nLab Kleisli object

Definition

Given a monad $t \colon a \to a$ 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 $RMod(-,t)$ that takes an object $x$ of $K$ to the category of right $t$-modules $a \to x$.

This means that there is a “free” 1-morphism $f_t \colon a \to a_t$ and a 2-morphism $\alpha \colon f_t t \Rightarrow f_t$ that induce an isomorphism $K(a_t,x) \cong RMod(x,t)$: given a right $t$-module $r \colon a \to x, \rho \colon r t \to r$, there is a unique morphism $h \colon a_t \to x$ whose composite with $f_t$ (respectively $\alpha$) is equal to $r$ (resp. $\rho$).

Examples

Example

The motivating example is that of ordinary Kleisli categories $C_T$ for monads in the 2-category Cat, hence for ordinary monadic endofunctors $T$ on categories $C$.

Example

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

Example

For a monad $T \colon A ⇸ A$ 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 monad unit and multiplication of $T$.

Every functor $B \to A$ yields a monad $A(f,f)$ in $Prof$, 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 $Prof$ with a bijective-on-objects functor $A \to B$.

Properties

Remark

By 2-category theoretic formal duality (see at opposite 2-category):

1. Kleisli objects in a 2-category $K$ are equivalently Eilenberg-Moore objects in $K^{op}$.

2. Kleisli objects for monads in $K^{co}$ are equivalently Kleisli objects for comonads in $K$.

Remark

Kleisli objects for a monad $t$ in $K$ are equivalently a particular sort of weighted 2-colimits, namely the lax colimits of the lax functor $\ast \to K$ corresponding to $t$.

Related concepts

• Eilenberg-Moore object

References

An original article is

• Ross Street, The formal theory of monads, Journal of Pure and Applied Algebra 2 2 (1972) 149-168 [doi:10.1016/0022-4049(72)90019-9]

Generalization from monads to more general categories enriched in a bicategory (where “Kleisli objects” are then called “collages”) is in

• Richard Garner, Michael Shulman, Enriched categories as a free cocompletion,

  Advances in Mathematics 289 (2016) 1-94 [arXiv:1301.3191, doi:10.1016/j.aim.2015.11.012]