612 research outputs found
-theoretic counterexamples to Ravenel's telescope conjecture
At each prime and height , we prove that the telescopic and
chromatic localizations of spectra differ. Specifically, for
acting by Adams operations on , we prove that the
-localized algebraic -theory of is not -local. We also show that Galois
hyperdescent, -invariance, and nil-invariance fail for the
-localized algebraic -theory of -local
-rings. In the case and we make complete
computations of , for certain finite Galois extensions
of the -local sphere. We show for that the algebraic -theory
of the -local sphere is asymptotically -local.Comment: 100 pages. Comments very welcom
Double Copy from Tensor Products of Metric BV-algebras
Field theories with kinematic Lie algebras, such as field theories featuring
colour-kinematics duality, possess an underlying algebraic structure known as
BV-algebra. If, additionally, matter fields are
present, this structure is supplemented by a module for the BV-algebra. We explain this perspective, expanding on our previous
work and providing many additional mathematical details. We also show how the
tensor product of two metric BV-algebras yields
the action of a new syngamy field theory, a construction which comprises the
familiar double copy construction. As examples, we discuss various scalar field
theories, Chern-Simons theory, self-dual Yang-Mills theory, and the pure spinor
formulations of both M2-brane models and supersymmetric Yang-Mills theory. The
latter leads to a new cubic pure spinor action for ten-dimensional
supergravity. We also give a homotopy-algebraic perspective on
colour-flavour-stripping, obtain a new restricted tensor product over a wide
class of bialgebras, and we show that any field theory (even one without
colour-kinematics duality) comes with a kinematic -algebra.Comment: v2: 97 pages, references added, typos fixed, comments welcom
Categorical structures for deduction
We begin by introducing categorized judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and first-order logic as special kinds of categorized judgemental theories. We believe our analysis sheds light on both the topics, providing a new point of view. In the case of type theory, we provide an abstract definition of type constructor featuring the usual formation, introduction, elimination and computation rules. For first-order logic we offer a deep analysis of structural rules, describing some of their properties, and putting them into context.
We then put one of the main constructions introduced, namely that of categorized judgemental dependent type theories, to the test: we frame it in the general context of categorical models for dependent types, describe a few examples, study its properties, and use it to model subtyping and as a tool to prove intrinsic properties hidden in other models.
Somehow orthogonally, then, we show a different side as to how categories can help the study of deductive systems: we transport a known model from set-based categories to enriched categories, and use the information naturally encoded into it to describe a theory of fuzzy types. We recover structural rules, observe new phenomena, and study different possible enrichments and their interpretation. We open the discussion to include different takes on the topic of definitional equality
Quiver presentations and isomorphisms of Hecke categories and Khovanov arc algebras
We prove that the extended Khovanov arc algebras are isomorphic to the basic
algebras of anti-spherical Hecke categories for maximal parabolics of symmetric
groups. We present these algebras by quiver and relations and provide the full
submodule lattices of Verma modules
Categorical Coherence from Term Rewriting Systems
International audienceThe celebrated Squier theorem allows to prove coherence properties of algebraic structures, such as MacLane’s coherence theorem for monoidal categories, based on rewriting techniques. We are interested here in extending the theory and associated tools simultaneously in two directions. Firstly, we want to take in account situations where coherence is partial, in the sense that it only applies for a subset of structural morphisms (for instance, in the case of the coherence theorem for symmetric monoidal categories, we do not want to strictify the symmetry). Secondly, we are interested in structures where variables can be duplicated or erased. We develop theorems and rewriting techniques in order to achieve this, first in the setting of abstract rewriting systems, and then extend them to term rewriting systems, suitably generalized in order to take coherence in account. As an illustration of our results, we explain how to recover the coherence theorems for monoidal and symmetric monoidal categories
Functorial String Diagrams for Reverse-Mode Automatic Differentiation
We formulate a reverse-mode automatic differentiation (RAD) algorithm for (applied) simply typed lambda calculus in the style of Pearlmutter and Siskind [Barak A. Pearlmutter and Jeffrey Mark Siskind, 2008], using the graphical formalism of string diagrams. Thanks to string diagram rewriting, we are able to formally prove for the first time the soundness of such an algorithm. Our approach requires developing a calculus of string diagrams with hierarchical features in the spirit of functorial boxes, in order to model closed monoidal (and cartesian closed) structure. To give an efficient yet principled implementation of the RAD algorithm, we use foliations of our hierarchical string diagrams
Symmetry TFTs for Non-Invertible Defects
Given any symmetry acting on a -dimensional quantum field theory, there is
an associated -dimensional topological field theory known as the
Symmetry TFT (SymTFT). The SymTFT is useful for decoupling the universal
quantities of quantum field theories, such as their generalized global
symmetries and 't Hooft anomalies, from their dynamics. In this work, we
explore the SymTFT for theories with Kramers-Wannier-like duality symmetry in
both d and d quantum field theories. After constructing the
SymTFT, we use it to reproduce the non-invertible fusion rules of duality
defects, and along the way we generalize the concept of duality defects to
\textit{higher} duality defects. We also apply the SymTFT to the problem of
distinguishing intrinsically versus non-intrinsically non-invertible duality
defects in d.Comment: 119 pages, 46 figures; v2: references added, typos corrected; v3:
publication versio
An axiomatic approach to differentiation of polynomial circuits
Reverse derivative categories (RDCs) have recently been shown to be a suitable semantic framework for studying machine learning algorithms. Whereas emphasis has been put on training methodologies, less attention has been devoted to particular model classes: the concrete categories whose morphisms represent machine learning models. In this paper we study presentations by generators and equations of classes of RDCs. In particular, we propose polynomial circuits as a suitable machine learning model class. We give an axiomatisation for these circuits and prove a functional completeness result. Finally, we discuss the use of polynomial circuits over specific semirings to perform machine learning with discrete values
Light-matter interaction in the ZXW calculus
In this paper, we develop a graphical calculus to rewrite photonic circuits
involving light-matter interactions and non-linear optical effects. We
introduce the infinite ZW calculus, a graphical language for linear operators
on the bosonic Fock space which captures both linear and non-linear photonic
circuits. This calculus is obtained by combining the QPath calculus, a
diagrammatic language for linear optics, and the recently developed qudit ZXW
calculus, a complete axiomatisation of linear maps between qudits. It comes
with a 'lifting' theorem allowing to prove equalities between infinite
operators by rewriting in the ZXW calculus. We give a method for representing
bosonic and fermionic Hamiltonians in the infinite ZW calculus. This allows us
to derive their exponentials by diagrammatic reasoning. Examples include phase
shifts and beam splitters, as well as non-linear Kerr media and Jaynes-Cummings
light-matter interaction.Comment: 27 pages, lots of figures, a previous version accepted to QPL 202
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