This is a second attempt after that my previous post on the same subject has been closed since, I was said, "not a real question". Pas de problème ! I hope that it is now a little bit clearer than before.
It is apparent that metric and normed (vector) spaces have profound structural differences. Loosely speaking, a metric space is a "bare" set $X$ together with a function $d: X \times X \to X$ satisfying some prescribed properties, while a normed vector space is an algebraic structure $V$ along with a function $n: |V| \to |V|$ fulfilling a few other axioms (I'm using $|V|$ for the carrier of $V$): The former is called a distance, the latter a norm. But names do not always reflect the essential nature of concepts and they happen to mark differences that can be partly smoothed out provided that one can move uphill and watch the valley from the mountain peak.
In this line of thought, we can all agree, I think, that some (basic) issues, results and constructions local to the theory of metric spaces are conceptually close to (basic) issues, constructions and results local to the theory of normed (vector) spaces: This is the case, e.g., with the most usual choices of the class of morphisms to associate with the categories $\bf Met$ (metric spaces with Lipschitz maps as arrows) and $\bf Nor$ (normed spaces with bounded linear transformations as arrows) and with their "short variants"; with binary products and coproducts (and indeed with arbitrary nonempty products and coproducts if we let metrics and norms be valued in $[0,+\infty]$ rather than in $\mathbb{R}_0^+$); or even with some of the conditions required, with the notation from above, by the definition itself of $d$ and $n$ (I'm especially thinking of the triangle inequality). These similarities are all the more striking, I think, if we enlarge our scope and bring other normed structures into the picture (e.g. normed groups [1], valuated rings [2], and normed algebras): Let me refer to these, in what follows, as normed algebraic structures (it is not that important, I think, to know what it is formally meant by a normed algebraic structure). One is then naturally prompted to raise the question: Does there exist some higher perspective (say, some sort of abstract theoretical framework, hopelessly shaped on the language of categories), where algebraic normed structures are all special instances of one same archetypical structure? But this is not my question. Instead, my question is
whether or not anything similar has been already done in the literature in relation to the construction of a unified framework where metric spaces AND algebraic normed structures can be all regarded as instances of the one same archetypical structure.
Here, the term "structure" must be formally intended as "model of a finitary first-order theory (in one or many sorts) interpreted over $\bf Set$ or another category $\bf C$ with some appropriate properties" (the underlying formal language, say, is that of the NBG class theory).
Notes. [1] See, e.g., N.H. Bingham and A.J. Ostaszewski, Normed groups: dichotomy and duality, CDAM Research Report Series, LSE-CDAM-2008-10, London School of Economics, 2008 and references therein. [2] I'm using the term "valuation" as a synonymous of "absolute value".
