Summary: Elementary Axioms for Local Maps of Toposes
Steven Awodey \Lambda Lars Birkedal y
November 18, 1999
Dedicated to Saunders Mac Lane on his 90th birthday.
We present a complete elementary axiomatization of local maps of
We recall the definition of a local map of toposes [9, 10, 7] (see in particular [7,
Definition 1.1. Let E and F be elementary toposes. A geometric morphism
f = (f \Lambda ; f \Lambda ) : E ! F is local if it is bounded and the direct image functor f \Lambda
has a right adjoint f ! which is full and faithful.
There are many examples of local maps of toposes, the classical one being
(the structure map of sheaves on) the spec of a local ring (arising, e.g., from
localization at a point). See, e.g.,  for many other topological and presheaf
examples. See  for an example of a (localic) local map between realizability
toposes; this example is the one that gave rise to this work.
Suppose (\Delta; \Gamma) : E ! F is a local map of toposes. Then since the right
adjoint, call it r, of \Gamma is full and faithful, it follows easily that the inverse image