nLab
Kleisli object

Definition

Given a monad $t:a\to a$ in a 2-category $K$, the Kleisli object $a_t$ of $t$ is, if it exists, the universal right $t$-module or $t$-opalgebra. Equivalently, $a_t$ represents the functor $\mathrm{RMod}(-,t)$ that takes an object $x$ of $K$ to the category of right $t$-modules $a\to x$.

This means that there is a morphism $f_t:a\to a_t$ and a 2-cell $\lambda :f_{t}t\Rightarrow f_t$ that induce an isomorphism $K(a_t,x)\cong \mathrm{RMod}(x,t)$: given a right $t$-module $r:a\to x,\alpha :rt\to r$, there is a unique morphism $a_t\to x$ whose composite with $f_t$ (repsectively $\lambda$) is equal to $r$ (resp. $\alpha$).

Examples

• The motivating example is that of Kleisli categories for monads in Cat.

• In a (locally ordered) bicategory of relations, the Kleisli object of a monad $t$ is part of a splitting of $t$ as an idempotent.

• For a monad $T:A⇸A$ in the bicategory Prof of profunctors, its Kleisli object consists of a category $A_T$ equipped with a bijective-on-objects functor $A\to A_T$. The category $A_T$ has the same objects as $A$, with hom-sets $A_T(a,b)=T(a,b)$. Identities and composition are given by the unit and multiplication of $T$.

  Every functor $B\to A$ yields a monad $A(f,f)$ in $\mathrm{Prof}$, whose Kleisli object is part of the (bijective on objects, fully-faithful) factorization $B\to A_{A(f,f)}\to A$ of $f$.

  Because of this, we can identify a monad on $A$ in $\mathrm{Prof}$ with a bijective-on-objects functor $A\to B$.

Remarks

• A Kleisli object in a 2-category $K$ is the same as an Eilenberg-Moore object in $K^{\mathrm{op}}$; see opposite 2-category. Kleisli objects for monads in $K^{\mathrm{co}}$ can be identified with Kleisli objects for comonads in $K$.

• A Kleisli object for a monad $t$ in $K$ can equivalently be defined as a particular sort of weighted 2-colimit, namely the lax colimit of the lax functor $*\to K$ corresponding to $t$.

References

• R. Street, The formal theory of monads, J. Pure Appl. Alg. 2, 149–168 (1972)