Caramello developed the theory of "toposes as bridges", which consists in methods and techniques for unifying different mathematical theories and transferring information between them by using toposes. This theory is based on the duality of
sites and
Grothendieck toposes, and on the notion of
classifying topos of a geometric
first-order theory, exploiting the diversity of possible presentations of each topos by infinitely many sites or theories. Caramello's theory involves several components: on the one hand, establishing
equivalences between
toposes presented in different ways; on the other hand, calculating or expressing topos invariants in terms of the various types of presentations considered, in order to produce correspondences between properties or elements of these various presentations. The theory of "toposes as bridges" can be considered a meta-mathematical theory of the relations between different theories and her program contributes to realizing the unifying potential of the notion of topos already glimpsed by
Alexander Grothendieck. Caramello organized international conferences in topos theory, "Topos à l'IHES" (2015). and "Toposes in Como" (2018) She is an editor of the journal
Logica Universalis and is running a blog and forum about toposes. ==Awards and recognition==