Loops

The loop diagram, consisting of one object and one arrow

is least studied of all small categorical diagrams. The limit of a loop f on an object X is its object Fix(f) of fixpoints . Its colimit is the universal invariant Universal(f). These ideas were used to characterise while-loops in Fixpoint and Loop Constructions as Colimits .

When the canonical composite

Fix(f) ---> X ---> Universal(f)
is an isomorphism then iteration of f will always converge to a fixpoint, and we say that f is a convergent loop. Examples can be constructed in Sets, domains, metric spaces, etc. This supports a direct definition of a tail recursive list object in terms of convergence of the loop which performs one step in the recursion Tail recursion through universal invariants .

Page Last Updated: Monday, 03-Nov-2003 16:47:53 EST

Please feel free to send any comments.