categories: Re: Addendum

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

> Instead of 1->3, put an arrow from i to j whenever i <= j <= 2i.

I keep forgetting to make a list of what to remember.  Peter Johnstone
kindly put me out of my misery on this noncategory.

Since I don't seem to be having much luck making the example more
complicated, maybe making it simpler might work.

The ring of integers mod 3 is a one-object monoidal category in the
usual way, with multiplication as composition and addition as the monoid.

Every arrow is clearly both a square coalgebra and a cubical coalgebra,
i.e. we have three of each.

Claim: There are no final square coalgebras, but 1 and 2 are final
cubical coalgebras.

Proof.  Square coalgebra homomorphisms f from 2x to x (as square
coalgebras) are those that satisfy xf = (2f)(2x).  But 2x2 = 1 (mod
3) so every f solves this.  Hence there are three square coalgebra
homomorphisms from 2x to x, whence no x is a final square coalgebra.

Cubical coalgebra homomorphisms f from y to x (as cubical coalgebras)
must satisfy xf = (3f)y = 0.  But for x other than 0, f = 0 is the only
solution.  So the two nonzero cubical coalgebras are final.

The same example (unless I've forgottten yet another thing) solves the
corresponding problem for initial algebras.

Vaughan