categories: Re: Addendum

[Vaughan Pratt <pratt@cs.stanford.edu>]

>How about the positive integers with * as sum, with 1->3 as the only
>nonidentity arrow?  The unique cubical coalgebra is  1  but there is no
>square coalgebra.

Oops, that gets associativity at the cost of * no longer being a functor
(pointed out to me by Peter Selinger).

Ok, let me dig myself in deeper by making my example more complicated.
Instead of 1->3, put an arrow from i to j whenever i <= j <= 2i.
Now every i is a square coalgebra but no i is a cubical coalgebra.

Now adjoin a new object oo (infinity), with x+oo = oo+x = oo for all x,
and the identity at oo as the only new arrow.  oo is both a square *and*
a cubical coalgebra.  Since it is disconnected from the other square
coalgebras there can't be a final such.  But oo is the only cubical
coalgebra, with only one self-map, making it a final cubical coalgebra.

Vaughan