nLab
Kleisli category of a comonad

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

• For an idempotent comonad the co-Kleisli category is the coreflective subcategory of its modal types.

• For $P= Jet$ a jet comonad, then morphisms in its coKleisli category are differential operators.

• The Kleisli category of a comonad can be used to model call-by-name? programming languages.

Related concepts

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

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)