  The homotopical approach to intensional type theory views proofs of equality
as paths. We explore what is required of an object $I$ in a topos to give such
a path-based model of type theory in which paths are just functions with domain
$I$. Cohen, Coquand, Huber and M\"ortberg give such a model using a particular
category of presheaves. We investigate the extent to which their model
construction can be expressed in the internal type theory of any topos and
identify a collection of quite weak axioms for this purpose. This clarifies the
definition and properties of the notion of uniform Kan filling that lies at the
heart of their constructive interpretation of Voevodsky's univalence axiom.
(This paper is a revised and expanded version of a paper of the same name that
appeared in the proceedings of the 25th EACSL Annual Conference on Computer
Science Logic, CSL 2016.)

Logic in Computer Science