Kleisli category of a comonad in nLab

Contents

Contents

Idea

Formally dually to how a monad has a Kleisli category so also a comonad P:𝒞→𝒞P \colon \mathcal{C}\to\mathcal{C} has a Kleisli category: its objects are the objects of 𝒞\mathcal{C}, a morphism f:c 1→c 2f \colon c_1 \to c_2 in the Kleisli category is a morphism

f˜:P(c 1)⟶c 2 \tilde f \colon P(c_1) \longrightarrow c_2

in 𝒞\mathcal{C}, and the composition of two such in the Kleisli category is represented by the morphism in 𝒞\mathcal{C} given by

f 2∘f 1˜:P(c 1)⟶P(P(c 1))⟶P(f˜ 1)P(c 2)⟶f˜ 2c 3. \widetilde{f_2 \circ f_1} \colon P(c_1) \longrightarrow P(P(c_1)) \stackrel{P(\tilde f_1)}{\longrightarrow} P(c_2) \stackrel{\tilde f_2}{\longrightarrow} c_3 \,.

Sometimes the term co-Kleisli category is encountered, but this is redundant, since there is only one notion of Kleisli category for a comonad.

Examples

General

Specific

• Kleisli category
• Eilenberg-Moore category
• monad (in computer science)

References

The equivalence of categories between the Kleisli category over a given comonad with the Kleisli category of an adjoint monad (if it exists):

• Mark Kleiner, Adjoint monads and an isomorphism of the Kleisli categories, Journal of Algebra

  Volume 133 1 (1990) 79-82 [doi:10.1016/0021-8693(90)90069-Z]

Some introductory material on comonads, coalgebras and co-Kleisli morphisms:

• Paolo Perrone, Notes on Category Theory with examples from basic mathematics, Chapter 5. (arXiv)