A characterisation of elementary fibrations

TL;DR

This paper characterizes elementary fibrations in category theory using structures called transporters, generalizing previous results and relating to identity types in Martin-Löf type theory.

Contribution

It provides a new characterization of elementary fibrations based on fiber structures called transporters, extending prior work on faithful fibrations.

Findings

Transporters characterize elementary fibrations.

The characterization generalizes previous results for faithful fibrations.

Resemblance to identity type semantics in Martin-Löf type theory.

Abstract

Grothendieck fibrations provide a unifying algebraic framework that underlies the treatment of various form of logics, such as first order logic, higher order logics and dependent type theories. In the categorical approach to logic proposed by Lawvere, which systematically uses adjoints to describe the logical operations, equality is presented in the form of a left adjoint to reindexing along a diagonal arrows in the base. Taking advantage of the modular perspective provided by category theory, one can look at those Grothendieck fibrations which sustain just the structure of equality, the so-called elementary fibrations, aka fibrations with equality. The present paper provides a characterisation of elementary fibrations based on particular structures in the fibres, called transporters. The characterisation is a substantial generalisation of the one already available for faithful fibrations. There is a close resemblance between transporters and the structures used in the semantics of the identity type of Martin-L\"of type theory. We close the paper by comparing the two.