The (∞,1)-accidentopos model of unintentional type theory

Dependent type theory associates to any elements x, y of a type A an identity type x =A y, the type of proofs of equality of these elements. In PER Martin-Löf’s extensional type theory, the identity type is a subsingleton inhabited precisely when x and y are judgmentally equal. Semantically, types are thus sets equipped with equivalence relations given by their identity types. Groupoid Martin-Löf’s intensional type theory (ITT), in contrast, does not explicitly prohibit this type from having other elements; Hofmann and Streicher showed in [4] that any closed intensional type A can be interpreted as a groupoid JAK, where terms x, y : A are objects JxK, JyK ∈ JAK, and Jx =A yK is the discrete groupoid homJAK(JxK, JyK). In general, the identity type itself can have non-trivial morphisms, resulting in an infinite tower of non-trivial identity types. This observation has given rise to the Homotopy Type Theory project [1], which has provided new semantics of Infinity-Groupoid Martin-Löf’s intensional type theory in simplicial sets [5], or globular strict [8] or weak ∞-groupoids. This work explores the lesser-known unintentional type theory (UTT), which