Relative Full Completeness for Bicategorical Cartesian Closed Structure

Abstract

The glueing construction, defined as a certain comma category, is an important tool for reasoning about type theories, logics, and programming languages. Here we extend the construction to accommodate '2-dimensional theories' of types, terms between types, and rewrites between terms. Taking bicategories as the semantic framework for such systems, we define the glueing bicategory and establish a bicategorical version of the well-known construction of cartesian closed structure on a glueing category. As an application, we show that free finite-product bicategories are fully complete relative to free cartesian closed bicategories, thereby establishing that the higher-order equational theory of rewriting in the simply-typed lambda calculus is a conservative extension of the algebraic equational theory of rewriting in the fragment with finite products only.

A categorical framework for congruence of applicative bisimilarity in higher-order languages

“…It would be instructive to better understand potential links between the two. Finally, let us mention recent work which, just like ours, establishes abstract versions of standard constructions and theorems in programming language theory like type soundness [6] or gluing [18,19].…”

Coherence and normalisation-by-evaluation for bicategorical cartesian closed structure

“…First, Theorem 1.1 drastically reduces the difficulty of constructing structure in an arbitrary cartesian closed bicategory. This will have applications in the development of a theory of opetopic structures [26] in the cartesian closed bicategories of generalised species [27] and cartesian distributors [29], as well as in the study of the equational theory of rewriting in the STLC [32,33].…”

“…Finally, we build the framework for future applications. The bicategorical glueing construction developed for this proof has already been put to use in the study of the equational theory of rewriting [33]. More broadly, this paper initiates the study of the 'bicategorical semantics' of simple type theories, and of the application of categorical normalisation arguments to coherence.…”

“…Theorem 4.9 ( [33,66]). Let (B, Π n (−), =▷) and (C, Π n (−), =▷) be cc-bicategories and assume that C has pullbacks.…”