Typical reflection principles in set theory are concerned with the height of the universe, or the relative height of certain stages. The resemblance between stages or between the universe itself and some of these stages is a very useful guiding principle that serves us to motivate large cardinal statements and many consequences of forcing axioms.
It is natural to wonder about similar reflection principles concerned instead with the width of the universe. In our paper Inner-model reflection principles, Neil Barton, Gunter Fuchs, Joel David Hamkins, Jonas Reitz, Ralf Schindler and I consider precisely this kind of reflection. Say that the inner-model reflection principle holds if and only if for any set 
We establish the consistency of the principle relative to ZFC. In fact, we build a model of the stronger ground-model reflection principle, where we further require that any such first-order 
The paper studies the principle under large cardinals and forcing axioms, and compares it with other statements considered in recent years, such as the maximality principle or the inner model hypothesis. The most technically involved and interesting results in the paper show that inner-model reflection and even ground-model reflection hold in certain fine-structural inner models but also that this requires large cardinals, and that the large cardinal requirements differ for both principles (precisely a proper class of Woodin cardinals is needed for ground-model reflection).
Curiously, the paper started as a series of informal exchanges in response to a question on math.stackexchange.
See also here. The paper will appear in Studia Logica. Meanwhile, it can be accessed on the arXiv, or in my papers page.
A kind of library 