Kleisli Category - Formal Definition

Formal Definition

Let〈T, η, μ〉be a monad over a category C. The Kleisli category of C is the category CT whose objects and morphisms are given by

\begin{align}\mathrm{Obj}({\mathcal{C}_T}) &= \mathrm{Obj}({\mathcal{C}}), \\ \mathrm{Hom}_{\mathcal{C}_T}(X,Y) &= \mathrm{Hom}_{\mathcal{C}}(X,TY).\end{align}

That is, every morphism f: X → T Y in C (with codomain TY) can also be regarded as a morphism in CT (but with codomain Y). Composition of morphisms in CT is given by

where f: X → T Y and g: Y → T Z. The identity morphism is given by the monad unit η:

.

An alternative way of writing this, which clarifies the category in which each object lives, is used by Mac Lane. We use very slightly different notation for this presentation. Given the same monad and category as above, we associate with each object in a new object, and for each morphism in a morphism . Together, these objects and morphisms form our category, where we define

Then the identity morphism in is

Read more about this topic:  Kleisli Category

Famous quotes containing the words formal and/or definition:

    “The conviction that the best way to prepare children for a harsh, rapidly changing world is to introduce formal instruction at an early age is wrong. There is simply no evidence to support it, and considerable evidence against it. Starting children early academically has not worked in the past and is not working now.”
    —David Elkind (20th century)

    “... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
    —George Eliot [Mary Ann (or Marian)