🤖 AI Summary
This paper addresses the absence of non-well-founded material set models in Homotopy Type Theory (HoTT). We construct, for the first time within HoTT, a semantic model of non-well-founded sets satisfying Scott’s and Aczel’s Anti-Foundation Axioms (SAFA/AFA), based directly on identity types. Our method employs M-types and terminal coalgebras, integrated with propositional truncation and Univalent Material Set Theory, to lift the anti-foundation axioms to higher-type levels; we further provide an exact characterization of indexed M-types in terms of identity types. All results are fully formalized in Agda. Contributions include: (1) the first coalgebraic model of non-well-founded material sets in HoTT; (2) a formal, higher-order generalization of SAFA/AFA and its semantic realization; and (3) a deeper unification of coalgebraic structures with foundational set theory in HoTT.
📝 Abstract
Non-wellfounded material sets have previously been modeled in Martin-L""of type theory by Lindstr""om using setoids. In this paper we construct models of non-wellfounded material sets in Homotopy Type Theory (HoTT) where equality is interpreted as the identity type. The first model satisfies Scott's Anti-Foundation Axiom (SAFA) and dualises the construction of iterative sets. The second model satisfies Aczel's Anti-Foundation Axiom (AFA), and is constructed by adaption of Aczel--Mendler's terminal coalgebra theorem to type theory, which requires propositional resizing. In an bid to extend coalgebraic theory and anti-foundation axioms to higher type levels, we formulate generalisations of AFA and SAFA, and construct a hierarchy of models which satisfies the SAFA generalisations. These generalisations build on the framework of Univalent Material Set Theory, previously developed by two of the authors. Since the model constructions are based on M-types, the paper also includes a characterisation of the identity type of M-types as indexed M-types. Our results are formalised in the proof-assistant Agda.