Topology-free type structures with conditioning events

Starting without any topological assumption, we establish the existence of the universal type structure in presence of—possibly uncountably many and topologically unrestricted—conditioning events, namely, a type structure that is non-redundant, belief-complete, terminal, and unique up to measurable type isomorphism, by performing a construction in the spirit of the hierarchical one in Heifetz and Samet (J Econ Theory 82:324–341, 1998). In particular, we obtain the result by exploiting arguments from category theory and the theory of coalgebras, thus, making explicit the mathematical structure underlying all the constructions of large interactive structures and obtaining the belief-completeness of the structure [unattainable via the standard hierarchical construction à la (Heifetz and Samet in J Econ Theory 82:324–341, 1998)] as an immediate corollary of known results from these fields. Additionally, we show how our construction, with its lack of topological and cardinality assumptions on the family of conditioning events, can be employed in various game-theoretical contexts.