Topological Representation of Geometric Theories

Using Butz and Moerdijk's topological groupoid representation of a topos with enough points, a `syntax-semantics' duality for geometric theories is constructed. The emphasis is on a logical presentation, starting with a description of the semantical topological groupoid of models and isomorphisms of a theory and a direct proof that this groupoid represents its classifying topos. Using this representation, a contravariant adjunction is constructed between theories and topological groupoids. The restriction of this adjunction yields a contravariant equivalence between theories with enough models and semantical groupoids. Technically a variant of the syntax-semantics duality constructed in [Awodey and Forssell, arXiv:1008.3145v1] for first-order logic, the construction here works for arbitrary geometric theories and uses a slice construction on the side of groupoids---reflecting the use of `indexed' models in the representation theorem---which in several respects simplifies the construction and allows for an intrinsic characterization of the semantic side.Comment: 32 pages. This is the first pre-print version, the final revised version can be found at http://onlinelibrary.wiley.com/doi/10.1002/malq.201100080/abstract (posting of which is not allowed by Wiley). Changes in v2: updated comment

Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective

We establish, generalizing Di Nola and Lettieri's categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-intepretability holding for particular classes of formulas: irreducible formulas, geometric sentences and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang's MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.Comment: 54 page