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 719

Options

Let \(\mathcal{C}\) be an arbitrary category and let \(\mathcal{P}\) be a preorder, thought of as a category. Consider the following statements:

- For any two functors \(F, G : \mathcal{C} \rightarrow \mathcal{P}\), there is at most one natural transformation \(F \rightarrow G\).
- For any two functors \(F, G : \mathcal{P} \rightarrow \mathcal{C}\), there is at most one natural transformation \(F \rightarrow G\).

For each, if it is true, say why; if it is false, give a counterexample.

## Comments

1) Any \(\alpha_c\) in \( \mathcal{P} \) is unique, because \( \mathcal{P} \) is a preorder meaning at most 1 arrow exists between a pair of objects.

2) There is nothing in \( \mathcal{C} \) restricting the number of arrows between a pair of objects.

`1) Any \\(\alpha_c\\) in \\( \mathcal{P} \\) is unique, because \\( \mathcal{P} \\) is a preorder meaning at most 1 arrow exists between a pair of objects. ![One Natural Transformation](https://docs.google.com/drawings/d/e/2PACX-1vSrQVA_5UJFCXyLjDq4aNZehvlM4rWuM7o8UxKPAzl4YihE1daYEkvbws_5G9089Dej9B4N5Z7JcoMB/pub?w=607&h=383) 2) There is nothing in \\( \mathcal{C} \\) restricting the number of arrows between a pair of objects. ![Two Natural Transformations](https://docs.google.com/drawings/d/e/2PACX-1vR_x8gK_X_cK3UcOTO8PAiMeu2bILBjwg8Fq6fz5g9PaWxkr2XY55VRM1pkAtEsd-k2XWXTmOZOvfrw/pub?w=786&h=406)`