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63
Localic completion of generalized metric spaces I
, 2005
"... Abstract. Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero selfdistance law and the triangle inequality. We describe a complet ..."
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Abstract. Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero selfdistance law and the triangle inequality. We describe a completion of gms’s by Cauchy filters of formal balls. In terms of Lawvere’s approach using categories enriched over [0, ∞], the Cauchy filters are equivalent to flat left modules. The completion generalizes the usual one for metric spaces. For quasimetrics it is equivalent to the Yoneda completion in its netwise form due to Künzi and Schellekens and thereby gives a new and explicit characterization of the points of the Yoneda completion. Nonexpansive functions between gms’s lift to continuous maps between the completions. Various examples and constructions are given, including finite products. The completion is easily adapted to produce a locale, and that part of the work is constructively valid. The exposition illustrates the use of geometric logic to enable pointbased reasoning for locales. 1.
Yoneda structures from 2toposes
"... Abstract. A 2categorical generalisation of the notion of elementary topos is provided, and some of the properties of the yoneda structure [SW78] it generates are explored. Results enabling one to exhibit objects as cocomplete in the sense definable within a yoneda structure are presented. Examples ..."
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Abstract. A 2categorical generalisation of the notion of elementary topos is provided, and some of the properties of the yoneda structure [SW78] it generates are explored. Results enabling one to exhibit objects as cocomplete in the sense definable within a yoneda structure are presented. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also contains some expository material on the theory of fibrations internal to a finitely complete 2category [Str74b] and provides a selfcontained development of the necessary background material on yoneda structures.
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also ..."
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
Characterisations of Morita equivalent inverse semigroups
, 2009
"... For a fixed inverse semigroup S, there are two natural categories of left actions of S: the category Fact of unitary actions of S on sets X meaning actions where SX = X, and the category Étale of étale actions meaning those unitary actions equipped with a function p: X → E(S), to the set of idempote ..."
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For a fixed inverse semigroup S, there are two natural categories of left actions of S: the category Fact of unitary actions of S on sets X meaning actions where SX = X, and the category Étale of étale actions meaning those unitary actions equipped with a function p: X → E(S), to the set of idempotents of S, such that p(x)x = x and p(sx) = ses ∗ , where s ∗ denotes the inverse of s. The category Étale can be regarded as the classifying topos of S. There is a forgetful functor U from Étale to Fact that forgets étale structure and simply remembers the action. Associated with these two types of actions are appropriate notions of Morita equivalence which we term Morita equivalence and strong Morita equivalence, respectively. We prove three main results: first, strong Morita equivalence is the same as Morita equivalence; second, the forgetful functor U has a right adjoint R, and the category of EilenbergMoore algebras of the monad M = RU is equivalent to the category of presheaves on the Cauchy completion C(S) of S; third, we show that equivalence bimodules, which witness strong Morita equivalence, can be viewed as abstract atlases, thus connecting with the pioneering work of V. V. Wagner on the theory of inverse semigroups and Anders Kock’s more recent work on pregroupoids.
LOCALIZATIONS FOR CONSTRUCTION OF QUANTUM COSET SPACES
, 2003
"... Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on n ..."
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Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on noncommutative localizations, we introduce localizations compatible with coactions. Coinvariants of these localized coactions give local information about quotients. We define Zariski locally trivial quantum group algebraic principal and associated bundles. Compatible localizations induce localizations on the categories of Hopf modules. Their interplay with the functor of taking coinvariants and its left adjoint is stressed out. Using localization approach, we constructed a natural class of examples of quantum coset spaces, related to the quantum flag varieties of type A of other authors. Noncommutative Gauss decomposition via quasideterminants reveals a new structure in noncommutative matrix bialgebras. In the quantum case, calculations with quantum minors yield the structure theorems. Notation. Ground field is k and we assume it is of characteristic zero. If we deal just with one kHopf algebra, say B, the comultiplication is ∆ : B → B ⊗ B, unit map η: k → B, counit ǫ: B → k, multiplication µ: B ⊗ B → B, and antipode (coinverse) is
Noncommutative Localization in noncommutative geometry
, 2008
"... The aim of these notes is to collect and motivate the basic localization toolbox for the geometric study of “spaces”, locally described by noncommutative rings and their categories of onesided modules. We present the basics of Ore localization of rings and modules in much detail. Common practical t ..."
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The aim of these notes is to collect and motivate the basic localization toolbox for the geometric study of “spaces”, locally described by noncommutative rings and their categories of onesided modules. We present the basics of Ore localization of rings and modules in much detail. Common practical techniques are studied as well. We also describe a counterexample for a folklore test principle. Localization in negatively filtered rings arising in deformation theory is presented. A new notion of the differential Ore condition is introduced in the study of localization of differential calculi. To aid the geometrical viewpoint, localization is studied with emphasis on descent formalism, flatness, abelian categories of quasicoherent sheaves and generalizations, and natural pairs of adjoint functors for sheaf and module categories. The key motivational theorems from the seminal works of Gabriel on localization, abelian categories and schemes are quoted without proof, as well as the related statements of Popescu, Watts, Deligne and Rosenberg. Cohn universal localization does not have good flatness properties, but may be described well at the ring level. We conjecture that the latter feature may be important for gluing in noncommutative geometry whenever the flat descent fails. Cohn localization is here related to the quasideterminants of Gelfand and Retakh; and this may help understanding both subjects.
Truth functionality and measurebased logics
 Fuzzy Sets, Logics and Reasoning about Knowledge
, 1999
"... We present a truthfunctional semantics for necessityvalued logics, based on the forcing technique. We interpret possibility distributions (which correspond to necessity measures) as informational states, and introduce a suitable language (basically, an extension of classical logic, similar to Pave ..."
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We present a truthfunctional semantics for necessityvalued logics, based on the forcing technique. We interpret possibility distributions (which correspond to necessity measures) as informational states, and introduce a suitable language (basically, an extension of classical logic, similar to Pavelka’s language). Then we define the relation of “forcing ” between an informational state and a formula, meaning that the state contains enough information to support the validity of the formula. The subsequent step is the definition of a manyvalued truthfunctional semantics, by simply taking the truth value of a formula to be the set of all informational states that force the truth of the formula. A proof system in sequent calculus form is provided, and validity and completeness theorems are proved. 1
Toposes pour les vraiment nuls
 Advances in Theory and Formal Methods of Computing
, 1996
"... Restriction to geometric logic can enable one to define topological structures and continuous maps without explicit reference to topologies. This idea is illustrated with some examples and used to explain 1 ..."
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Restriction to geometric logic can enable one to define topological structures and continuous maps without explicit reference to topologies. This idea is illustrated with some examples and used to explain 1