Motivation: Let $\Gamma = (V,E)$ be a directed graph. To each edge $e \in E$, choose a value $\kappa^e \in \mathbb R$, representing the cost of transporting one unit of "stuff" through the edge. Let $\operatorname{Paths}(\Gamma,\kappa)$ denote the path category over $\Gamma$, where the objects are vertices, and the morphisms are finite paths via directed edges.
Because of the assignment of edge-costs, there is a corresponding cost functional on paths. The cost of a path is simply the sum of the edge-costs along the path: $K(\gamma) := \sum_{e \in \gamma} \kappa^e$. Furthermore, this induces a generalized metric on the vertex set $V$, where $$d(v,v') := \inf_{\gamma : v \to v'} K(\gamma).$$
This category is implicitly of central interest to probabilists who study first-passage percolation, where the graph is usually assumed to be homogeneous (e.g., $\Gamma \cong \mathbb Z^d$), and the edge-costs are i.i.d. non-negative random variables.
An analogous category is also of interest in Riemannian geometry and general relativity, where the paths correspond to geodesics between two points, the cost functional is the Riemannian length functional, and the distance function is defined as above. Positive costs correspond to space-like distances, and negative costs correspond to time-like distances.
These are clearly two examples of the more general structure of a "metrized category", which I'm hoping that somebody has studied in depth.
Definition: Let us say that a category $\mathcal C$ is metrized if its object set $(V,d)$ is a generalized metric space; its hom-set $H$ is equipped with a real-valued "cost function" $K : H \to \mathbb R$ satisfying $K(1_v) = 0$ and $K(\gamma' \circ \gamma) = K(\gamma') + K(\gamma)$; and if the metric and cost function are related by the identity $d(v,v') = \inf_{\gamma : v \to v'} K(\gamma)$ for all $v,v' \in V$.
The concept of a generalized metric space is ambiguous, so let me make explicit my assumptions. A generalized metric satisfies the assumptions of reflexivity ($d(v,v) = 0$), transitivity (the triangle inequality: $d(v,v') + d(v',v'') \le d(v,v'')$), and the non-degeneracy condition that $d(v,v') > -\infty$ for all $v,v' \in V$. The directed nature of the category means that we do not require the metric to be symmetric.
Question 1: is there a common name for such a "metrized category" in the literature? Does the category of metrized categories $\operatorname{MetCat}$ satisfy the usual properties (products, limits, etc.)? How does a metrized category relate to Lawvere metric spaces, which are "precisely the categories enriched in the monoidal poset $([0,\infty], \ge)$, where the monoidal product is taken to be addition."
In addition to the above "reference request" question, let me also offer a more substantial question:
Question 2: what does the Yoneda lemma imply about metrized categories?
