  Definability is a key notion in the theory of Grothendieck fibrations that
characterises when an external property of objects can be accessed from within
the internal logic of the base of a fibration. In this paper we consider a
generalisation of definability from properties of objects to structures on
objects, introduced by Shulman under the name local representability.
  We first develop some general theory and show how to recover existing notions
due to B\'{e}nabou and Johnstone as special cases. We give several examples of
definable and non definable notions o structure, focusing on algebraic weak
factorisation systems, which can be naturally viewed as notions of structure on
codomain fibrations. Regarding definability, we give a sufficient criterion for
cofibrantly generated awfs's to be definable, generalising a construction of
the universe for cubical sets, but also including some very different looking
examples that do not satisfy tininess in the internal sense, that exponential
functors have a right adjoint. Our examples of non definability include the
identification of logical principles holding for the interval objects in
simplicial sets and Bezem-Coquand-Huber cubical sets that suffice to show a
certain definition of Kan fibration is not definable.
