# (Co)products and exponentials for Lawvere metric 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).
• There are generic answers applicable to categories enriched over complete and cocomplete symmetric monoidal closed categories such as, yes, $\textbf{Cost}$. Are you looking/hoping for a simplified description in this special case? Commented Apr 26, 2022 at 9:42
• It'd be great to have a simplified description, but it'd also be good just to know some references where I can find those constructions that work in more general situations. I have a feeling that some of this stuff is mentioned in BASIC CONCEPTS OF ENRICHED CATEGORY THEORY but I'm not sure which terminology to look for (I'm still quite new to enriched category theory). Commented Apr 26, 2022 at 9:51
• I wrote an answer to your question about enriched coproducts some months ago. If you could review it and say what is still not clear then I might be able to write something more helpful this time. Commented Apr 26, 2022 at 10:37

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 $$a\in X$$,

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

• Remark: there is no nontrivial product in usual (symmetrical) metric spaces, because $d(a\odot b,a) = d(a\odot b,b) = 0$. The only products that exist are $a\odot a = a$. Commented Apr 26, 2022 at 12:47
• Wonderful, thank you Fosco. I wonder if I can be greedy and ask if similar knowledge can be gained about when all exponential objects exist and what form they have (I'm guessing they are given by the right adjoint of the usual functor involving products). Commented Apr 26, 2022 at 13:19
• @Dabouliplop indeed! I noticed it shortly after having written my answer :D Commented Apr 26, 2022 at 14:01
• @RichardSouthwell as you can see, products do not exist in nontrivial forms... that's why the notion has had little development. Can you explain better why you're interested in construction inside a metric space? Commented Apr 26, 2022 at 14:02