# nLab Kleisli category of a comonad

### 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

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

## References

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)