arXiv subjects

Joseph Hua

Publications and source records attributed to Joseph Hua.

3 recordsLinked to original sources

arXiv2026

Internal Algebraic Type Theory

This thesis brings us closer to applying computer-assisted, internal, type-theoretic reasoning to a category, with examples in the category of cubical sets, the category of groupoids, and the category of categories. The steps we make towards this general goal are both in furthering the type theoretic analysis of these examples, as well as implementing computer-assisted syntax-semantic reasoning as part of the HoTTLean project. One key component of this work is the consideration of new exponentiability conditions with respect to a class of maps in a category, and the development of polynomial functors for these maps, making the methods of "algebraic type theory" possible in this very general setting.

math.CT↗

arXiv2026

Polynomial functors in {\pi}-clans for the semantics of type theory

The category of contexts underlying a model of Martin-L\"of type theory with Unit-, $\Sigma$-, and $\Pi$-types need not be locally Cartesian closed, but is necessarily a $\pi$-clan. We exploit this $\pi$-clan structure to build the theory of polynomial functors. This paper presents two equivalent notions of strict semantics for MLTT in this weaker setting, respectively "elementary models" - reformulating categories with families - and "algebraic models" - reformulating natural models. These components fit into a practical sequence of steps for constructing models of MLTT: building an elementary model, extracting a $\pi$-clan from the elementary model, and then using polynomial functors built on the $\pi$-clan structure to convert the elementary model into an algebraic one.

math.CT↗

arXiv2026

Path Types in Algebraic Type Theory

A new approach to the semantics of identity types in intensional Martin-L\"of type theory is proposed, assuming only a category with finite limits and an interval. The specification of \emph{extensional} identity types in the original presentation of natural models paralleled that of the other type formers $\Sigma$ and $\Pi$, but the treatment of the \emph{intensional} case there was less uniform. It was later reformulated to an account based on polynomials; here a further improvement in the style of the other type formers is achieved by employing an interval, in order to give a single pullback specification of a model with \emph{path types}. The interval is also used to specify a (Hurewicz) fibration structure on the universe of the model. It is shown that the combination of these two conditions suffices to model the intensional identity rules, assuming only finite limits. The addition of an interval also relates the current treatment to that of cubical type theory.

math.CT↗