Re: free algebras in ASD

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

Paul Taylor wrote in part:

>Toby Bartels asked me:

>>My intuition is that polynomials with overt discrete coefficients
>>should have overt discrete initial algebras,
>>while those with compact Hausdorff coefficients
>>should have compact Hausdorff final coalgebras.
>>Have you any thoughts about that question?

>In fact, what I have to say about this (in the setting of the
>existing established theory for locally compact spaces) is little
>more than Toby's "intuition".  The existence of these spaces
>would follow from the limit--colimit coincidence, which is sketched
>in Remark 10.16 of
>    "Geometric and higher-order logic in ASD"
>    www.PaulTaylor.EU/ASD/loccpct#geohol

Yes, there it is!  That's what I get for not reading them in order.  (^_^)

>The symmetry between
>     =>  T  /\   =    some   overt    discrete   free     algebra
>and  <=  F  \/   !=   all   compact   Hausdorff  cofree   coalgebra
>is very strong in this, but not perfect, because
>     N   is   overt    discrete    Hausdorff      not compact
>   2^N   is   compact  Hausdorff   not discrete   overt
>I have not managed to isolate convincingly the precise point where
>the symmetry breaks down.

I was about to say that this comparison is not really fair,
since N is the initial algebra of X |-> X + 1,
while 2^N is the final coalgebra of X |-> 2 x X,
but I guess that the final coalgebra of X |-> X + 1
is also overt but not discrete.

Perhaps the asymmetry is simply between initial algebras and final colagebras.
One is a colimit and the other is a limit;
there are already several asymmetries between these,
such as that products distribute over sums but not vice versa.
Indeed, if final coalgebras preserve "some"s,
this might be more than just a bad pun.


--Toby