239 research outputs found
"Le present est plein de l’avenir, et chargé du passé" : Vorträge des XI. Internationalen Leibniz-Kongresses, 31. Juli – 4. August 2023, Leibniz Universität Hannover, Deutschland. Band 2
[No abstract available]Deutschen Forschungsgemeinschaft (DFG)/Projektnr. 517991912VGH VersicherungNiedersächsisches Ministerium für Wissenschaft und Kultur (MWK
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Selected Topics in Gravity, Field Theory and Quantum Mechanics
Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
Clones of pigmented words and realizations of special classes of monoids
Clones are generalizations of operads forming powerful instruments to
describe varieties of algebras wherein repeating variables are allowed in their
relations. They allow us in this way to realize and study a large range of
algebraic structures. A functorial construction from the category of monoids to
the category of clones is introduced. The obtained clones involve words on
positive integers where letters are pigmented by elements of a monoid. By
considering quotients of these structures, we construct a complete hierarchy of
clones involving some families of combinatorial objects. This provides clone
realizations of some known and some new special classes of monoids as among
others the variety of left-regular bands, bounded semilattices, and regular
band monoids.Comment: 41 page
Koszul Operads Governing Props and Wheeled Props
In this paper, we construct groupoid coloured operads governing props and
wheeled props, and show they are Koszul. This is accomplished by new biased
definitions for (wheeled) props, and an extension of the theory of Groebner
bases for operads to apply to groupoid coloured operads. Using the Koszul
machine, we define homotopy (wheeled) props, and show they are not formed by
polytope based models. Finally, using homotopy transfer theory, we construct
Massey products for (wheeled) props, show these products characterise the
formality of these structures, and re-obtain a theorem of Mac Lane on the
existence of higher homotopies of (co)commutative Hopf algebras.Comment: Questions or comments are most welcome
Scalar enrichment and cotraces in bicategories
It is known that every monoidal bicategory has an associated braided monoidal category of scalars. In this thesis we show that every monoidal bicategory which is closed both monoidally and compositionally, can be enriched over the monoidal 2-category of scalar enriched categories. This enrichment provides a number of key insights into the relationship
between linear algebra and category theory.
The enrichment replaces every set of 2-cells with a scalar, and we show that this replacement can be given in terms of the cotrace, first defined by Day and Street in the context of profunctors. This is analogous to the construction of the Frobenius inner product between linear maps, which is constructed in terms of the trace of linear maps. In linear algebra it is also possible to define the trace in terms of the Frobenius inner product. We show that the cotrace can be defined in terms of the enrichment, and in doing so we prove that the cotrace is an enriched version of the ‘categorical trace’ studied by Ganter and Kapranov, and Bartlett. Thus, we unify the concept of a categorical trace with the concept of a cotrace.
Finally, we study the relationship between the trace and the cotrace for compact closed bicategories. We show that the trace and cotrace have a structured relationship and share many of the properties of the linear trace including – but not limited to – dual invariance and linearity. Motivating examples are given throughout. We also introduce a decorated string diagram language to simplify some of the proofs
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