categories: Alternative characterisation of 'flat sublocale' wanted

["Christopher Townsend" <cft71@hotmail.com>]

If C is an ordered symmetric monoidal closed category (i.e. has a tensor 
such that X \tensor (_) has a right adjoint) then the category ICMon(C) of 
idempotent commutative monoids over C (such that the monoidal operation 
preserves the order in a sensible way) forms a category that can behave like 
the category of frames. Take for example C= Sup lattices, or C = Preframes 
and ICMon(C) is, in both cases, equivalent to the category of frames.

I was therefore trying to look at ICMon(C)^op for any symmetric monoidal C, 
and see how well Locale theory can be done in this category. Does anyone 
have any suggestions about how a flat sublocale may be axiomatised in this 
setting?

Thanks


Christopher

[i: X_0 >-> X (a regular monic in the category of locales, i.e. a sublocale) 
is flat iff the corresponding nucleus preserves finite joins]







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