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.
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
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
Paul Taylor, Dept of Computing, Imperial College, London SW7 2BZ, UK
+44 1 589 5111 x 4980 email@example.com