Bi-presheaves

[Andrew Stacey <andrew.stacey@math.ntnu.no>]

Many thanks to all those who replied to my question.  The replies were
extremely useful.  I would like to sum up what I've been told to see if I've
understood it correctly.

1. What I described is known as the 'Isbell envelope', and has been known
about for quite some time - if my reading of Lawvere's emails is correct then
the idea dates back to his thesis, after which Isbell worked on the idea and
named it the 'double envelope', consequently Lawvere renamed it the 'Isbell
envelope'.  However, whilst it is known, it is classed as 'folklore' which
I interpret to mean 'everyone knows about it, but no-one has written anything
particular on it' so there's no easy reference to which I could direct someone
(particularly someone like myself not well-versed in the lore of category
theory).

2. It is also a special case of a 'profunctor', which also goes by the names
'distributor', 'commune', and 'bimodule'.  Rather, what I'm describing is
something that can be built out of particular profunctors and natural
transformations - the 'lax factorisation' of Jeff's email.

3. They are also related to Chu spaces - something else completely new to me!

A quick check on MathSciNet provides me with a reasonable reading list.  It
seems, at first glance, easier to find information on Chu spaces than
profunctors, and certainly easier to find it on profunctors than 'Isbell
envelopes'.

What I actually intend doing is fairly simple and I suspect that my audience
(if any) will be more from the non-categorical side of mathematics so I'm
looking more for "here's where this concept has occurred before" rather than
"here's where you can find all the theorems that we need".

For the record, these came up when looking at the various different categories
of "smooth space" that I've encountered.  I'm really a differential topologist
(I hope that that admission doesn't get me expelled from the list) but I like
applying the techniques of differential topology to things that aren't really
smooth manifolds.  This leads to the question of what they actually are and,
as I'm sure everyone here knows, there have been several candidates proposed.
In trying to compare them all, I've been looking for a unified way of
describing them to make it easier to see the differences.  That's where this
notion of two functors and a "composition" came up.  There's an extra part,
which I didn't say originally, in that there are often conditions that these
functors have to satisfy.  That is, I'm really looking at a full subcategories
of the Isbell envelope where some constraints are satisfied.

It probably doesn't class as much of a grand project but it is helping me
learn a little category theory so I hope you all approve of it in that regard!

So many thanks again to all those who replied, and if anyone has any further
words of wisdom to impart then I'm happy to learn more.

Andrew Stacey