Exercise 2.40#

As a preorder $$\bf{Cost}^{op}$$ is $$([0, ∞], \leq)$$. In this preorder smaller numbers are “better” (greater) as in golf.

The monoidal unit remains 0, and the monoidal product remains +.

Note that (confusingly) we use the symbol $$≤_{X^{op}}$$ to mean the same as ≥ (where ≥ has its normal meaning in a linear order); see the first example in Opposite category. See also “is parent of” as discussed in Relation (mathematics). Given an arrow, the source is the first argument to the operator (e.g. ≤) and the target is the second argument. Visually:

It’s not enough to draw a diagram with arrows to communicate (by definition non-commutative) relationships. Let’s say you were planning a gift exchange with family members, and you sent out this list:

George → Mary
Mary → Sandeep
Sandeep → George

Does George buy a gift for Mary, or Mary buy a gift for George? Without the giver and receiver labeled, there’s no way to know which is correct. If you send out this list, someone may end up without a gift while someone else gets two! Only one arrow needs to be labeled, in the style of a legend:

$Giver \; \underrightarrow{buys for} \; Receiver$

Looking ahead, you can see this graph as describing morphisms in Set where the sets are singletons (see Singleton (mathematics)).