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First-Order Logical Duality Henrik Forssell
 

Summary: First-Order Logical Duality
Henrik Forssell
A dissertation submitted in partial fulfillment of the
requirements for the degree of
PhD in Logic, Computation and Methodology
Carnegie Mellon University
2008
Abstract
Generalizing Stone duality for Boolean algebras, an adjunction between Bool-
ean coherent categories--representing first-order syntax--and certain topo-
logical groupoids--representing semantics--is constructed. The embedding
of a Boolean algebra into a frame of open sets of a space of 2-valued mod-
els is replaced by an embedding of a Boolean coherent category, B, into a
topos of equivariant sheaves on a topological groupoid of set-valued models
and isomorphisms between them. The latter is a groupoid representation
of the topos of coherent sheaves on B, analogously to how the Stone space
of a Boolean algebra is a spatial representation of the ideal completion of
the algebra, and the category B can then be recovered from its semantical
groupoid, up to pretopos completion. By equipping the groupoid of sets and
bijections with a particular topology, one obtains a particular topological

  

Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University

 

Collections: Mathematics