33 research outputs found
First-Order Logical Duality
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.Comment: Final pre-print version. 62 page
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
Ionads
The notion of Grothendieck topos may be considered as a generalisation of
that of topological space, one in which the points of the space may have
non-trivial automorphisms. However, the analogy is not precise, since in a
topological space, it is the points which have conceptual priority over the
open sets, whereas in a topos it is the other way around. Hence a topos is more
correctly regarded as a generalised locale, than as a generalised space. In
this article we introduce the notion of ionad, which stands in the same
relationship to a topological space as a (Grothendieck) topos does to a locale.
We develop basic aspects of their theory and discuss their relationship with
toposes.Comment: 24 pages; v2: diverse revisions; v3: chopped about in face of
trenchant and insightful referee feedbac
The Universal Theory of First Order Algebras and Various Reducts
First order formulas in a relational signature can be considered as
operations on the relations of an underlying set, giving rise to multisorted
algebras we call first order algebras. We present universal axioms so that an
algebra satisfies the axioms iff it embeds into a first order algebra.
Importantly, our argument is modular and also works for, e.g., the positive
existential algebras (where we restrict attention to the positive existential
formulas) and the quantifier-free algebras. We also explain the relationship to
theories, and indicate how to add in function symbols.Comment: 30 page
Morita Equivalence
Logicians and philosophers of science have proposed various formal criteria
for theoretical equivalence. In this paper, we examine two such proposals:
definitional equivalence and categorical equivalence. In order to show
precisely how these two well-known criteria are related to one another, we
investigate an intermediate criterion called Morita equivalence.Comment: 30 page
Functorial Data Migration
In this paper we present a simple database definition language: that of
categories and functors. A database schema is a small category and an instance
is a set-valued functor on it. We show that morphisms of schemas induce three
"data migration functors", which translate instances from one schema to the
other in canonical ways. These functors parameterize projections, unions, and
joins over all tables simultaneously and can be used in place of conjunctive
and disjunctive queries. We also show how to connect a database and a
functional programming language by introducing a functorial connection between
the schema and the category of types for that language. We begin the paper with
a multitude of examples to motivate the definitions, and near the end we
provide a dictionary whereby one can translate database concepts into
category-theoretic concepts and vice-versa.Comment: 30 page