Re: KT Chen's smooth CCC, a correction

[Bill Lawvere <wlawvere@buffalo.edu>]

Dear Ronnie and Colleagues,

Your comments are extremely interesting.  Thank you very much for raising
in so striking a manner the question of the relation between general
monoidal structures and cartesian closed structures.
Below are some observations which show, I think,
that everybody should be interested in this relation because it is
manyfold and fruitful.

(1)	While cartesian closed structures have the virtue of being unique,
general monoidal closed structures have the virtue of not being unique.
Thus, for example, the cartesian closed presheaf toposes (with their
exactness properties and combinatorial truth object) often have a further
monoidal closed structure given by Brian Day's convolution with respect to
a pro-co-monoidal structure on the site. Cubical as well as simplicial
sets have both cartesian and non-cartesian closed structures, and that is
'true', not merely 'convenient'.

(2)	Another category having both cartesian and non-cartesian monoidal
structures is the real interval from zero to infinity with 'x dominates y'
as the morphism from x to y. (Actually, this category is derived by
collapsing a natural topos of dynamical systems in 'Taking categories
seriously' TAC Reprints.) Categories enriched with respect to the
non-cartesian structure here (see 'Metric Spaces' TAC reprints) arise
every day in analysis and the rich insights of enrichment theory (Functor
categories, bi-module composition, free categories, etcetera) should be
systematically applied to the advance of analysis and geometry, while on
the other hand metric examples inspire further developments of enrichment
theory. Cauchy (who never worked on idempotent splitting in ordinary
categories and additive categories in the way that Freyd and Karoubi did)
does not deserve to have his name brandished as a joke to scare one's
uncomprehending colleagues in analysis. The kind of completeness that is
inspired by two-sided intervals (unlike the one-sided intervals
inaccurately alluded to in common discussions of 'density') indeed reduces
to the one attributed to Cauchy in the particular example of Metric
Spaces. The author hoped that observation would contribute to the advance
of analysis and the development of enrichment theory, not to the supply of
buzzwords.

     In fact, there is an insufficiently known branch of analysis called
'Idempotent Analysis', which deals largely with composition of bi-modules,
or more precisely, with the relation between the two closed structures on
the infinite interval. Of course, that monoidal category is isomorphic to
the unit interval under multiplication (still cartesian closed too) which
induces many of the relations between probablility and entropy.

(3)	Perhaps the most common relation between non-cartesian monoidal
categories and cartesian categories arises when a structure such as vector
space is interpreted in a cohesive background. I am sticking to my story
that cohesive backgrounds are basically cartesian closed, due to the
ubiquitous role of diagonal maps and also due to the fact that, for
example, bornological vector spaces have an obvious monoidal closed
structure, whereas topological vector spaces have none. The rumor that
topological vector spaces might have a tensor with an adjoint hom is part
of the disinformation that makes functional analysis look more difficult
than it is. A more accurate account of the relation between non-Mackey
convergence and closed structure can be found in C. Houzel's paper on
Grauert finiteness, Mathematische Annalen, vol. 205, 1973, 13-54:
essentially, the topological categories are merely enriched in the
genuinely monoidal closed bornological ones. Similarly, the idea that not
all dual spaces are complete seems to be based on a misguided generality
in the notion of Cauchy nets (they should be bounded).

(4)	Although pointed spaces are somewhat entrenched in algebraic
topology, there is an improvement suggested by your own work, Ronnie.
Consider the category whose objects are arrows S ---> E where E is a space
(object of a cartesian closed cohesive background category) and S is a
discrete space. This category is even a topos if the category of E's was,
as is the larger category of arrows between general pairs of spaces. The
first category is actually an adjoint retract of the second, correcting
the discontinuity that arises from the traditional limitation S = 1.
Intuitively, in the case where the pair of spaces is a subspace inclusion,
the adjoint collapses the subspace to a point if the subspace is
connected, but if it is not connected, does not artificially merge its
components. There are many applications of this corrected construction of
the space which results from 'neglecting' a subspace, both in algebraic
topology and in functional analysis, too numerous to discuss here.

Bill



On Wed, 27 Aug 2008, R Brown wrote:

> Dear Bill and Colleagues,
>
> I would like to explain my own interest in function spaces and function
> objects since it has a different origin to what Bill explains and a
> different direction which could be of interest for comment and
> investigation.

...