thoughts arising from a letter of Lawvere

thoughts arising from a letter of Lawvere

[Max Kelly]

Sorting through old papers after the move to our new house, I came
across a communication to Categories by Bill Lawvere, dated 21 Nov 2001,
entitled
"categories:K-spaces and Hurewich", and concerned with the history of
k-spaces and related concepts.
I thought the following bit of history worth contributing.

Before studying monoidal closed categories in his well-known doctoral
thesis, Brian Day wrote a very pleasant Masters thesis on monoidal
closed structures on variants of topological spaces. For some reason
this never got published - perhaps it was not thought original enough at
the time - but it
contained the perfect way of introducing k-spaces; and not just
hausdorff ones - restricting to those is an error.

One starts with the category Top of topological    spaces, and the
category Comp of compact hausdorff spaces. Based on Comp, one forms
Steenrod's category of quasi-spaces: a quasi-space is a set X along
with, for each A in Comp, a subset of Set(A,X) whose elements may be
called the "allowable" maps - one imposes a few evident axioms on these.
The quasi-spaces form a category Qu, whose morphisms from X to Y are
those set-maps whose composites with allowables are allowable.

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.

Of course one is not obliged to use Comp in defining one's quasi-spaces;
write Qu' for the quasi-spaces based instead on Top. Now Top --> Qu' is
fully faithful, and reflective: we know the reflexion explicitly. Again
Qu' is cartesian-closed, although Top is not. This is how Brian and I
proved those results in [On topological quotient maps preserved by
pullbacks or products, Proc. Cambridge Phil. Soc.67, 1970, 553 - 558].
We did the pulling back in the cartesian closed Qu', applied the
reflexion, and wrote down the condition for preservation.

We feared, however, that topologists would be frightened off by these
"abstract categorical notions"; so we went through all that we had done,
translating it into the usual language of topology, before we submitted
it for publication. The readers, with our motives and techniques so
concealed, must have thought it black magic.

Of course, as Bill Lawvere said, the whole "quasi" business should be
done abstractly, and turns out to involve subcategories of presheaf
categories, with associated toposes like that of Peter Johnstone.

I see that this letter has become very long. I must apologize: but so
much of our history is getting lost forever.

Max Kelly.

Re: thoughts arising from a letter of Lawvere

[Martin Escardo]

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

Re: thoughts arising from a letter of Lawvere

[Max Kelly]

Martin Escardo (Categories 8 Jan.) is correct in surmising that the
results of Brian Day's 1968 M.Sc. thesis are included, in a much more
general form, as an example in his 1972 article in JPAA. My point was only
the history of the matter; the 1972 paper is one of three which together
expound the content of Brian's Ph.D. thesis, and appeared four years later
than his Masters degree. In the interim the adjunction between topological
spaces and Spanier's quasi-spaces, and the relation of this to
compactly-generated spaces, had been observed and published by two or
three other of our colleagues; Booth was one, and there was at least one
more whose name escapes me. Brian was then rueful about not submitting the
M.Sc. thesis for publication in 1968; as his supervisor, I shared his rue.

Martin also seeks more information about the hidden category theory
behind my 1970 paper with Brian. There was never anything like a
"categorical version" that got turned into a "topological" one;
we translated as we went. I believe my 2 Jan. letter to Categories
contains enough hints to make a reconstruction straightforward, even if
a bit long and tedious.

Max Kelly.

Re: thoughts arising from a letter of Lawvere

[Ross Street]

Sammy Eilenberg told me he wrote a paper based on the Day-Kelly paper

	On topological quotient maps preserved by pullbacks or products,
	Proc. Cambridge Phil. Soc. 67 (1970) 553-558.

I believe Sammy's paper was for an MAA expository series.  It sounded
a beautiful approach but I did not see the manuscript and forget the
details.  While the paper was in the universal categorical spirit it
made only one parenthetic remark that something was a category. Sammy
claimed this was why the paper was rejected.  What a great pity!

I would love to have a copy of the preprint. Does anyone else know about it?

--Ross