# Differences

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hask [2016/05/10 18:27] nikolaj |
hask [2016/10/16 15:09] (current) nikolaj |
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==== Note ==== | ==== Note ==== | ||

>**todo:** | >**todo:** | ||

- | >Hask as category: | ||

- | > | ||

> objects are types | > objects are types | ||

> | > | ||

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==== Discussion ==== | ==== Discussion ==== | ||

+ | Hask is the syntactic subset of Haskell which permits a whole bunch of operations that are seen in basic category theory. | ||

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With $\to$ and $\times$ etc., Hask is //almost// Cartesian closed. A particular problem is the polymorphic term 'undefined', which is defined to be term of every type. It prevents, for example, initial objects (i.e. there no analog to the "empty set"). Or when it comes to setting up the categorical product, the projections $\pi_1,\pi_2$ couldn't distinguish between $'(\text{undefined},\text{undefined})'$ and just $\text{undefined}$, spoiling uniqueness. See [[http://www.haskell.org/haskellwiki/Hask|Hask]] for more examples. | With $\to$ and $\times$ etc., Hask is //almost// Cartesian closed. A particular problem is the polymorphic term 'undefined', which is defined to be term of every type. It prevents, for example, initial objects (i.e. there no analog to the "empty set"). Or when it comes to setting up the categorical product, the projections $\pi_1,\pi_2$ couldn't distinguish between $'(\text{undefined},\text{undefined})'$ and just $\text{undefined}$, spoiling uniqueness. See [[http://www.haskell.org/haskellwiki/Hask|Hask]] for more examples. | ||

- | Hask is the largest category of types in Haskell but you can find more well behaved categories. See also the [[http://hackage.haskell.org/package/category-extras|category-extras]] package. | + | As far as some basic category theoretical concepts go, Hask is the largest collection of types in Haskell but by more restriction, you can find some actual and nicely behaved categories. See also the [[http://hackage.haskell.org/package/category-extras|category-extras]] package. |

Another deficit is that when it comes to passing functions as arguments, Haskell sees more than just Hask morphisms. A nicer definition of the largest category of Haskell would be if arrows weren't Haskell functions identified by function extensionally | Another deficit is that when it comes to passing functions as arguments, Haskell sees more than just Hask morphisms. A nicer definition of the largest category of Haskell would be if arrows weren't Haskell functions identified by function extensionally |