Exercise 2-73

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.

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