Kleisli category
Noun
Meaning
Commutative diagram of function composition in a Kleisli category. Given a monad
(
T
,
η
,
μ
)
=
(
T
:
C
→
C
,
η
:
id
C
→
T
,
μ
:
T
2
→
T
)
{\displaystyle (T,\eta,\mu )=(T:{\mathcal {C}}\rightarrow {\mathcal {C}},\ \eta :{\mbox{id}}_{\mathcal {C}}\rightarrow T,\ \mu :T^{2}\rightarrow T)}
, consider a Kleisli category
K
{\displaystyle {\mathcal {K}}}
over that monad. Morphisms
f
~
:
A
→
B
{\displaystyle {\tilde {f}}:A\rightarrow B}
,
g
~
:
B
→
C
{\displaystyle {\tilde {g}}:B\rightarrow C}
, and
h
~
=
g
~
∘
f
~
:
A
→
C
{\displaystyle {\tilde {h}}={\tilde {g}}\circ {\tilde {f}}:A\rightarrow C}
in
K
{\displaystyle {\mathcal {K}}}
correspond to morphisms
f
:
A
→
T
B
{\displaystyle f:A\rightarrow TB}
,
g
:
B
→
T
C
{\displaystyle g:B\rightarrow TC}
, and
h
:
A
→
T
C
{\displaystyle h:A\rightarrow TC}
in
C
{\displaystyle {\mathcal {C}}}
, respectively. The composition rule is
g
~
∘
f
~
=
(
μ
cod
(
g
)
∘
T
g
∘
f
)
∼
{\displaystyle {\tilde {g}}\circ {\tilde {f}}={(\mu _{{\text{cod}}(g)}\circ Tg\circ f)}^{\sim }}
. The Kleisli category shares the same objects as its underlying category. The morphisms of the Kleisli category (e.g.:
f
~
{\displaystyle {\tilde {f}}}
and
g
~
{\displaystyle {\tilde {g}}}
) are embellished versions of the morphisms of its underlying category (e.g.: f and g), and they are derived from those of the underlying category by means of applying a monad to their codomains.
Kleisli category (plural Kleisli categories)
(category theory) A category naturally associated to any monad T, and equivalent to the category of free T-algebras.