I've been thinking up some examples of \\( \mathbf{Cost}\\)-functors. I was inspired by John's use of the term **short map** and its connection to the word **short cut**.

**Example 1**

Let \\(\mathcal X \\) be the \\(\mathbf{Cost}\\)-category where the \\(\mathrm{Ob}(\mathcal{X})\\) are landmarks in my city and \\(\mathcal{X}(x, x')\\) is the time it takes to walk between each landmark.

Let \\(\mathcal Y\\) be the \\(\mathbf{Cost}\\)-category where the \\(\mathrm{Ob}(\mathcal{Y})\\) are again landmarks in my city and \\(\mathcal{Y}(y, y')\\) is the time it takes to bike between each landmark.

Then define \\(F: \mathcal X \to \mathcal Y\\) to be the identity on objects (i.e. it maps each landmark to itself). The property

\[ \mathcal{X}(x,x') \leq \mathcal{Y}(F(x),F(x')) \]

is satisfied because it takes longer to walk between locations than to bike. In other words biking is a shortcut to walking!

**Example 2**

This example is a throwback to our discussions about pie! Consider the set of ingredients needed to make a pie \\( S = \\{\textrm{flour}, \textrm{water}, \textrm{butter}, \textrm{crust}, \textrm{filling}, \textrm{pie}\\}\\). And suppose that it takes 30 minutes to turn flour, water, and butter into a crust, and 60 minutes to turn crust and filling into pie.

Let \\(\mathcal X\\) be the \\(\mathbf{Cost}\\)-category where the objects are the elements of \\(\mathbb N[S]\\) and \\(\mathcal X (x, x')\\) is the time it takes to turn the list of ingredients described by \\(x\\) into the list of ingredients described by \\(x'\\) or \\(\infty\\) if this is impossible. Some examples:


\[ \mathcal{X}( [\textrm{flour}] + [\textrm{water}] + [\textrm{butter}], [\textrm{crust}]) = 30 \]

\[\mathcal{X}( [2\textrm{flour}] +2 [\textrm{water}] + 2[\textrm{butter}] + [\textrm{filling}] , [\textrm{crust}] + [\textrm{pie}]) = 120\]

\[\mathcal{X}([\textrm{pie}], [\textrm{pie}]) = 0 \]

\[\mathcal{X}([\textrm{water}], [\textrm{pie}]) = \infty \]


Now suppose I am a busy person who prefers to buy my pie crust from the store instead of making it from scratch. Now the set of ingredients I need to make a pie is \\( T = \\{\textrm{crust}, \textrm{filling}, \textrm{pie}\\}\\). I can create another \\(\mathbf{Cost}\\)-category \\(\mathcal{Y}\\) where the objects are the elements of \\(\mathbb{N}[T]\\) and \\(\mathcal{Y}(y, y')\\) is again given by the time it takes to make transform the ingredients of \\(y\\) into the ingredients of \\(y'\\).

Lastly we need to define a \\(\mathbf{Cost}\\)-functor \\(F: \mathcal{X} \to \mathcal{Y}\\). On objects define \\(F\\) so that the ingredients needed to make a crust maps to a whole crust; crusts, fillings, and pies map to themselves; and any extra ingredients disappear. Formally:

\[F (a [\textrm{flour}] + b [\textrm{water}] + c [\textrm{butter}] + d [\textrm{crust}] + e [\textrm{filling}] + f[\textrm{pie}]) = (\textrm{min}(a,b,c) + d) [\textrm{crust}] + e [\textrm{filling}] + f[\textrm{pie}]) \]


\\(F\\) satisfies the property

\[ \mathcal{X}(x,x') \leq \mathcal{Y}(F(x),F(x')) \]

because it takes longer to make a pie from scratch than it does to buy pre-made ingredients from the store!