Fixpoint and loop constructions as colimits

Abstract

The constructions of fixpoints and while-loops in a category of domains can be derived from the colimit, loop(f) of a diagram which consists of a single endomorphism f : D → D. If f is increasing then the colimiting map is the least-fixpoint function Y and

$$loop\left( f \right) = fix\left( f \right)$$

the subobject of fixpoints. If f=cond(b, g, 1) is the conditional of a while-program then

$$loop\left( f \right) = \left( {D_{\neg b} + loop_\infty \left( g \right)} \right)_ \bot$$

the lifted sum of the terminating values (where b is false) and the infinite loops.

Supported by grants GR/E 78487 and GR/F 07866 from SERC, and OGPIN 016 from NSERC.