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 Failed to parse (unknown function\turnstile): U\turnstile F
for the statement U is left adjoint to F.
Now, let Failed to parse (unknown function\turnstile): U\turnstile F . 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)