RE: Subobject Classifier Algorithm

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

Of course the initial responses  given by Fred and me were notthemselves algorithmic, but why can they not be a prelude to such?Fred recalled the sequence of UMPs that defines Omega and I recalled what results in the case of a finite site. The hope is that experts in Maple and the like will be able to solve the sort of problem needed to generate displays. That is, given a finite presentation of a category C,find a presentation of the resulting Heyting algebra with the action ofC on it. Of course as stated this includes the Burnside problem , the Post problem etc . Hence the need for recognizing solvable subproblems.Even  for the class of graphic monoids, where the structure is finite, it growsexponentially , which has been used in the past by other computer scientists as an excuse not to consider it.  But if we are given a few fixed finite C, we can consider the class of categories discretely fiberedover them (equivalent to objects of the toposes) and try to see whetherOmega is computable in the above sense for them.
I  said contravariant 2-valued functors and of course these factor through the poset reflection. Calculating that poset reflection is a search problem  wrt 
the underlying graph, but actually wrt composition as well since we have todo it for the slice categories C/A.Bill> Date: Fri, 25 Feb 2011 11:20:36 +0000
> From: pt11@PaulTaylor.EU
> To: categories@mta.ca
> Subject: categories: Subobject Classifier Algorithm
> 
> Ellis Cooper asked,
> 
>> 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?
> 
> Thanks to Bill and Fred for describing the constructions.
> 
> I would suggest, however, that it is rather stretching the meaning
> of the word "algorithm" to describe them as such.   What kind of
> machine might be able to perform these operations?
> 
> A propos of this question, it is well known both to new students
> of category theory and to those who like to use the subject to
> discuss Life, The Universe And Everything, that:
> 
> (1)   a topos is a cartesian closed category with
> (2)   an internal Heyting algebra Omega,
> 

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