# 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

μ:UFUF − > UF μX = UεFX

ι:1 − > UF ιX = ηX

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.

• 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:

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.

• 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