Abject apologies for being a bit late to this party; had a bit of trouble getting a pumpkin (can’t be a Norwegian myth).
All of the suggested starting categories have a separator and this has serious consequences for Isbell conjugation. (To throw a link to the discussion over there, after working some of this out on paper I tried a few MathSciNet searches to see if anyone else had come up with it; several of the searches I tried used MSC codes to try to restrict the results to something useful. For some reason, the words “conjugation” and “dual” are fairly common in mathematics.)
So let be a category with a separator, say . Let be a presheaf on (oh how I’d love to be able to do macros in iTeX); that is, as I’ve learnt, a contravariant functor from to .
For objects in we therefore get a set map
functorial in both and . We can turn this around slightly to get a set map
still functorial in both and . In terms of elements (gosh, did I really use that word?), this is
In particular, for a -morphism we get a commutative diagram
where the right-hand vertical arrow is the obvious one. Hence we can define a new functor by
There is an obvious natural transformation which consists of surjections.
Let us denote Isbell conjugation by primes. From the natural transformation we obtain a natural transformation . I claim that this is a natural isomorphism.
Let us start with injectivity.
Let be an object of and let . This is a natural transformation so for an object of we have a map of sets
In particular, we get .
Suppose that are such that . Then there is some object of such that . As these are set maps, there is some such that . These are now morphisms in . As is a separator, there is some morphism such that .
Now as and are natural transformations,
and similarly for . Hence the element distinguishes and . Thus the map
is injective.
This holds with in place of and we have a commuting diagram.
with both horizontal maps being inclusions.
Now
where the map is . There is also a map
given by evaluation on the identity. The composition is the identity on and thus . Moreover, this isomorphism is that appearing in the natural transformation . Thus in the above diagram, the right-hand vertical map is an isomorphism. The left-hand map is thus injective.
Now let us consider surjectivity.
Let . This is a natural transformation . We need to show that it factors through . Let be an object in and suppose that are such that in . Then there is a morphism such that . Using again the fact that is a natural transformation, and similarly for . Thus in . Hence the maps defined by and differ. That is, the images of and in are distinct.
This shows that factors through and hence, as consists of surjections, we can build a natural transformation , such that under , maps to . Thus is surjective.
We therefore conclude that the natural transformation is a natural isomorphism.
We have
we also have a pairing compatible with these inclusions; that is (on objects)
(inclusion as is a separator).
From this it is clear that is given by
Lo and behold! A Frölicher space! Or rather, A Frölicher -object.
Thus stability under Isbell conjugation is a very strong condition indeed; it forces quasi-representability and also the saturation condition. This suggests to me that there is a qualitative difference between quasi-representability and non-quasi-representable. As I conjectured, there is a functor from all presheaves to quasi-representable ones, and of course then one can (may I say “should”?) saturate with respect to duality. So every presheaf has an underlying Frölicher object, but this assignment is by no means injective. In fact, contrary to what I thought, the fibres of this functor do have some interesting stuff in them.
For example, if we take the simplified version of Urs’ favourite functor; namely 1-forms, then the associated Frölicher space is the single point (since ).
It turns out, therefore, that Urs and I are looking at orthogonal concepts!
One possible area for further study is to see what happens if you vary the separator. It doesn’t have to be a singleton point in our categories of interest. For example, if we enlarge our category of “known smooth stuff” to include , the direct limit of the s, and to be 1-forms, then by taking as the separator we find that
is injective, and so is quasi-representable. Of course, it is no longer quasi-representable in the strictest sense because we do not have an injection
for any set . We only get this type of quasi-representability if our separator is a singleton point.
So perhaps we should define quasi--representable where is a separator in the category , by which we mean that the natural transformation which on objects is
consists of injections.
However, varying the singleton won’t change the fact that Isbell conjugation (now thinking of one of our smoothish categories) results in strictly quasi-representable presheaves since the argument above made no assumptions on the separator so it certainly does work for the singleton point.
Right, it’s lunchtime so I’m stopping here.
Andrew
Re: Space and Quantity
I have restructured a bit and added a section (number 2 at the moment) on the general abstract nonsense of space and quantity. I followed Lawvere’s Taking categories seriously, but varied slightly. Please check.
(I haven’t said anything about the dotted arrows that Lawvere has on p. 17. I guess these are supposed to allude to the kind of “saturation” or “completion” operation which we keep talking about?)
I have added a statement, right now proposition 5, which says that for all and any -algebra we have
I think this is essentially obvious, using the fact that the GCA underlying is freely generated in positive degree. But please check.