Abstract
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.
Highlights
1.1 AimsPitts showed in [12] and [13], following earlier work by Staton, that the category of cubical sets is equivalent to a category based on nominal sets, called the category of 01-substitution sets, or 01Sub.We will construct an algebraic weak factorisation system on 01Sub such that the Ralgebras are precisely the Kan fibrations together with a choice of Kan filling operation
The author will show how these new path objects can be understood via Riehl’s notion of algebraic model structure and give a more complete proof that this gives a model of type theory including the computational rule for identity
Cubical sets were developed by Bezem, Coquand and Huber in [3] as a constructive model of homotopy type theory, inspired by the simplicial set model due to Voevodsky and other homotopical models
Summary
Pitts showed in [12] and [13], following earlier work by Staton, that the category of cubical sets is equivalent to a category based on nominal sets, called the category of 01-substitution sets, or 01Sub. We will construct an algebraic weak factorisation system on 01Sub such that the Ralgebras are precisely the Kan fibrations together with a choice of Kan filling operation. The author will show how these new path objects can be understood via Riehl’s notion of algebraic model structure (from [15]) and give a more complete proof that this gives a model of type theory including the computational rule for identity These ideas (via an earlier version of this paper and direct communication with the author), together with some rephrasing and simplification have already been used by Cohen, Coquand, Huber and Mortberg to implement identity types in a new variant of cubical sets (see [6, Section 9.1])
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