  We provide bicategorical analogs of several aspects of the notion of geometry
in the sense of the theory of spectrum. We first introduce a notion of local
right biadjoint, and prove it to be equivalent to a notion of bistable
pseudofunctor, categorifying an analog 1-categorical result. We also describe
further laxness conditions, giving some properties of the already known lax
familial pseudofunctors. We also describe 2-dimensional analogs of
orthogonality and factorization systems, and use them to construct examples of
bistable pseudofunctors through inclusion of left objects and left maps. We
apply the latter construction to several examples of factorization systems for
geometric morphisms to produce geometry-like situations for Grothendieck topoi.
In particular we prove a new (terminally connected, pro-etale) factorization
theorem for geometric morphisms, which corresponds to a certain local geometry
involved in the general construction of spectra.
