Products of families of types in the C-systems defined by a universe category

Abstract

We introduce the notion of a $(\Pi,\lambda)$-structure on a C-system and show that C-systems with $(\Pi,\lambda)$-structures are constructively equivalent to contextual categories with products of families of types. We then show how to construct $(\Pi,\lambda)$-structures on C-systems of the form $CC({\cal C},p)$ defined by a universe $p$ in a locally cartesian closed category $\cal C$ from a simple pull-back square based on $p$. In the last section we prove a theorem that asserts that our construction is functorial. This version introduces some changes compared to the previous one to ensure rigorous compatibility with arXiv:1409.7925v3.