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From: Martin Escardo <m.escardo@cs.bham.ac.uk>
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Subject: Re: thoughts arising from a letter of Lawvere
Date: Wed, 8 Jan 2003 10:37:02 +0000
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I (and a colleague) wonder whether what Max Kelly is referring to is
what Brian Day published in an incredibly concise way in pages 4-5 of
the paper "A reflection theorem for closed categories", J. Pure
Appl. Algebra 2 (1972), no. 1, 1--11.

Max Kelly writes:
 > [...]
 > This is of course classical; but what Brian had is the following. There
 > is an evident functor f: Top --> Qu; just call A --> X allowable if it
 > is continuous. There is an equally evident functor
 > g: Qu --> Top; call a subset open if its characteristic function into
 > the Sierpinski space 2 lies in Qu. We have the adjunction g --| f. As
 > with any adjunction, we have an equivalence between the full subcategory
 > of Top where the counit is invertible and the full subcategory of Qu
 > where the unit is invertible.
 >
 > The subcategory of Top here, of course reflective in Top, is the
 > category of k-spaces, better called the "compactly-generated" spaces; it
 > is also a coreflective full subcategory of Qu. Others have noticed this
 > since and published it; but certainly subsequent to Brian's 1968 (I
 > think) Master's thesis.
 > [...]

(We would also be interested in having a copy of the version of the
paper "On quotients maps preserved by product and pullback" before the
translation from category theory to topology (referred to in the
deleted part of Kelly's message). Is that still available?)

Martin Escardo