categories: k-spaces and Hurewicz

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

In response to Ronnie Brown's inquiries about "embeddability"
and cartesian closed categories, the following three publications
may be of interest:

F.W. Lawvere
Volterra's Functionals and the Covariant Cohesion of Space
Rendiconti del Circolo Matematico di Palermo (2) Supplemento No. 64, 2000,
pp 201-204.

    This paper is partly about the history of the problem
with which Ronnie is concerned, but I only later became aware of the
significance of the following two papers:

Ralph H. Fox
On Topologies for Function Spaces
Bulletin of the American Mathematical Society, vol. 51, 1945, pp 429-432

    This paper is often cited, but note that it states explicitly that
it was written in response to a question in a letter by W. Hurewicz.

David Gale
Compact Sets of Functions and Function Rings
Proceedings of the American Mathematical Society, vol. 1, 1950, pp 303-308

    Here David Gale states that the definition of k-space was due to
W. Hurewicz.


    Thus it appears that both the statement of the problem, as well as
its standard solution were given by W. Hurewicz.  The relevance to
homotopy theory as well as to functional analysis was recognized over
fifty years ago.

    There are actually many similar categories;  an axiomatic approach
(rather than a pragmatic one) is required in order to systematize the
relation  between them.  They can be "normalized", as Peter Johnstone did
for the sequential case, to become toposes; this should clarify the
comparisons as well as provide categories with much more "convenient"
exactness properties.

    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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