# (Co)products and exponentials for Lawvere metric spaces

category-theoryenriched-category-theorymetric-spaces

A Lawvere metric space can be thought of as a $$\textbf{Cost}$$-enriched category where $$\textbf{Cost}=([0,\infty],\geq,0,+)$$ is a symmetric monoidal preorder (see Chapter 2 of Fong and Spivak for example). I have several questions about $$\textbf{Cost}$$-enriched categories, and would appreciate help with any of them:

1. How do I define the product of objects in a $$\textbf{Cost}$$-enriched category ?
2. How do I define the coproduct of objects in a $$\textbf{Cost}$$-enriched category ?
3. How do I define the exponential of objects in a $$\textbf{Cost}$$-enriched category ?
4. Are there classes of $$\textbf{Cost}$$-enriched categories which always have the above structure ? (like categories of functors into $$\textbf{Set}$$ are always bicartesian closed in the $$\textbf{Set}$$-enriched case).

I can think of three characterizations of products in a $$Cost$$-enriched category $$X$$ (the case of coproducts is dual):

1. adjointness: $$X$$ has binary products if the functor $$\Delta_X : X \to X\otimes X$$ has a right adjoint

$$\_\odot\_ : X\otimes X \to X.$$ The object $$X\otimes X$$ is the product metric space, which has the correct universal property to make $$Cost$$-categories a monoidal category. Note that this is slightly more than what you asked for, because this way you define when all products exist, not just for a single pair $$(x,y)$$.

2. universal property: given $$x,y\in X$$, there exists a point $$p=x\odot y$$ with the following property: let $$\alpha_x, \alpha_y$$ be respectively the distances $$d(p,x), d(p,y)\in [0,\infty]$$; then, for every other $$z\in X$$, the distance $$d(z,p)$$ is terminal among all real numbers such that $$\begin{cases} h + \alpha_x \ge d(z,x) \\ h + \alpha_y \ge d(z,y) \end{cases}$$ all in all this means that $$d(z,p)=\max\{d(z,x)-d(p,x),d(z,y)-d(p,y)\}$$. (I think it's just a matter of unwinding the universal property of an adjoint to see that $$p = x\odot y$$).

3. representability: observe that a generic category admits the binary product of two objects $$x,y$$ when the functor $$a\mapsto X(a,x)\times X(a,y)$$ is representable, i.e. when (taking into account that $$X$$ is enriched over $$[0,\infty]^\text{op}$$)

$$X(a,x\odot y) = X(a,x)\lor X(a,y)$$

i.e. when (taking into account that the hom-object $$X(u,v)$$ is the real number $$d(u,v)$$) for every $$a\in X$$ one has $$d(a,x\odot y) = \max\{d(a,x), d(a,y)\}$$.

Certainly the three definitions of $$x\odot y$$ are equivalent; I think I understand the first only in terms of the third. Also, the third generalises easily (well, not so easily if you want your metric to be finite...) to the case of $$\kappa$$-ary products: given a family $$\{x_i\mid i\in\kappa\}$$ of elements of $$X$$, $$\bigodot_i x_i$$ is a point of $$X$$ such that for every $$a\in X$$,

$$\textstyle d(a,\bigodot_ix) = \sup_i d(a,x_i)$$