[FOM] new paper
Steve Awodey
awodey at andrew.cmu.edu
Thu Aug 19 14:49:48 EDT 2010
Some readers of this list may be interested in the following new
preprint:
First-Order Logical Duality
Steve Awodey & Henrik Forssell
available at:
http://www.andrew.cmu.edu/user/awodey/preprints/fold.pdf
Abstract:
From a logical point of view, Stone duality for Boolean algebras
relates
theories in classical propositional logic and their collections of
models. The
theories can be seen as presentations of Boolean algebras, and the
collections
of models can be topologized in such a way that the theory can be
recovered
from its space of models. The situation can be cast as a formal duality
relating two categories of syntax and semantics, mediated by homming
into a
common dualizing object, in this case $2$. In the present work, we
generalize
the entire arrangement from propositional to first-order logic. Boolean
algebras are replaced by Boolean categories presented by theories in
first-order logic, and spaces of models are replaced by topological
groupoids
of models and their isomorphisms. A duality between the resulting
categories of
syntax and semantics, expressed first in the form of a contravariant
adjunction, is established by homming into a common dualizing object,
now
$\Sets$, regarded once as a boolean category, and once as a groupoid
equipped
with an intrinsic topology. The overall framework of our investigation
is
provided by topos theory. Direct proofs of the main results are given,
but the
specialist will recognize toposophical ideas in the background.
Indeed, the
duality between syntax and semantics is really a manifestation of that
between
algebra and geometry in the two directions of the geometric morphisms
that lurk
behind our formal theory. Along the way, we construct the classifying
topos of
a decidable coherent theory out of its groupoid of models via a
simplified
covering theorem for coherent toposes.
Regards,
Steve Awodey
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