  As several different formal systems with inequivalent syntax may describe
equivalent semantics, it is possible to find `completions' to more expressive
syntaxes that are semantically invariant. Doctrine theory, in the sense of
Lawvere, is the natural categorical framework in which to express completions
for first-order logic. We study the suitability of a fibred generalisation of
the ideal completion of a preorder to act as a completion for doctrines to the
syntax of geometric logic. In contrast to other completions of doctrines
considered in the literature, our completion takes a Grothendieck topology as a
second argument. As a result, the geometric completion is idempotent, as well
as being `semantically invariant' for any doctrine whose models can be
expressed as a category of continuous flat functors, encompassing a wide class
of the most commonly considered doctrines.
  We also relate the geometric completion to other completions of doctrines
considered in the literature: first, by studying the behaviour of the geometric
completion when the second argument is omitted, and then by studying the
interaction with those completions of doctrines that complete to a fragment of
geometric logic. Throughout, we reference how these completions of doctrines
yield completions of categories via the syntactic category construction. We
demonstrate that it is equivalent to represent logical theories by either
doctrines or syntactic categories, in so far as they have equivalent
classifying toposes.
