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 $U\dashv F$ for the statement $U$ is left adjoint to $F$.

Now, let $U\dashv F$. We set $T=UF$. Then we have natural transformations

$\mu :UFUF->UF$ $\mu _{X}=U\epsilon _{FX}$

$\iota :1->UF$ $\iota _{X}=\eta _{X}$

such that $\mu$ is associative and $\iota$ is the unit of $\mu$.

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.

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.

Some adjunctions we already know

• initial/terminal are adjunctions.
• (co)-products are adjunctions.
• Actually, all (co)limits are adjunctions.

Some adjunctions we don't know yet

• Existential and universal qualifiers as adjunctions.
• Powersets and im(f) -| f^\inv

Properties of adjoints

RAPL: Right Adjoints Preserve Limits

Recognizing adjoints

Theorem (Freyd: The Adjoint Functor Theorem)

Why should we care in CS?

Monads