An abstract view on syntax with sharing

The notion of term graph encodes a refinement of inductively generated syntax in which regard is paid to the the sharing and discard of subterms. Inductively generated syntax has an abstract expression in terms of initial algebras for certain endofunctors on the category of sets, which permits one to go beyond the set-based case, and speak of inductively generated syntax in other settings. In this article, we give a similar abstract expression to the notion of term graph. Aspects of the concrete theory are redeveloped in this setting, and applications beyond the realm of sets discussed.