Conditionally reversible computations and weak universality in category theory

Main attention is directed to the notion of weak universality in category theory. While the definitions based on the ordinary universal constructions usually hold up to isomorphisms, that is, unconditionnally reversible arrows, weakly universal constructions may be seen “positively” as defined up to conditionnally reversible arrows. It is shown that weak universality is closely connected with intensional equality, typically considered in categories used in computer science. As a possible application of weakly universal categorical constructions we suggest the notion of conditionnally reversible computation in the theory of computations.