Abstract
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.
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