad-natural-transformations

# See also

#definition