Talk:Kleisli category

Latest comment: 8 years ago by Beroal in topic Kleisli monoid

Non-monadic definition edit

Hi. I have an alternative description of a Kleisli category which does not rely on monads.

Given a category C <ObjC, HomC, idC, compC>, and a mapping T from ObjC to ObjC (so T is not a functor as it does not go from a category to another one), a Kleisli category over <C, T> is a category <ObjC, λ a b. HomC(a, T b), idK, compK> (so it shares the same objects than C, and its morphisms are defined from those of C) such that

∀ f g h. compK(f, compC(g, h)) = compC(compK(f, g), h).

From that, we can extend T to a functor and define (η X) as (idK X) and (μ X) as (compK(idC (T (T X)), idC (T X))), and show that the monad laws are verified (I did it in the Coq proof assistant).

Sedrikov (talk) 8 January 2013‎

Kleisli monoid edit

In particular, the Kleisli construction can be applied to monoids. If   is a monoid,   is a monoid homomorphism,  ,  ,  ,  , and  , the Kleisli monoid has the same underlying set as   but with identity   and multiplication  . GeoffreyT2000 (talk) 03:10, 17 May 2015 (UTC)Reply

By monoid you mean a category with 1 object? --Beroal (talk) 18:39, 17 May 2015 (UTC)Reply