Polynomial functors in π-clans for the semantics of type theory

Abstract

The category of contexts underlying a model of Martin-Löf 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.