200 research outputs found
An Algebraic Weak Factorisation System on 01-Substitution Sets: A Constructive Proof
We will construct an algebraic weak factorisation system on the category of
01 substitution sets such that the R-algebras are precisely the Kan fibrations
together with a choice of Kan filling operation. The proof is based on Garner's
small object argument for algebraic weak factorization systems. In order to
ensure the proof is valid constructively, rather than applying the general
small object argument, we give a direct proof based on the same ideas. We use
this us to give an explanation why the J computation rule is absent from the
original cubical set model and suggest a way to fix this
Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom
This paper presents a type theory in which it is possible to
directly manipulate -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\u27s 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
Axioms for modelling cubical type theory in a Topos
The homotopical approach to intensional type theory views proofs of equality as paths. We explore what is required of an interval-like object I in a topos to give a model of type theory in which elements of identity types are functions with domain I. Cohen, Coquand, Huber and Mörtberg give such a model using a particular category of presheaves. We investigate the extent to which their model construction can be expressed in the internal type theory of any topos and identify a collection of quite weak axioms for this purpose. This clarifies the definition and properties of the notion of uniform Kan filling that lies at the heart of their constructive interpretation of Voevodsky’s univalence axiom. Furthermore, since our axioms can be satisfied in a number of different ways, we show that there is a range of topos-theoretic models of homotopy type theory in this style.Engineering and Physical Sciences Research Council (Doctoral Training Award)This is the final version of the article. It first appeared from Schloss Dagstuhl via http://dx.doi.org/10.4230/LIPIcs.CSL.2016.2
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