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monad [2015/02/17 18:47] nikolaj |
monad [2015/10/05 16:23] nikolaj |
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| @#55EE55: postulate | @#55EE55: $\mu\circ T\eta=\mu\circ\eta T=1_T$ | | | @#55EE55: postulate | @#55EE55: $\mu\circ T\eta=\mu\circ\eta T=1_T$ | | ||
- | ==== Discussion ==== | + | ----- |
- | === Idea === | + | === Discussion === |
A monad is functor together with two natural transformations that fulfill some algebraic relations. | A monad is functor together with two natural transformations that fulfill some algebraic relations. | ||
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== Algebraic characterization == | == 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 triple given by | + | 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 |
$M$, | $M$, | ||
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$\mu:TT\xrightarrow{\bullet}T$ | $\mu:TT\xrightarrow{\bullet}T$ | ||
- | an 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 $S\otimes T:= ST$. | + | 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 === | === Reference === | ||
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[[http://en.wikipedia.org/wiki/Monad_%28category_theory%29|Monad (category theory)]], | [[http://en.wikipedia.org/wiki/Monad_%28category_theory%29|Monad (category theory)]], | ||
[[http://de.wikipedia.org/wiki/Monade_%28Informatik%29|Monade (Informatik)]] | [[http://de.wikipedia.org/wiki/Monade_%28Informatik%29|Monade (Informatik)]] | ||
- | ==== Parents ==== | + | |
+ | ----- | ||
=== Context === | === Context === | ||
=== Requirements === | === Requirements === |