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!