thirding

thirding

[Peter Freyd]

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.