4 April 2006

1:00 - 2:30

M Bunge

Michael coverings are comprehensive

Abstract:

The motivational example for the comprehension scheme came from logic 
(Lawvere'68), and this suggested a certain terminology ("types", "formulas", 
proofs", "truth", "comprehension"). An example with categories as types 
(Gray '69, Street-Walters '73) exhibited 2-comprehension as the familiar 
Grothendieck construction associated with a functor F:B-->Sets.

The purpose of this talk is to consider a new instance of 2-comprehension 
with Grothendieck toposes as types, in such a way that the comprehensive 
factorization of a geometric morphism agrees with the hyperpure, (Michael) 
complete spread factorization (Bunge-Funk, Math.Proc. Cambr. Phil. Soc, to 
appear 2006).

This goal leads to the introduction of a notion of "0-distribution" (or 
cocontinuous functor)
M:E-->Loc_0, with Loc_0 the category of 0-dimensional locales. The extension 
(or rather, display) of a 0-distribution M is constructed as the Michael 
complete spread associated with M. In the absence of the connected 
components functor, we resort to a locale of quasicomponents functor, whose 
points (if any) we interpret as completly prime filters of clopens (or of 
definable subobjects in the sense of Barr-Pare '80 when the base topos S is 
arbitrary). The "comprehensive factorization" discussed at length in 
'Singular Coverings of Toposes" (Bunge-Funk, Springer LNM, to appear 2006), 
is retrieved by restricting to discrete locales and geometric morphisms with 
a locally connected domain.

(This is joint and ongoing work with J. Funk.)