It looks like you're new here. If you want to get involved, click one of these buttons!

- All Categories 2.4K
- Chat 505
- Study Groups 21
- Petri Nets 9
- Epidemiology 4
- Leaf Modeling 2
- Review Sections 9
- MIT 2020: Programming with Categories 51
- MIT 2020: Lectures 20
- MIT 2020: Exercises 25
- Baez ACT 2019: Online Course 339
- Baez ACT 2019: Lectures 79
- Baez ACT 2019: Exercises 149
- Baez ACT 2019: Chat 50
- UCR ACT Seminar 4
- General 75
- Azimuth Code Project 111
- Statistical methods 4
- Drafts 10
- Math Syntax Demos 15
- Wiki - Latest Changes 3
- Strategy 113
- Azimuth Project 1.1K
- - Spam 1
- News and Information 148
- Azimuth Blog 149
- - Conventions and Policies 21
- - Questions 43
- Azimuth Wiki 718

Options

Does \( F := \lfloor −/3\rfloor \) have a right adjoint \( R : \mathbb{N} \rightarrow \mathbb{N} \) ?

If not, why?

If so, does its right adjoint have a right adjoint?

## Comments

This seems to now be Exercise 80 in the latest draft (April 20th).

\(F\) does indeed have a right adjoint \(R : \mathbb{N} \to \mathbb{N}\); it is defined by \(R(x) = 3(x + 1) - 1\). (One can alternately write \(3x + 2\), but this version changes less as you change the modulus.)

\(R\) does

nothave a right adjoint of its own. Suppose it did, and call it \(R'\). Then \(0 \le R'(0)\), since 0 is the minimum of \(\mathbb{N}\). So we expect \(R(0) \le 0\); however, \(R(0) = 3(0 + 1) - 1 = 2\), which is strictly greater than \(0)\). Therefore, \(R\) cannot have a right adjoint.`This seems to now be Exercise 80 in the latest draft (April 20th). \\(F\\) does indeed have a right adjoint \\(R : \mathbb{N} \to \mathbb{N}\\); it is defined by \\(R(x) = 3(x + 1) - 1\\). (One can alternately write \\(3x + 2\\), but this version changes less as you change the modulus.) \\(R\\) does _not_ have a right adjoint of its own. Suppose it did, and call it \\(R'\\). Then \\(0 \le R'(0)\\), since 0 is the minimum of \\(\mathbb{N}\\). So we expect \\(R(0) \le 0\\); however, \\(R(0) = 3(0 + 1) - 1 = 2\\), which is strictly greater than \\(0)\\). Therefore, \\(R\\) cannot have a right adjoint.`

I fixed the typo you caught; can renumber this and other exercises as needed when I have a bit of time.

`I fixed the typo you caught; can renumber this and other exercises as needed when I have a bit of time.`