Lawvere F.W. Metric spaces, generalized logic and closed categories

Canada.: Theory and Applications of Categories (TAC), 2002 (No. 1), P.p.: 1-37, EnglishOriginally published as:
Rendiconti del Seminario Matematico e Fisico di Milano, XLIII (1973), 135-166.
Republished in:
Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37Author Commentary:
Enriched Categories in the Logic of Geometry and Analysis.
Because parts of the following 1973 article have been suggestive to workers in several areas, the editors of TAC have kindly proposed to make it available in the present form. The idea on which it is based can be developed considerably further, as initiated in the 1986 article [1]. In the second part of this brief introduction I will summarize, for those familiar with the theory of enriched categories, some of the more promising of these further developments and possibilities, including suggestions coming from the modern theory of metric spaces which have not yet been elaborated categorically. (The 1973 and 1986 articles had also a didactic purpose, and so include a detailed introduction to the theory of enriched categories itself.)
While listening to a 1967 lecture of Richard Swan, which included a discussion of the relative codimension of pairs of subvarieties, I noticed the analogy between the triangle inequality and a categorical composition law. Later I saw that Hausdorff had mentioned the analogy between metric spaces and posets. The poset analogy is by itself perhaps not sufficient to suggest a whole system of constructions and theorems appropriate for metric spaces, but the categorical connection is! This connection is more fruitful than a mere analogy, because it provides a sequence of mathematical theorems, so that enriched category theory can suggest new directions of research in metric space theory and conversely, unusual for two subjects so old (1966 and 1906 respectively).Closed categories, strong categories, strong functors, closed functors.
Functor Categories, Yoneda embedding, adequacy, comprehension scheme.
Bimodules, Kan quantification, Cauchy completeness.
Free V-categories.
Further remarks.