Homotopy Model Theory, The Journal of Symbolic Logic, 86/4 (Dec. 2021), p. 1301-1323.
doi:10.1017/jsl.2019.19
Any structure for a first-order language can be turned into a simplicial set. This prompts a bundle of connections between model theory and homotopy theory. An adjunction result is proved between a subcategory of simplicial sets and the category of o-minimal structures.
Models As Universes, Notre Dame Journal of Formal Logic 58/1, 2017, p. 47-78.
Every model of ZFC can be shown to contain, in a sense to be precised, another model of ZFC. The paper explains that fact and draws conclusions based on it about logical consequence from ZFC. A modal logic of internal models of ZFC is set out at the end.
Sets and Descent, in A. Sereni & F. Boccuni (eds), Objectivity, Realism and Proof, Springer, Boston Studies in the History and Philosophy of Science 318, 2016, p. 123-142.
Algebraic Set Theory (AST) is an influential reconsideration of Zermelo-Frankel set theory (ZFC) in category-theoretic terms. This paper gets back to the original formulation by Joyal & Moerdijk. It explains in detail how the first axioms of AST (in this formulation) set up a framework linked to modern algebraic geometry. As a result, AST is shown to accomplish, not only an original and fruitful combination of set theory with category theory, but the genuine graft of a deeply geometric idea onto the usual setting of ZFC.
Sur une application possible du concept d’homotopie à la théorie des modèles, Annales de la Faculté des Sciences de Toulouse, Série 6, XXII/5, 2013, p. 1017-1043.
This paper endeavors to show the possible application to model theory of concepts coming from modern homotopy theory. In particular, the concept of simplicial set can be brought into play to describe the formulas of a first-order language L, the definable subsets of an L-structure, as well as the type spaces of a theory expressed in L. A comparison is sketched between categories of models (in the model-theoretic sense) and model categories (in the homotopy-theoretic sense).