1,364 research outputs found
Linear Logic and Noncommutativity in the Calculus of Structures
In this thesis I study several deductive systems for linear logic, its fragments, and some noncommutative extensions. All systems will be designed within the calculus of structures, which is a proof theoretical formalism for specifying logical systems, in the tradition of Hilbert's formalism, natural deduction, and the sequent calculus. Systems in the calculus of structures are based on two simple principles: deep inference and top-down symmetry. Together they have remarkable consequences for the properties of the logical systems. For example, for linear logic it is possible to design a deductive system, in which all rules are local. In particular, the contraction rule is reduced to an atomic version, and there is no global promotion rule. I will also show an extension of multiplicative exponential linear logic by a noncommutative, self-dual connective which is not representable in the sequent calculus. All systems enjoy the cut elimination property. Moreover, this can be proved independently from the sequent calculus via techniques that are based on the new top-down symmetry. Furthermore, for all systems, I will present several decomposition theorems which constitute a new type of normal form for derivations
Emergence of Riemannian geometry and the massive graviton
We overview a new mechanism whereby classical Riemannian geometry emerges out
of the differential structure on quantum spacetime, as extension data for the
classical algebra of differential forms. Outcomes for physics include a new
formula for the standard Levi-Civita connection, a new point of view of the
cosmological constant as a very small mass for the graviton of around
ev, and a weakening of metric-compatibility in the presence of
torsion. The same mechanism also provides a new construction for quantum
bimodule connections on quantum spacetimes and a new approach to the quantum
Ricci tensor.Comment: 14 pages, Amslatex, conference proceedings introduction; only fixed a
few typos compared to previous versio
Sheaves that fail to represent matrix rings
There are two fundamental obstructions to representing noncommutative rings
via sheaves. First, there is no subcanonical coverage on the opposite of the
category of rings that includes all covering families in the big Zariski site.
Second, there is no contravariant functor F from the category of rings to the
category of ringed categories whose composite with the global sections functor
is naturally isomorphic to the identity, such that F restricts to the Zariski
spectrum functor Spec on the category of commutative rings (in a compatible way
with the natural isomorphism). Both of these no-go results are proved by
restricting attention to matrix rings.Comment: 13 pages; final versio
-algebraic drawings of dendroidal sets
In recent years the theory of dendroidal sets has emerged as an important
framework for higher algebra. In this article we introduce the concept of a
-algebraic drawing of a dendroidal set. It depicts a dendroidal set as an
object in the category of presheaves on -algebras. We show that the
construction is functorial and, in fact, it is the left adjoint of a Quillen
adjunction between combinatorial model categories. We use this construction to
produce a bridge between the two prominent paradigms of noncommutative geometry
via adjunctions of presentable -categories, which is the primary
motivation behind this article. As a consequence we obtain a single mechanism
to construct bivariant homology theories in both paradigms. We propose a
(conjectural) roadmap to harmonize algebraic and analytic (or topological)
bivariant K-theory. Finally, a method to analyse graph algebras in terms of
trees is sketched.Comment: 28 pages; v2 expanded version with some improvements; v3 revised and
added references; v4 some changes according to the suggestions of the
referees (to appear in Algebr. Geom. Topol.
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