#### Howdy, Stranger!

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

Options

# A proof with labelled statements

edited May 2018

The original version of the following proof can be found here

Puzzle 90. What's a $$\mathbf{Cost}^{\text{op}}$$-category, and what if anything are they good for?

The story doesn't look very good for them :(

Theorem. If $$\mathcal{X}$$ is a $$\mathbf{Cost}^{\text{op}}$$-enriched category, then:

$$\tag{a} \forall a,b. \mathcal{X}(a,b) = 0 \text{ or } \mathcal{X}(a,b) = \infty$$ Proof.

If every element $$\mathcal{X}(a,b) = 0$$, we are done.

Next assume to the contrary. We must show that for an arbitrary $$a$$ and $$b$$ that $$\mathcal{X}(a,b) = \infty$$.

So observe there must be some $$\hat{a}$$ and $$\hat{b}$$ such that $$\mathcal{X}(\hat{a},\hat{b}) > 0$$.

It must be $$\mathcal{X}(\hat{a},\hat{b}) = \infty$$. To see this, we know from the laws of enriched categories (part (b)) that:

\tag{b} \begin{align} \mathcal{X}(\hat{a},\hat{b}) + \mathcal{X}(\hat{b},\hat{a}) & \leq \mathcal{X}(\hat{a},\hat{a}) \\ \implies \mathcal{X}(\hat{a},\hat{b}) & \leq \mathcal{X}(\hat{a},\hat{a}) \end{align} However, then we have

\tag{c} \begin{align} \mathcal{X}(\hat{a},\hat{b}) + \mathcal{X}(\hat{a},\hat{a}) & \leq \mathcal{X}(\hat{a},\hat{b}) \\ \implies 2 \mathcal{X}(\hat{a},\hat{b}) & \leq \mathcal{X}(\hat{a},\hat{b}) \end{align} This can only happen if $$\mathcal{X}(\hat{a},\hat{b}) = \infty$$ or $$\mathcal{X}(\hat{a},\hat{b}) = 0$$. But we know $$\mathcal{X}(\hat{a},\hat{b}) > 0$$ so it must be $$\mathcal{X}(\hat{a},\hat{b}) = \infty$$ .

Next observe from the enriched category theory law (b) that:

$$\tag{d} \mathcal{X}(a,\hat{a}) + \mathcal{X}(\hat{a},\hat{b}) \leq \mathcal{X}(a,\hat{b})$$ So it must be that $$\mathcal{X}(a,\hat{b}) = \infty$$. But then

$$\tag{e} \mathcal{X}(a,\hat{b}) + \mathcal{X}(\hat{b},b) \leq \mathcal{X}(a,b)$$ Hence $$\mathcal{X}(a,b) = \infty$$. $$\qquad \square$$

[Edit: Updated following Christopher's suggestion. Thanks Chris!]

• Options
1.
edited May 2018

This example has some nice features.

• The math statements are clearly centered and tagged with names.
• Justification for the next for statement is provided with appropriate embedded math statements.
• It uses the QED box
• It provides an edit note indicating that it was updated based on a subsequent comment, there is a link provided to that comment.
• It starts with a restatement of the problem
• It indicates how to 'quote' someone else.

Links to comments can be obtained via the 'gear' icon in the upper right corner of the comment.

Comment Source:This example has some nice features. - The math statements are clearly centered and tagged with names. - Justification for the next for statement is provided with appropriate embedded math statements. - It uses the QED box - It provides an edit note indicating that it was updated based on a subsequent comment, there is a link provided to that comment. - It starts with a restatement of the problem - It indicates how to 'quote' someone else. Links to comments can be obtained via the 'gear' icon in the upper right corner of the comment.