text: This paper introduces a new family of models of intensional Martin-L "of type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and on cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.
  @cite_2: This paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.
summary: Spitters [Section 3] SpittersB:cubstt uses a somewhat different formulation of Moore path in the cubical topos @cite_2 . His notion is the reflexive-transitive closure of the usual path types given by the bounded interval. For better properties and to get a closer relationship with our version, one would like to quotient these cubical Spitters-Moore'' path objects up to degenerate paths; but the undecidability of degeneracy seems to stop one being able to do that while retaining the (uniform) Kan-fibrancy of such path objects. Here we can side-step such issues, since notably our models manage to avoid using a notion of Kan fibrancy at all.