  This thesis concerns embeddings and self-embeddings of foundational
structures in both set theory and category theory.
  The first part of the work on models of set theory consists in establishing a
refined version of Friedman's theorem on the existence of embeddings between
countable non-standard models of a fragment of ZF, and an analogue of a theorem
of Gaifman to the effect that certain countable models of set theory can be
elementarily end-extended to a model with many automorphisms whose sets of
fixed points equal the original model. The second part of the work on set
theory consists in combining these two results into a technical machinery,
yielding several results about non-standard models of set theory relating such
notions as self-embeddings, their sets of fixed points, strong rank-cuts, and
set theories of different strengths.
  The work in foundational category theory consists in the formulation of a
novel algebraic set theory which is proved to be equiconsistent to New
Foundations (NF), and which can be modulated to correspond to intuitionistic or
classical NF, with or without atoms. A key axiom of this theory expresses that
its structures have an endofunctor with natural properties.
