Kleisli Category

Idea

In category theory, a Kleisli category is a category associated with a monad that captures the notion of “computation with effects.” It provides a way to treat morphisms that return an object wrapped in a monad as regular arrows between objects.

Definition

Let $(T,\eta ,\mu )$ be a monad on a category $\mathcal{C}$. The Kleisli category of $T$, denoted ${\mathcal{C}}_T$, is defined as follows:

• Objects: The objects of ${\mathcal{C}}_T$ are exactly the objects of $\mathcal{C}$. For every object $X\in \text{ob}(\mathcal{C})$, there is a corresponding object $X\in \text{ob}({\mathcal{C}}_T)$.
• Morphisms: A morphism $f:X\to Y$ in ${\mathcal{C}}_T$ is defined as a morphism $f:X\to TY$ in the base category $\mathcal{C}$.
• Identity: For any object $X$, the identity morphism $1_X$ in ${\mathcal{C}}_T$ is the unit of the monad at $X$ in the base category: ${\eta }_X:X\to TX$.
• Composition: Given morphisms $f:X\to Y$ and $g:Y\to Z$ in ${\mathcal{C}}_T$ (which are $f:X\to TY$ and $g:Y\to TZ$ in $\mathcal{C}$), their Kleisli composition $g{\circ }_{T}f:X\to Z$ is defined using the monad multiplication $\mu$: $g{\circ }_{T}f={\mu }_Z\circ Tg\circ f$

The following diagram illustrates this composition in the base category $\mathcal{C}$:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
X \arrow[r, "f"] \arrow[rr, "g \circ_T f", bend right=40] & TY \arrow[r, "Tg"] & T^2Z \arrow[r, "\mu_Z"] & TZ
\end{tikzcd}
\end{document}

The Kleisli Adjunction

The Kleisli category is the “smallest” (initial) resolution of a monad into an adjunction. There exists a canonical adjunction $F_T⊣U_T$ between $\mathcal{C}$ and ${\mathcal{C}}_T$:

1. Free Functor $F_T:\mathcal{C}\to {\mathcal{C}}_T$:
   • On objects: $F_T(X)=X$.
   • On morphisms: $F_T(f)={\eta }_Y\circ f$ (where $f:X\to Y$).
2. Forgetful Functor $U_T:{\mathcal{C}}_T\to \mathcal{C}$:
   • On objects: $U_T(X)=TX$.
   • On morphisms: $U_T(f)={\mu }_Y\circ Tf$ (where $f:X\to TY$ in $\mathcal{C}$).

This adjunction induces the original monad $T=U_T{F}_T$.

Significance

The Kleisli category is extensively used in computer science (particularly in functional programming) to model various computational effects such as state, exceptions, and non-determinism. Morphisms in the Kleisli category represent “effectful programs,” while the composition represents the sequential execution of these programs.

References

• maclane1978-categories
• riehl2016-category-theory
• kleisli1965-kleisli-category