C-system of a module over a monad on sets

TL;DR

This paper constructs C-systems from monads and modules over sets, providing a mathematical framework for the initial steps of the semantics of dependent type theories, especially in the context of nominal signatures.

Contribution

It introduces a method to build C-systems from any monad and its left module, extending the semantic understanding of dependent type theories.

Findings

Constructs C-systems from monads and modules over sets.

Describes sub-quotients of C-systems in terms of objects from R and LM.

Applies to monads of expressions in nominal signatures, linking to dependent type theories.

Abstract

This is the second paper in a series that aims to provide mathematical descriptions of objects and constructions related to the first few steps of the semantical theory of dependent type systems. We construct for any pair $(R,LM)$, where $R$ is a monad on sets and $LM$ is a left module over $R$, a C-system (contextual category) $CC(R,LM)$ and describe a class of sub-quotients of $CC(R,LM)$ in terms of objects directly constructed from $R$ and $LM$. In the special case of the monads of expressions associated with nominal signatures this construction gives the C-systems of general dependent type theories when they are specified by collections of judgements of the four standard kinds.