nLab Kleisli category of a comonad

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Context

2-Category theory

2-category theory

Structures on 2-categories

Higher algebra

higher algebra

universal algebra

Contents

Idea

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

$\tilde f \colon P(c_1) \longrightarrow c_2$

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

$\widetilde{f_2 \circ f_1} \colon P(c_1) \longrightarrow P(P(c_1)) \stackrel{P(\tilde f_1)}{\longrightarrow} P(c_2) \stackrel{\tilde f_2}{\longrightarrow} c_3 \,.$

Examples

Last revised on September 22, 2017 at 16:10:02. See the history of this page for a list of all contributions to it.