Okay, I've finally had a moment to relax and think. I _did_ get it backwards. My answer annoyed me at the time, because it went against my mnemonic for right and left adjoints. I'm much happier now!

[Cole D. Pruitt](https://forum.azimuthproject.org/discussion/comment/16584/#Comment_16584) wrote:

> I'm afraid I'm still confused about the answers to **Puzzles 12 and 13** that are re-quoted here in the lecture material - I get a right adjoint \\(g(n) = \lfloor n/2 \rfloor \\), not \\( \lceil n/2 \rceil\\) as is given in the lecture. Or (much less likely!) there's an error in the lecture, but I figure someone would have caught it by now.

You're the one who caught the error! I've fixed it. Here's the correct story:

But if you did the puzzles, you saw that \\(f\\) has a "right adjoint" \\(g : \mathbb{N} \to \mathbb{N}\\). This is defined by the property

$$ f(a) \le b \textrm{ if and only if } a \le g(b) . $$

or in other words,

$$ 2a \le b \textrm{ if and only if } a \le g(b) .$$

Using our knowledge of fractions, we have

$$ 2a \le b \textrm{ if and only if } a \le b/2 $$

but since \\(a\\) is a natural number, this implies

$$ 2a \le b \textrm{ if and only if } a \le \lfloor b/2 \rfloor $$

where we are using the [floor function](https://en.wikipedia.org/wiki/Floor_and_ceiling_functions) to pick out the largest integer \\(\le b/2\\). So,

$$ g(b) = \lfloor b/2 \rfloor. $$

Moral: the right adjoint \\(g \\) is the "best approximation from below" to the nonexistent inverse of \\(f\\).

If you did the puzzles, you also saw that \\(f\\) has a left adjoint! This is the "best approximation from above" to the nonexistent inverse of \\(f\\): it gives you the smallest integer that's \\(\ge n/2\\).

So, while \\(f\\) has no inverse, it has two "approximate inverses". The left adjoint comes as close as possible to the (perhaps nonexistent) correct answer while making sure to never choose a number that's _too small_. The right adjoint comes as close as possible while making sure to never choose a number that's _too big_.

The two adjoints represent two opposing philosophies of life: _make sure you never ask for too little_ and _make sure you never ask for too much_. This is why they're philosophically profound. But the great thing is that they are defined in a completely precise, systematic way that applies to a huge number of situations!

If you need a mnemonic to remember which is which, remember left adjoints are "left-wing" or "liberal" or "generous", while right adjoints are "right-wing" or "conservative" or "cautious".

[Cole D. Pruitt](https://forum.azimuthproject.org/discussion/comment/16584/#Comment_16584) wrote:

> I'm afraid I'm still confused about the answers to **Puzzles 12 and 13** that are re-quoted here in the lecture material - I get a right adjoint \\(g(n) = \lfloor n/2 \rfloor \\), not \\( \lceil n/2 \rceil\\) as is given in the lecture. Or (much less likely!) there's an error in the lecture, but I figure someone would have caught it by now.

You're the one who caught the error! I've fixed it. Here's the correct story:

But if you did the puzzles, you saw that \\(f\\) has a "right adjoint" \\(g : \mathbb{N} \to \mathbb{N}\\). This is defined by the property

$$ f(a) \le b \textrm{ if and only if } a \le g(b) . $$

or in other words,

$$ 2a \le b \textrm{ if and only if } a \le g(b) .$$

Using our knowledge of fractions, we have

$$ 2a \le b \textrm{ if and only if } a \le b/2 $$

but since \\(a\\) is a natural number, this implies

$$ 2a \le b \textrm{ if and only if } a \le \lfloor b/2 \rfloor $$

where we are using the [floor function](https://en.wikipedia.org/wiki/Floor_and_ceiling_functions) to pick out the largest integer \\(\le b/2\\). So,

$$ g(b) = \lfloor b/2 \rfloor. $$

Moral: the right adjoint \\(g \\) is the "best approximation from below" to the nonexistent inverse of \\(f\\).

If you did the puzzles, you also saw that \\(f\\) has a left adjoint! This is the "best approximation from above" to the nonexistent inverse of \\(f\\): it gives you the smallest integer that's \\(\ge n/2\\).

So, while \\(f\\) has no inverse, it has two "approximate inverses". The left adjoint comes as close as possible to the (perhaps nonexistent) correct answer while making sure to never choose a number that's _too small_. The right adjoint comes as close as possible while making sure to never choose a number that's _too big_.

The two adjoints represent two opposing philosophies of life: _make sure you never ask for too little_ and _make sure you never ask for too much_. This is why they're philosophically profound. But the great thing is that they are defined in a completely precise, systematic way that applies to a huge number of situations!

If you need a mnemonic to remember which is which, remember left adjoints are "left-wing" or "liberal" or "generous", while right adjoints are "right-wing" or "conservative" or "cautious".