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A monad over a category
 is a triple
 , where  is an endofunctor of
 ,  is a natural transformation from the identity functor on
 , and  is a natural transformations from  to  , such that the following two properties hold:
These laws are illustrated in the following diagrams.
As an application, monads have been successfully applied in the field of functional programming. A pure functional program can have no side effects, but some computations are frequently much simpler with such behavior. Thus a mathematical model of computation such as a monad is needed. In this case, monads serve to represent state transformations, mutable variables, and interactions between a program and its environment. For further information in this regard, see http://www.nomaware.com/monads/html/.
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"monad" is owned by mathcam. [ full author list (2) | owner history (1) ]
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(view preamble)
Cross-references: information, variables, transformations, state, represent, side, functional, field, application, properties, identity functor, natural transformation, endofunctor, category
There are 8 references to this entry.
This is version 8 of monad, born on 2002-02-24, modified 2005-01-25.
Object id is 2614, canonical name is Monad.
Accessed 5696 times total.
Classification:
| AMS MSC: | 68Q70 (Computer science :: Theory of computing :: Algebraic theory of languages and automata) | | | 18C15 (Category theory; homological algebra :: Categories and theories :: Triples , algebras for a triple, homology and derived functors for triples) |
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Pending Errata and Addenda
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![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ T^3(\mathcal{C}) \ar[rr]^{T\mu_{... ...\ T^2(\mathcal{C}) \ar[rr]_{\mu_{\mathcal{C}}} && T(\mathcal{C}) } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/2614/l2h/img12.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ T^3(\mathcal{C}) \ar[rr]^{T\mu_{... ...\ T^2(\mathcal{C}) \ar[rr]_{\mu_{\mathcal{C}}} && T(\mathcal{C}) } } \end{xy}$")
![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ T(\mathrm{id}_{\mathcal{C}}(A)) ... ...}} \ar[ddll]_{\mathrm{id}^{\mathcal{C}}} \ \ && T(\mathcal{C}) } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/2614/l2h/img13.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ T(\mathrm{id}_{\mathcal{C}}(A)) ... ...}} \ar[ddll]_{\mathrm{id}^{\mathcal{C}}} \ \ && T(\mathcal{C}) } } \end{xy}$")
