Enriched functors

A \(\mathcal{V}\) functor \(\mathcal{X}\xrightarrow{F}\mathcal{Y}\) between two \(\mathcal{V}\) categories

A function \(Ob(\mathcal{X})\xrightarrow{F}Ob(\mathcal{Y})\) subject to the constraint:

\(\forall x_1,x_2 \in Ob(\mathcal{X}): \mathcal{X}(x_1,x_2) \leq \mathcal{Y}(F(x_1),F(x_2))\)

Monotone maps, considering the source and target preorders as Bool-categories, are in fact Bool-functors.
The monotone map constraint, that ?\(x_1\ \leq_X\ x_2 \implies F(x_1)\leq_Y F(x_2)\), translates to the enriched category functor constraint, that ?\(\mathcal{X}(x_1,x_2) \leq \mathcal{Y}(F(x_1),F(x_2))\).

A Cost-functor is also known as a Lipschitz function.
Therefore a Lipschitz function is one under which ?the distance between any pair of points does not increase.

The concepts of opposite/dagger/skeleton extend from preorders to \(\mathcal{V}\) categories.
- An opposite \(\mathcal{V}\) category \(\mathcal{X}\) has the same objects and \(\mathcal{X}^{op}(x,y):=\mathcal{X}(y,x)\)
- A dagger category is identical to its opposite.
- A skeletal \(\mathcal{V}\) category is one in which \(I \leq \mathcal{X}(x,y) \land I \leq \mathcal{X}(y,x)\) implies \(x = y\)
Recall an extended metric space \((X,d)\) is a Lawvere metric space with two extra properties.
Show that a skeletal dagger Cost-category is an extended metric space

Solution(1)

The skeletal dagger cost category has a set of objects, \(Ob(\mathcal{X})\) which we can call ?points.
For any pair of points, we assign a hom-object in \([0,\infty]\) (we can call this a ?distance function).
Skeletal property enforces the constraint ?\(d(x,y)=0 \iff x=y\).
The second enriched category property enforces ?the triangle inequality.
Because we have a dagger category, our distance function is ?forced to be symmetric.
Just like the information of a preorder is discarded (to yield a set) when we only consider skeletal dagger preorders, information must be discarded from Cost-categories to yield a Lawvere metric space.