categories: "catamorphism"

[Frank Atanassow <franka@cs.uu.nl>]

What is the definition/scope of the term "catamorphism"?

In the calculational school of programming, I think people agree that:

  For every F-algebra A, the unique F-homomorphism from an initial F-algebra to
  A is a catamorphism.

but:

  Is the converse true also, or is `catamorphism' more general: "a
  catamorphism is the unique arrow from an initial _object_ (in any category,
  not just categories of algebras)." I think the answer to that question is
  no, but is there a snappy name for such arrows?

  What about the universal arrows in the free algebra construction? Are those
  `catamorphisms' or not? If not, is there a name for those arrows?

and furthermore:

  In the case of initial F-algebras, for F an endofunctor on category C, is the
  catamorphism the arrow in the category of F-algebras, or the corresponding
  one in C? In other words, is `catamorphism' a property of the arrow or the
  arrow + the category? If you compare with `monomorphism', since we speak of
  being `monic _in_ C', I presume the latter, and thus only one arrow would be
  `catamorphic', but which? Judging from the literature, I am inclined to
  say that the arrow in C is the catamorphism, but I would like a second
  opinion.

-- 
Frank Atanassow, Information & Computing Sciences, Utrecht University
Padualaan 14, PO Box 80.089, 3508 TB Utrecht, Netherlands
Tel +31 (030) 253-3261 Fax +31 (030) 251-379