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Stable Categories form a CCC

If you just got two messages with screensful of TYPES addresses, I apologize.
Let's hope I've figured out how to prevent it for the future.

Regards, Albert
Moderator, types@theory.lcs.mit.edu
----------------------------------------------------
Date: Mon, 1 Aug 88 12:59:24 BST
From: Paul Taylor <mcvax!doc.ic.ac.uk!pt@uunet.UU.NET>

On 20 January 1988, in my first contribution to types, I announced that
the category of stable domains (posets with directed join such that
every down-set is a continuous lattice) and stable maps (preserving
directed joins and connected meets) is (1) a category of algebras for
a monad on locally connected spaces and (2) cartesian closed. I have
just submitted a paper to JPAA containing these results.

I have now proved that the following (2-)category is cartesian closed:
   objects:  locally small categories with small filtered colimits
      and a set of objects of which from which any object can be
      constructed using filtered colimits, such that every slice
      category is a continuous category in the sense of Johnstone-Joyal.
   morphisms: functors preserving filtered colimits, which on every
      slice have left adjoints
   2-cells: cartesian natural transformations
These are called (weak) stable categories.  A strong stable category
has and strong stable functors preserve equalisers in addition; this
2-category is also cartesian closed.

The ingredients of the proof (which is rather complicated) are:
   1)  the limit-colimit coincidence generalised to categories with
      filtered colimits
   2)  the factorisation of a stable functor into a homomorphism
      (which preserves filtered colimits and has a left adjoint) and
      an isotomy (which is an equivalence on each slice)
      --- this corresponds to the trace in the work of Berry and Girard
      and to spectra in Diers' work.
   3)  the correspondence between cartesian natural transformations
      and rigid adjunctions between traces  --- the generalises the
      inclusion of traces characterisation of the Berry order.
   4)  a discussion of rigid adjunctions and comonads.

Draft copies of the proof will be available shortly for anyone who
in interested.

Paul Taylor,  Dept of Computing,  Imperial College,  London SW7 2BZ,  UK
+44 1 589 5111 x 4980        pt@doc.ic.ac.uk