Related papers: Parametricity and Semi-Cubical Types
We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms…
We propose a new cubical type theory, termed (self-deprecatingly) the naive cubical type theory, and study its semantics using the universe category framework, which is similar to Uemura's categories with representable morphisms. In…
Inference on the parametric part of a semiparametric model is no trivial task. If one approximates the infinite dimensional part of the semiparametric model by a parametric function, one obtains a parametric model that is in some sense…
The notion of a categorical quotient can be generalized since its standard categorical concept does not recover the expected quotients in certain categories. We present a more general formulation in the form of $\mathcal{F}$-quotients in a…
We present a refinement of the Calculus of Inductive Constructions in which one can easily define a notion of relational parametricity. It provides a new way to automate proofs in an interactive theorem prover like Coq.
Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djokovi\'{c}'s and Winkler's relations play an important role in the theory of partial cubes. These structures are employed in the…
In this paper we give a brief review of semiparametric theory, using as a running example the common problem of estimating an average causal effect. Semiparametric models allow at least part of the data-generating process to be unspecified…
We introduce a general notion of $J$-tribe, and construct the $J$-tribe of $J$-frames in a given tribe $\mathcal{T}$, where $J$ a suitable generalized direct category. This construction applies to semi-cubical diagrams for a category of…
We develop formulas that define permutahedral commutation coherence relations of all orders. To illustrate the result geometrically, we begin by defining a rigid transformation of the $(n+1)$-permutahedron into a $n$-cube of dimensions $1…
Reynold's abstraction theorem is now a well-established result for a large class of type systems. We propose here a definition of relational parametricity and a proof of the abstraction theorem in the Calculus of Inductive Constructions…
We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms…
A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…
Reynolds' parametricity originally equips types with proof-irrelevant binary propositional relations over the types. But such relations can also be taken proof-relevant or unary, and described either in an indexed or fibred way.…
We give a polymorphic account of the relational algebra. We introduce a formalism of ``type formulas'' specifically tuned for relational algebra expressions, and present an algorithm that computes the ``principal'' type for a given…
A survey of properties of the adjunction involving a semisymmetrization functor, which was suggested by J.D.H. Smith, and which maps the category of quasigroups with homotopies to the category of semisymmetric quasigroups with…
We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin-L\"{o}f type theory, two-level type theory and cubical type theory. We establish basic results in the semantics of type theory:…
Quotients and comprehension are fundamental mathematical constructions that can be described via adjunctions in categorical logic. This paper reveals that quotients and comprehension are related to measurement, not only in quantum logic,…
This work arose from efforts to generalise the usual cubical boundary by using different 'weights' for opposite faces, but still to obtain a chain complex, and this method was found to generalise. We describe a variant of the classical…
An early result in the theory of Natural Dualities is that an algebra with a near unanimity (NU) term is dualizable. A converse to this is also true: if V(A) is congruence distributive and A is dualizable, then A has an NU term. An…
We present a formalization of a version of Abadi and Plotkin's logic for parametricity for a polymorphic dual intuitionistic/linear type theory with fixed points, and show, following Plotkin's suggestions, that it can be used to define a…