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.

> 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.