We consider notions of metrized categories, and then approximate categorical structures defined by a function of three variables generalizing the notion of 2-metric space. We prove an embedding theorem giving sufficient conditions for an approximate categorical structure to come from an inclusion into a metrized category.
Keywords: metric, $2$-metric space, category, functor, Yoneda embedding, bimodule, path, triangle
2010 MSC: Primary 18A05; Secondary 54E35, 08A72
Theory and Applications of Categories, Vol. 32, 2017, No. 44, pp 1522-1562.