# The definitive axiomatisation for ASD (maybe)

## Paul Taylor

(Manchester)

ASD has provided the complete axiomatisation of the category

of computably based locally compact locales and (with the "underlying set"

axiom) that of locally compact locales over an elementary topos. Key

to this (and also to the Heine--Borel theorem in recursive topology)

is the requirement that the adjunction $\Sigma^{-}\dashv\Sigma^{-}$ be

monadic.

The generalisation of this theory beyong local compactness has been

problematic. Somehow, this monadic condition, which ensures that

(certain) subobjects carry the subspace topology, needs to be generalised.

Over a number of years, I experimented with several possible axioms,

which looked very pretty but had neither models nor consistency proofs.

Now I believe I have the solution - the original monadicity property,

except now with all finite limits in the category. The test of this

will be the characterisation of the free model, which appears to be

the monadic completion of Scott's category of equilogical spaces.

(See also www.cs.man.ac.uk/~pt/ASD/extension.html)

**Tuesday 28th November 2006, 14:00**

Board Room

Department of Computer Science