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monad [2015/02/17 18:47] nikolaj |
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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 === | ||
