Hi Matti. It’s great to see this linking into your ideas. Yes, naturalness of the ‘higher topos’ definitions is the key here. From kneemo’s remarks, a cyclic generalisation of Heyting algebra would be a good place to start. By necessity, this means dealing with generalised complementation, and I agree it is important to keep Grothendieck’s vision in mind while doing this. Heh, I wish I could discipline myself to struggle through his whole letter to Quillen…

Re BLOGGER: they are being painful, but they keep promising to sort this out.

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If I have understood correctly, logical deductions could be seen as “category-theoretical reductionism” to initial objects by truth preserving paths or as reverse paths from terminal objects.

The following example concretizes Grothendienck’s vision. Consider the categories of sets and open sets in plane, call then Set and Open. In Set negation of statement is represented as the complement of set. In Open the interior of complement would represent it. Therefore plane with any number of punctures, which is open set, has always empty set as interior of complement so that negation of empty set is not unique and double negation can lead from one punctured plane to another punctured plane.

I encountered so called p-adic logics and matrix logics for some time ago and the problem of defining the negation or its generalization in a natural manner. For instance, should one replace negation with n-valued cyclic map? That would mean that initial and final objects are replaced by vertices of n-polygon so that one would have a new variant of category-theoretical reductionism!

Just thoughts;-)

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