categories: co-iteration?

["Al Vilcius" <avilcius@webpearls.com>]

For many (categorical) notions,
there is a useful (often fantastic) dual notion.

What about iteration?

In a category with finite coproducts,
we have a notion of iteration   f:A-->A+B
(written A -f-> A+B here)
which in the case of sets and partial functions, for example,
is completely specified by the Elgot equation
   A -f-> A+B -f"+1-> B =  A -f"-> B  recursive in  f"
(plus one more little, quite natural condition - see [Manes '92])
This awful looking mess, written using the infix morphism notation,
actually looks quite neat when you draw the diagram.

Now without meaning to start the "co"-wars again,
- is there a useful notion of co-iteration?
- what could it do for us, say in the category of partial functions?
- is there a simple algebra/coalgebra context?

Reference:
[Manes '92] E.G.Manes, "Predicate Transformer Semantics", CUP 1992

Al Vilcius
personal: al.r@vilcius.com
business: avilcius@webpearls.com