Kleisli category

Summary

The Kleisli category of a monad $T$ on a category $\mathcal{C}$ has the same objects as $\mathcal{C}$. A morphism from $X$ to $Y$ in it is a morphism $X \to T Y$ of $\mathcal{C}$. Composition uses the multiplication of the monad to flatten $T T$ to $T$. It is equivalent to the full subcategory of free algebras in the Eilenberg–Moore category. Among all adjunctions that induce the monad, it gives the initial one. In computer science, Kleisli morphisms model programs with side effects, such as partiality, nondeterminism or state.