  We define exact weights on a triangulated category to be nonnegative
functions on objects satisfying a subadditivity condition with respect to exact
triangles. Such weights induce a metric on objects in the triangulated
category, which we call a path metric. Our exact weights generalize the rank
functions of J.\ Chuang and A.\ Lazarev and are analogous to the exact weights
for an exact category given by the first author and J.\ Scott and D.\ Stanley.
We show that cohomological functors from a triangulated category to an abelian
category with an additive weight induce an exact weight on the triangulated
category. We prove that triangle equivalences induce an isometry for the path
metrics induced by cohomological functors. In the perfectly generated or
compactly generated case, we use Brown representability to express the exact
weight on the triangulated category. We give three characterizations of
exactness for a weight on a triangulated category and show that they are
equivalent. We also define Wasserstein distances for triangulated categories.
Finally, we apply our work to derived categories of persistence modules and to
representations of continuous quivers of type $\mathbb{A}$.
