===== Monad =====
==== Collection ====
| @#55CCEE: context     | @#55CCEE: ${\bf C}$ ... category |
| @#FFBB00: definiendum | @#FFBB00: $\langle T,\eta,\mu\rangle$ in $\mathrm{it}$ |
| @#AAFFAA: inclusion   | @#AAFFAA: $T$ in ${\bf C}^{\bf C}$ |
| @#AAFFAA: inclusion   | @#AAFFAA: $\eta:1_{\bf C}\xrightarrow{\bullet}T$ |
| @#AAFFAA: inclusion   | @#AAFFAA: $\mu:TT\xrightarrow{\bullet}T$ |
| @#55EE55: postulate   | @#55EE55: $\mu\circ T\mu=\mu\circ\mu T$  |
| @#55EE55: postulate   | @#55EE55: $\mu\circ T\eta=\mu\circ\eta T=1_T$  |

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=== Discussion ===
A monad is functor together with two natural transformations that fulfill some algebraic relations. 

=== Elaboration ===
Written out in component form, the postulates read 

  * $\mu_X\circ T(\mu_X)=\mu_X\circ\mu_{TX}$ 

  * $\mu_X\circ T(\eta_X)=\mu_X\circ\eta_{TX}=1_{TX}$.

Picture the following chain of objects and notice that the axioms merely say the following: All ways ending up at $TX$ must be the same.

$X   \overset{\eta_X}{\rightarrow}   TX   \overset{T(\eta_{TX})\ \text{or}\ \eta_{TX}}{\rightarrow}\ \ \overset{\mu_X}{\leftarrow}   TTX   \overset{T(\mu_X)\ \text{or}\ \mu_{TX}}{\leftarrow}   TTTX$

=== Alternative definition ===
== From adjunctions ==
All monads arise by transferring some co-unit into the other category: Set $T:=GF$ and $\mu_Y:=G(\varepsilon_{FY})$ and you got yourself a monad $\langle T,\eta,\mu\rangle$. 

=== Terminology/Notation ===
The forward arrows $\eta_X$ are called „unit“ or, in programmer circles, „return“. The backwards arrows $\mu_X$ are called „join“ and from the [[counit-unit adjunction]] perspective, they are the arrow images of the functor applied to the „co-unit“ components.

== Algebraic characterization ==
Recall that $\bf{Set}$ can be equipped with a monoidal structure where units are singletons (= final objects in $\bf{Set}$, e.g. $1:=\{0\}$) and the product $\otimes$ can be taken to be the Cartesian product $M\otimes N:=M\times N$ (= categorical product for $\bf{Set}$). Here, a monoid object is a triple given by

$M$,

$e:1\to M$,

$*:M\times M\to M$.

Now a monad is given by

$T$,

$\eta:1_{\bf C}\xrightarrow{\bullet}T$,

$\mu:TT\xrightarrow{\bullet}T$ 

and is also a monoid object, namely in the category of endofunctors ${\bf C}^{\bf C}$, with the monoidal product $\otimes$ (not the categorical product) given by concatenation of functors $S\otimes T:= ST$.

=== Reference ===
Wikipedia: 
[[http://en.wikipedia.org/wiki/Monad_%28category_theory%29|Monad (category theory)]],
[[http://de.wikipedia.org/wiki/Monade_%28Informatik%29|Monade (Informatik)]]

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=== Context ===
=== Requirements ===
[[Functor]], [[Natural transformation]], [[Identity functor]]
=== Related ===
[[Monoid]]