# User:Richard Pinch/sandbox-16

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triple, standard construction, on a category $\mathfrak C$

a covariant functor $T: \mathfrak C \mathop \rightarrow \limits \mathfrak C$ endowed with natural transformations $\eta : {\mathop{\rm Id}\nolimits} _ {\mathfrak C} \mathop \rightarrow \limits T$ and $\mu : T ^ {2} \mathop \rightarrow \limits T$ (here ${\mathop{\rm Id}\nolimits} _ {\mathfrak C}$ denotes the identity functor on $\mathfrak C$) such that the following diagrams are commutative:

$$\begin{array}{crclc} T (X) & \mathop \rightarrow \limits ^ {T ( \eta _ {X} )} &T ^ {2} (X) & \mathop \leftarrow \limits ^ {\eta _ {T (X)}} &T (X) \\ {} &{} _ {1 _ {T (X)}} \searrow &\scriptsize {\mu _ {X}} \downarrow &\swarrow _ {1 _ {T (X)}} &{} \\ {} &{} &T (X) &{} &{} \\ \end{array}$$

$$\begin{array}{rcl} T ^ {3} (X) & \mathop \rightarrow \limits ^ {T ( \mu _ {X} )} &T ^ {2} (X) \\ \scriptsize {\mu _ {T (X)}} \downarrow &{} &\downarrow \scriptsize {\mu _ {X}} \\ T ^ {2} (X) & \mathop \rightarrow \limits _ {\mu _ {X}} &T (X) \\ \end{array}$$

For any pair of adjoint functors $F : \mathfrak C \mathop \rightarrow \limits \mathfrak D$ and $G: \mathfrak D \mathop \rightarrow \limits \mathfrak C$ with unit and co-unit of adjunction $\eta : {\mathop{\rm Id}\nolimits} _ {\mathfrak C} \mathop \rightarrow \limits GF$ and $\epsilon : FG \mathop \rightarrow \limits {\mathop{\rm Id}\nolimits} _ {\mathfrak C}$, respectively, the functor $T = GF: \mathfrak C \mathop \rightarrow \limits \mathfrak C$ endowed with $\eta : {\mathop{\rm Id}\nolimits} _ {\mathfrak C} \mathop \rightarrow \limits T$ and $\mu = G ( \epsilon _ {F} ): T ^ {2} \mathop \rightarrow \limits T$ is a triple on $\mathfrak R$.

Conversely, for any triple $(T, \eta , \mu )$ there exist pairs of adjoint functors $F$ and $G$ such that $T = GF$, and the transformations $\eta$ and $\mu$ are obtained from the unit and co-unit of the adjunction in the manner described above. The different such decompositions of a triple may form a proper class. In this class there is a smallest element (the Kleisli construction) and a largest element (the Eilenberg–Moore construction).

The Kleisli category of a monad $(T,\eta,\mu)$ on $\mathfrak C$ has an object $\bar x$ for each object $x$ of $\mathfrak C$ and a map $f^\flat : \bar x \to \bar y$ for each map $f:x \to Ty$ in $\mathfrak C$ with composition of $f^\flat$ with $g^\flat : \bar y \to \bar z$ defined by $g^\flat \circ f^\flat = (\mu_z \circ Tg \circ f)^\flat$.

### Examples

1) In the category of sets, the functor $T$ which sends an arbitrary set to the set of all its subsets has the structure of a monad. The transformation $\eta$ is the natural embedding of a set $X$ in the set of its subsets via singleton sets, and to each set of subsets of $X$ one associates the union of these subsets.

2) In the category of sets, every representable functor $H _ {A} (X) = H (A, X)$ carries a triple: The mapping $\eta _ {X} : X \mathop \rightarrow \limits H (A, X)$ associates to each $x \in X$ the constant function $f _ {x} : A \mathop \rightarrow \limits X$ with value $x$; the mapping $\mu _ {X} : H (A, H (A, X)) \simeq H (A \times A, X) \mathop \rightarrow \limits H (A, X)$ associates to each function of two variables its restriction to the diagonal.

3) In the category of topological spaces, each topological group $G$, with unit $e$, enables one to define a functor $T _ {G} (X) = X \times G$ that carries a triple: Each element $x \in X$ is taken to the element $(x, e)$ and the mapping $\mu : X \times G \times G \mathop \rightarrow \limits X \times G$ is defined by $\mu _ {X} (x, g, g ^ \prime ) = (x, gg ^ \prime )$.

4) In the category of modules over a commutative ring $R$, each (associative, unital) $R$- algebra $A$ gives rise to a triple structure on the functor $T _ {A} (X) = X \otimes _ {R} A$, in a manner similar to Example 3).

How to Cite This Entry:
Richard Pinch/sandbox-16. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-16&oldid=51747