In "METRIC SPACES, GENERALIZED LOGIC, AND CLOSED CATEGORIES" (Rend. del Sem. Mat. e Fis. di Milano 43 (1973)) Bill Lawvere suggests the notion of a "normed category" as a category enriched in a suitable (symmetric monoidal) closed category S(R). (Here R denotes the interval [0,\infty], ordered by >= and with + as tensor and truncated - as internal Hom.) Has anybody worked out the details? Presumably, the objects of S(R) are to be subsets of R. This is suggested by the claim that there ought to be a closed functor inf from S(R) to R that induces the passage from normed categories to metric spaces (and by the use of the same symbol "S" that is used to denote the category of sets). As with any commutative monoid, the power-set of R under inclusion carries a symmetric monoidal closed structure: A + B = { a+b | a\in A and b\in B } (point-wise) defines the tensor and C - A = { x\in R | A + {x} contained in C } (not point-wise!) defines the internal Hom. {0} is the unit object. Clearly, inf turns into a strong functor as required. But how am I to interpret "the fundamental property of a normed category" (top of p. 140), namely (*) |f| + |g| >= |fg|? This inequality seems to apply to ordinary (= Set-enriched) categories X that carry an extra structure, namely a function |-| from Mor(X) to R. Presumably, this function also ought to satisfy (**) 0 >= |id_Y| for any X-object Y. E.g., one could define |f| to be 0 for every isomorphism of X, and 1 for every other morphism. In general this does NOT yield an S(R)-enriched category in the sense defined above. In fact, the definition above only seems to support the equality |f| + |g| = |fg|. Can further morphisms be added to S(R) without destroying the monoidal closed structure? Yes, if A +{q} is contained in B one can interpret q as a morphism from A to B. (The original inclusions arise for q = 0.) All these new morphisms f satisfy x <= f(x). But in view of the preceding paragraph one would prefer to have morphisms g that satisfy x >= g(x). Since this latter condition does not allow any new morphisms with domain {0}, the monoidal closed structure cannot be maintained. I must be missing something obvious. E.g., the definition of S(R) above doesn't use the symmetric monoidal closedness of R itself at all. Could someone please point me in the right direction? Thank you! J"urgen Koslowski | If I don't see you no more in this world | I meet you in the next world | and don't be late! koslowj@math.ksu.edu | Jimi Hendrix (Voodoo Chile) aberne@dhvrrzn1.uni-hannover.d400.de ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++