nLab Kleisli object



2-Category theory

Higher algebra



Given a monad t:aat \colon a \to a in a 2-category KK, the Kleisli object a ta_t of tt is, if it exists, the universal right tt-module or tt-opalgebra. Equivalently, a ta_t represents the functor RMod(,t)RMod(-,t) that takes an object xx of KK to the category of right tt-modules axa \to x.

This means that there is a “free” 1-morphism f t:aa tf_t \colon a \to a_t and a 2-morphism α:f ttf t\alpha \colon f_t t \Rightarrow f_t that induce an isomorphism K(a t,x)RMod(x,t)K(a_t,x) \cong RMod(x,t): given a right tt-module r:ax,ρ:rtrr \colon a \to x, \rho \colon r t \to r, there is a unique morphism h:a txh \colon a_t \to x whose composite with f tf_t (respectively α\alpha) is equal to rr (resp. ρ\rho).



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


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


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

Every functor BAB \to A yields a monad A(f,f)A(f,f) in ProfProf, whose Kleisli object is part of the (bijective on objects, fully-faithful) factorization BA A(f,f)AB \to A_{A(f,f)} \to A of ff.

Because of this, we can identify a monad on AA in ProfProf with a bijective-on-objects functor ABA \to B.



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

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

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


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


An original article is

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

Last revised on January 23, 2024 at 15:17:41. See the history of this page for a list of all contributions to it.