Topology-Free Type Structures with Conditioning Events

TL;DR

This paper proves the existence of a universal type structure with conditioning events without topological assumptions, using category theory and coalgebras, confirming a longstanding conjecture.

Contribution

It introduces a topology-free construction of universal type structures with conditioning events, resolving a major open problem in the field.

Findings

Existence of a belief-complete, non-redundant universal type structure

Construction based on category theory and coalgebras

Affirmative answer to Battigalli & Siniscalchi's conjecture

Abstract

We establish the existence of the universal type structure in presence of conditioning events without any topological assumption, namely, a type structure that is terminal, belief-complete, and non-redundant, by performing a construction \`a la Heifetz & Samet (1998). In doing so, we answer affirmatively to a longstanding conjecture made by Battigalli & Siniscalchi (1999) concerning the possibility of performing such a construction with conditioning events. In particular, we obtain the result by exploiting arguments from category theory and the theory of coalgebras, thus, making explicit the mathematical structure underlining all the constructions of large interactive structures and obtaining the belief-completeness of the structure as an immediate corollary of known results from these fields.