RE: Subobject Classifier Algorithm

["F. William Lawvere" <wlawvere@hotmail.com>]

For sheaves on a finite site (which by a theorem of AGV in SGA4 vol 2 is the same as all presheaves on some smaller finite category C), take the category of all functors from C^op to bold 2. It is a finite poset, in fact, a Heyting algebra (indeed even  bi-Heyting) belying the old misconception that one deviates from Boole only for infinite sets. If for each given A  in C we do the same for C/A, we get the figures of shape A in the Omega of the topos. The adjoints to maps induced by A'->A give a concrete model of tense logic.
By the same AGV (not only these C/A' ->C/A but) any functor between finite categories induces a geometric morphism that is even "essential".
Actually, taking sheaves valued in the topos of finite sets would be interesting, providing a more objective version of number theorythan the abstract exponential rig traditionally called "natural".
Topos theory is bristling with potential examples that we "generalists" have been slow to take up.
Anyway, the above construction of Omega is manifestly exponential, hence an effort to find computable sub cases is clearly needed, as you suggest Ellis.
Bill



> Date: Wed, 23 Feb 2011 10:16:56 -0500
> To: categories@mta.ca
> From: xtalv1@netropolis.net
> Subject: categories: Subobject Classifier Algorithm
> 
> What are the general rules for calculating the sub-object classifier
> of a topos? Or, for what class of toposes is there an algorithm for
> calculating the sub-object classifier of its members?
> 
> Ellis D. Cooper
> 
> 

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