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*To*: categories@mta.ca*Subject*: categories: thirding*From*: Peter Freyd <pjf@saul.cis.upenn.edu>*Date*: Wed, 19 Jan 2000 14:57:34 -0500 (EST)*Sender*: cat-dist@mta.ca

Let S and C be endo-functors that commute, that is, there's a natural equivalence c:CS --> SC. If f: F -> SF is a final S-coalgebra then there a special map CF --> F, to wit, the induced Cf c_F coalgebra map from the S-coalgebra CF --> CSF --> SCF. Now specialize to the case that SX is X*X for an associative bifunctor and CX is X*X*X. Indeed, specialize further to the case that * is the ordered-wedge functor so that F is the closed interval, I. In this special case the induced map I v I v I --> I is an isomorphism and its inverse makes I a final cubical coalgebra. I don't know a general theorem that specializes to this result. Clearly I v I v I --> I can be used to obtain the thirding map on the interval that I needed for my definition of derivatives. And clearly there's nothing special about the number three. We obtain a special isomorphism from every iterated ordered-wedge of I back to I. In that previous posting on derivatives I wrote: Using that the closed interval is the final coalgebra for X v X v X we can define the thirding map t:I --> I in a manner similar to (and simpler than) the definition of the halving map. The problem, of course, was that there was no control on _which_ final cubical coalgebra structure was to be used.

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