Dear colleagues, Here are a few thoughts on the recent discussion of metric spaces: The whole general theory of enriched categories should in particular be focused on metric spaces and relatives. For example, enriched functor categories have a uniform definition, which in the case of metric spaces yields the sup metric. Of course, this does involve subtracting distances (but not in the sense of abelian groups, rather subtraction is the hom in the enriching category itself.) Dividing distances would be a dimensional mistake: the enriching category should be visualized as a possible conception of actual physical distance (not of numbers that might measure it), the addition and ordering being independent of any choice of unit. The same idea applies to the equivalent notion of nearness, as appropriate to the intrinsic measuring of convex sets (n(x,y)n(y,z) less than or equal to n(x,z), often complicated by using d = exp(-n/u) where u is a unit). With either point of view, there is no further definable operation on distance with respect to which to "divide". But on monoidal functors instead, there is of course the operation of composition. Not only categorists are enlightened by lax monoidal functors and such; subadditive functions are familiar to analysts. In their 1965 paper Eilenberg and Kelly included a definite "laxity" in the definition of monoidal (or closed) functor, since the goal was to induce good 2-functors between categories of enriched categories. For example, why should an additive category ever be considered as an ordinary category? - because the functor from abelian groups to sets is accompanied by a comparison transformation between tensor product and cartesian product. In other cases where the analogous comparison is an isomorphism, one might speak of strict monoidal functors. To deal with the fibered category of metric spaces whose morphisms have general Lipschitz constants (not just less than or equal to 1), the base monoid is conceived as acting by strict monoidal functors on distances. In particular, the hom of Banach spaces is normed in THIS monoid, not in the original one. The more general monoidal endofunctors yield much more refined notions of Lipschitz (and more refined notions of Hausdorff dimension) wherein they replace the special numbers as indices. But indices occur in another way: The Orlicz family forms a more natural parameterizable class of reflexive Banach spaces than the Lebesgue family. Here we encounter the important fact that the adjoint of a monoidal functor is not usually monoidal. The square root function is monoidal, but squaring is not. This suggests inverting the system of parameterizing the Lebesgue spaces (or more generally), so that the parameter for Hilbert space is the square root function, not 2. Since the origin of Hilbert space is in the Pythagorean metric on the product of two metric spaces, it is suggested to consider an infinite family of products. The first product that occurs to a category theorist for the category of enriched categories with given value category, is the tensor product which extends the given tensor product in the value category. This leads to the "sum" metric on the product of two metric spaces (this functor's right adjoint is the sup metric on distance-decreasing functions as mentioned above). Of course, there is also the cartesian product which in our case leads to the "max" metric. Infinite versions of these two products are a staple of analysis, but so are many intermediate products, which, as suggested above, should be parameterized by the monoidal endofunctors of the category of distance-values. Computational aspects might be approached in the following two traditional ways, which could even be combined. Paul Taylor's ascending reals seem to be related to the study of metric spaces in a topos, where the object of semi-continuous or one-sided Dedekind reals, the inf-completion of the non-negative rationals, is often the appropriate recipient of norms for continuously varying Banach spaces (even though the scalar multipliers remain the two-sided continuous Dedekind reals). The other, even older, idea was to replace closed sets by "located" sets which are actually Lipschitz functions, that is, objects in the enriched functor category that plays the role of the power set in generalized logic. Bill Lawvere ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************