Kleisli category of a comonad



2-Category theory

Higher algebra



Formally dually to how a monad has a Kleisli category so also a comonad P:𝒞𝒞P \colon \mathcal{C}\to\mathcal{C} has a (co-)Kleisli category: its objects are the objects of 𝒞\mathcal{C}, a morphism f:c 1c 2f \colon c_1 \to c_2 in the co-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 co-Kleisli category is represented by the morphism in 𝒞\mathcal{C} given by

f 2f 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 \,.



Some introductory material on comonads, coalgebras and co-Kleisli morphisms can be found in

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

Last revised on January 14, 2020 at 23:45:13. See the history of this page for a list of all contributions to it.