Michael wrote:

> So we are only looking at the possibility of production and not the actual process.

If we're taking option 1, then yes. If we're taking option 4, then no:

1. Given what I have, _is it possible_ to get what I want?
2. Given what I have, _how much will it cost_ to get what I want?
3. Given what I have, _how long will it take_ to get what I want?
4. Given what I have, _what is the set of ways_ to get what I want?

This is the difference between preorders and categories. In a preorder we only ask _is it possible_ to go from \$$x\$$ to \$$y\$$, and if so we write \$$y \le x\$$ (or maybe \$$x \le y\$$ if we change our conventions about the meaning of \$$\le\$$. In a category we discuss the _set of ways_ to go from \$$x\$$ to \$$y\$$; each way is called a **morphism** \$$f : x \to y\$$.

Fong and Spivak spend chapters 1 and 2 on preorders, and then move to categories.