User:Michiexile/MATH198/Lecture 7
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Last week we saw what an adjunction was. Here's one thing we can do with adjunctions.
One piece of notation we didn't cover last week was writing for the statement U is left adjoint to F.
Now, let . We set T = UF. Then we have natural transformations
μ:UFUF − > UF μ_{X} = Uε_{FX}
ι:1 − > UF ι_{X} = η_{X}
such that μ is associative and ι is the unit of μ.
These requirements remind us of the definition of a monoid  and this is not that much of a surprise. To see the exact connection, and to garner a wider spread of definitions.
Contents 
1 Algebraic objects in categories
$THEORY objects in Categories.
Now, we call something a monad in a category if it is a monoid object in the category of endofunctors of that category.
We thus define a monad in a category C to be a monoid in that category.
Thus: a monad with this definition in Haskell is:
 a type of kind .m :: * > *
 equipped with functions
 return :: a > m a
 join :: m m a > m a

 examples:
List: return x = [x] join (l:lsts) = l ++ join lsts Maybe: return x = Just x join (Just (Just x)) = Just x join _ = Nothing
Note: not quite what Haskell claims a monad to be. Other related concepts:
 Kleisli category & factorization of monads: if we start with T a
monad, then can we find an adjunction U  F to _somewhere_ such that T = UF? And the monoidal structure is given by U e_FX and eta_X?
 From the Kleisli Category to monadic bind.
 Monads and sequencing.
2 Some adjunctions we already know
 initial/terminal are adjunctions.
 (co)products are adjunctions.
 Actually, all (co)limits are adjunctions.
3 Some adjunctions we don't know yet
 Existential and universal qualifiers as adjunctions.
 Powersets and im(f)  f^\inv
4 Properties of adjoints
4.1 RAPL: Right Adjoints Preserve Limits
4.2 Recognizing adjoints
Theorem (Freyd: The Adjoint Functor Theorem)