Adjunction between convex spaces and Lawvere metric spaces

Question

In Section 1.21 Meng describes a functor $\varphi$ from the category of convex spaces, $\text{Conv},$ to the category of Lawvere metric spaces, cost-Cat. Here cost-Cat can also be thought of as the category of $\text{cost}=[0,\infty]^{op}$ enriched categories.

Explicitly, $\varphi$ sends a convex set $C$ to the Lawvere metric space $(X,\delta_X)$ with the same set of points $X=C$ but with Lawvere metric $\delta_X(x,x')=-\text{log }\text{sup } \{t \in [0,1]|\exists{y}\in C: x = t.x' + (1-t).y \}.$

In Section 1.26 Meng claims that $\varphi$ has a left adjoint $L.$ I would like an explicit description of the functor $L.$

Answer 1

A morphism of Lawvere metric spaces $f\colon X\to LC$ is a function satisfying $\sup\{t\in[0,1]\mid\exists z\in C:f(y)=tf(x)+(1-t)z\}\ge\exp(-d_X(y,x))$ for all $x,y\in X$.

This means that in addition to a closed segment $[x,y]\subseteq C$, there are also semi-open segments $[f(x),z)$ and $(w,f(y)]$ relative to which $f(y)$ and $f(x)$ have barycentric coordinates $\exp(-d_X(y,x))$ and $\exp(-d_X(x,y))$, respectively.

Moreover, $d_X(y,x)=0$ or $d_X(x,y)=0$ imply $f(x)=f(y)$.

Thus a morphism $f\colon X\to LC$ of Lawvere spaces determines for each unordered pair $\{x,y\}\subseteq X$ an interval (possible open, semi-open, or closed) in the convex space $C$ containing $f(x)$ and $f(y)$, within which they have certain barycentric corrdinates determined by $d_X(x,y)$ and $d_X(y,x)$, and with $f(x)=f(y)$ if $d_X(x,y)=0$ or $d_X(y,x)=0$.

So $RX$ is a convex structure freely extending the union of segments, one for each unordered pair $\{x,y\}$, within which $x$ and $y$ have barycentric coordinates specified by $d_X(x,y)$ and $d_X(y,x)$, with $x=y$ if $d_X(x,y)=0$ or $d_X(y,x)=0$. Note that any two of these segments, if they intersect, do so in a point of $X$.

One construction is as the quotient of the convex space generated by these (open, semi-open, or closed) segments, considered as independent segments, each having a pair of points at certain barycentric coordinates labeled by points of $X$, then gluing, i.e. taking the quotient of, the space along points that have labels $x$ and $y$ satisfying $d_X(x,y)=0$ or $d_X(y,x)=0$. Making this more explicit amounts to making more explcit the result of this quotient in the category of convex spaces.