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Exercise 57 - Chapter 3

edited June 2018 in Exercises

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

  1. For any two functors \(F, G : \mathcal{C} \rightarrow \mathcal{P}\), there is at most one natural transformation \(F \rightarrow G\).
  2. 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.

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  • 1.
    edited May 2018

    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

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

    Two Natural Transformations

    Comment Source: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)
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