Joins and meets are much more deeply connected to left and right adjoints than I've admitted so far. To dig a little deeper, let's prove this result I mentioned [last time](https://forum.azimuthproject.org/discussion/2013/lecture-14-chapter-1-adjoints-joins-and-meets/p1):

**Theorem.** Left adjoints preserve joins and right adjoints preserve meets. Suppose \\(f : A \to B\\) and \\(g : B \to A\\) are monotone functions between posets. Suppose that \\(f\\) is the left adjoint of \\(g\\), or equivalently, \\(g\\) is the right adjoint of \\(f\\). If the join of \\(a,a' \in A\\) exists then so does the join of \\(f(a), f(a') \in B\\), and

\[ f(a \vee a') = f(a) \vee f(a'). \]

If the meet of \\(b,b' \in B\\) exists then so does the meet of \\(g(b), g(b') \in A\\), and

\[ g(b \wedge b') = g(b) \wedge g(b'). \]

**Proof.** First, suppose \\( f : A \to B\\) is _any_ monotone function and the join \\(a \vee a'\\) exists. By definition, \\( a \vee a' \\) is the least upper bound of \\( a \\) and \\( a' \\), so

\[ a \le a \vee a' \textrm{ and } a' \le a \vee a' . \]

Since \\( f \\) is monotone this gives

\[ f(a) \le f(a \vee a') \textrm{ and } f(a') \le f(a \vee a') .\]

This says that \\( f(a \vee a') \\) is an upper bound of \\( f(a) \\) and \\( f(a') \\).

To finish the job, we need to bring in our assumption that \\( f \\) is the left adjoint of \\( g \\). Let's show that \\( f(a \vee a') \\) is the _least_ upper bound of \\( f(a) \\) and \\( f(a') \\). So, suppose \\( b \\) is any other upper bound of these guys:

\[ f(a) \le b \textrm{ and } f(a') \le b. \]

Since \\( f \\) is the left adjoint of \\( g \\), this means

\[ a \le g(b) \textrm{ and } a' \le g(b). \]

This says that \\( g(b) \\) is an upper bound of \\( a \\) and \\( a' \\). Since \\( a \vee a' \\) is the _least_ upper bound, we have

\[ a \vee a' \le g(b) . \]

Since \\( f \\) is the left adjoint of \\( g \\), this means

\[ f(a \vee a') \le b. \]

So, \\( f(a \vee a') \\) is indeed the _least_ upper bound of \\( f(a) \\) and \\( f(a') \\). This implies that \\( f(a) \vee f(a') \\) actually exists, and that

\[ f(a \vee a') = f(a) \vee f(a'). \]

The argument that right adjoints preserve meets works exactly the same way, with all the inequalities reversed. \\( \quad \blacksquare \\)

To check that you understand this material, come up with the similar argument for right adjoints when you're on a boring train or airplane trip - without looking at any notes.

It's worth noting a little spinoff of the argument I gave. If \\( f : A \to B \\) is _any_ monotone function between posets,

\[ f(a) \vee f(a') \le f(a \vee a') \]

when the joins here actually exist. Assuming that \\( f \\) is a left adjoint gives us the reverse inequality. Similarly, if \\( g : B \to A \\) is _any_ monotone function between posets,

\[ g(b \wedge b') \le g(b) \wedge g(b') \]

when the meets actually exist. Assuming that \\( g \\) is a right adjoint gives us the reverse inequality.

Our next step will be to strengthen the theorem we proved today. The idea doesn't just apply to "binary" meets and joins - that is, meets and joins of a pair of elements. It applies to meets and joins of _arbitrary_ subsets. And when we state it that way, a kind of converse is true too.

Suppose, for example, we have two posets that have all meets and joins. Then a monotone map between them preserves joins of arbitrary subsets _if and only if_ it's a left adjoint! And it preserves meets of arbitrary subsets _if and only if_ it's a right adjoint!

This is staggeringly beautiful. It's called the "Adjoint Functor Theorem for Posets". We'll prove it next time.

Why does it have that name? It's a special case of a more general theorem about adjoint functors for categories. You see, all our results about posets can be generalized to categories, since:

* posets are a special case of [categories](https://ncatlab.org/nlab/show/category#definitions),

* monotone functions between posets are a special case of [functors](https://ncatlab.org/nlab/show/functor#definition) between categories,

* Galois connections between posets are a special case of [adjoint functors](https://ncatlab.org/nlab/show/adjoint+functor#definition) between categories,

* left adjoint monotone functions are a special case of [left adjoint functors](https://ncatlab.org/nlab/show/adjoint+functor#definition),

* right adjoint monotone functions are a special case of [right adjoint functors](https://ncatlab.org/nlab/show/adjoint+functor#definition),

* meets are a special case of [limits](https://en.wikipedia.org/wiki/Limit_(category_theory)#Definition), and

* joins are a special case of [colimits](https://en.wikipedia.org/wiki/Limit_(category_theory)#Colimits).

The terms in blue are some of the most important concepts in category theory. Fong and Spivak's strategy is to help you develop an intuition for these concepts by working with posets before diving into more general categories. The [Adjoint Functor Theorem](https://ncatlab.org/nlab/show/adjoint+functor+theorem), in particular, is rather tricky to prove. But the version for posets is much easier.

**[To read other lectures go here.](http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory#Course)**

**Theorem.** Left adjoints preserve joins and right adjoints preserve meets. Suppose \\(f : A \to B\\) and \\(g : B \to A\\) are monotone functions between posets. Suppose that \\(f\\) is the left adjoint of \\(g\\), or equivalently, \\(g\\) is the right adjoint of \\(f\\). If the join of \\(a,a' \in A\\) exists then so does the join of \\(f(a), f(a') \in B\\), and

\[ f(a \vee a') = f(a) \vee f(a'). \]

If the meet of \\(b,b' \in B\\) exists then so does the meet of \\(g(b), g(b') \in A\\), and

\[ g(b \wedge b') = g(b) \wedge g(b'). \]

**Proof.** First, suppose \\( f : A \to B\\) is _any_ monotone function and the join \\(a \vee a'\\) exists. By definition, \\( a \vee a' \\) is the least upper bound of \\( a \\) and \\( a' \\), so

\[ a \le a \vee a' \textrm{ and } a' \le a \vee a' . \]

Since \\( f \\) is monotone this gives

\[ f(a) \le f(a \vee a') \textrm{ and } f(a') \le f(a \vee a') .\]

This says that \\( f(a \vee a') \\) is an upper bound of \\( f(a) \\) and \\( f(a') \\).

To finish the job, we need to bring in our assumption that \\( f \\) is the left adjoint of \\( g \\). Let's show that \\( f(a \vee a') \\) is the _least_ upper bound of \\( f(a) \\) and \\( f(a') \\). So, suppose \\( b \\) is any other upper bound of these guys:

\[ f(a) \le b \textrm{ and } f(a') \le b. \]

Since \\( f \\) is the left adjoint of \\( g \\), this means

\[ a \le g(b) \textrm{ and } a' \le g(b). \]

This says that \\( g(b) \\) is an upper bound of \\( a \\) and \\( a' \\). Since \\( a \vee a' \\) is the _least_ upper bound, we have

\[ a \vee a' \le g(b) . \]

Since \\( f \\) is the left adjoint of \\( g \\), this means

\[ f(a \vee a') \le b. \]

So, \\( f(a \vee a') \\) is indeed the _least_ upper bound of \\( f(a) \\) and \\( f(a') \\). This implies that \\( f(a) \vee f(a') \\) actually exists, and that

\[ f(a \vee a') = f(a) \vee f(a'). \]

The argument that right adjoints preserve meets works exactly the same way, with all the inequalities reversed. \\( \quad \blacksquare \\)

To check that you understand this material, come up with the similar argument for right adjoints when you're on a boring train or airplane trip - without looking at any notes.

It's worth noting a little spinoff of the argument I gave. If \\( f : A \to B \\) is _any_ monotone function between posets,

\[ f(a) \vee f(a') \le f(a \vee a') \]

when the joins here actually exist. Assuming that \\( f \\) is a left adjoint gives us the reverse inequality. Similarly, if \\( g : B \to A \\) is _any_ monotone function between posets,

\[ g(b \wedge b') \le g(b) \wedge g(b') \]

when the meets actually exist. Assuming that \\( g \\) is a right adjoint gives us the reverse inequality.

Our next step will be to strengthen the theorem we proved today. The idea doesn't just apply to "binary" meets and joins - that is, meets and joins of a pair of elements. It applies to meets and joins of _arbitrary_ subsets. And when we state it that way, a kind of converse is true too.

Suppose, for example, we have two posets that have all meets and joins. Then a monotone map between them preserves joins of arbitrary subsets _if and only if_ it's a left adjoint! And it preserves meets of arbitrary subsets _if and only if_ it's a right adjoint!

This is staggeringly beautiful. It's called the "Adjoint Functor Theorem for Posets". We'll prove it next time.

Why does it have that name? It's a special case of a more general theorem about adjoint functors for categories. You see, all our results about posets can be generalized to categories, since:

* posets are a special case of [categories](https://ncatlab.org/nlab/show/category#definitions),

* monotone functions between posets are a special case of [functors](https://ncatlab.org/nlab/show/functor#definition) between categories,

* Galois connections between posets are a special case of [adjoint functors](https://ncatlab.org/nlab/show/adjoint+functor#definition) between categories,

* left adjoint monotone functions are a special case of [left adjoint functors](https://ncatlab.org/nlab/show/adjoint+functor#definition),

* right adjoint monotone functions are a special case of [right adjoint functors](https://ncatlab.org/nlab/show/adjoint+functor#definition),

* meets are a special case of [limits](https://en.wikipedia.org/wiki/Limit_(category_theory)#Definition), and

* joins are a special case of [colimits](https://en.wikipedia.org/wiki/Limit_(category_theory)#Colimits).

The terms in blue are some of the most important concepts in category theory. Fong and Spivak's strategy is to help you develop an intuition for these concepts by working with posets before diving into more general categories. The [Adjoint Functor Theorem](https://ncatlab.org/nlab/show/adjoint+functor+theorem), in particular, is rather tricky to prove. But the version for posets is much easier.

**[To read other lectures go here.](http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory#Course)**