Re: Lawvere-Metrics and Banach Spaces

[F W Lawvere <wlawvere@buffalo.edu>]

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


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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
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