612 research outputs found

    KK-theoretic counterexamples to Ravenel's telescope conjecture

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    At each prime pp and height n+1≥2n+1 \ge 2, we prove that the telescopic and chromatic localizations of spectra differ. Specifically, for Z\mathbb{Z} acting by Adams operations on BP⟨n⟩\mathrm{BP}\langle n \rangle, we prove that the T(n+1)T(n+1)-localized algebraic KK-theory of BP⟨n⟩hZ\mathrm{BP}\langle n \rangle^{h\mathbb{Z}} is not K(n+1)K(n+1)-local. We also show that Galois hyperdescent, A1\mathbb{A}^1-invariance, and nil-invariance fail for the K(n+1)K(n+1)-localized algebraic KK-theory of K(n)K(n)-local E∞\mathbb{E}_{\infty}-rings. In the case n=1n=1 and p≥7p \ge 7 we make complete computations of T(2)∗K(R)T(2)_*\mathrm{K}(R), for RR certain finite Galois extensions of the K(1)K(1)-local sphere. We show for p≥5p\geq 5 that the algebraic KK-theory of the K(1)K(1)-local sphere is asymptotically L2fL_2^{f}-local.Comment: 100 pages. Comments very welcom

    Double Copy from Tensor Products of Metric BVâ– {}^{\color{gray} \blacksquare}-algebras

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    Field theories with kinematic Lie algebras, such as field theories featuring colour-kinematics duality, possess an underlying algebraic structure known as BV■{}^{\color{gray} \blacksquare}-algebra. If, additionally, matter fields are present, this structure is supplemented by a module for the BV■{}^{\color{gray} \blacksquare}-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■{}^{\color{gray} \blacksquare}-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 L∞L_\infty-algebra.Comment: v2: 97 pages, references added, typos fixed, comments welcom

    Categorical structures for deduction

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    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

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    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

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    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

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    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

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    Given any symmetry acting on a dd-dimensional quantum field theory, there is an associated (d+1)(d+1)-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 (1+1)(1+1)d and (3+1)(3+1)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 (1+1)(1+1)d.Comment: 119 pages, 46 figures; v2: references added, typos corrected; v3: publication versio

    An axiomatic approach to differentiation of polynomial circuits

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    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

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    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

    Reductions in Higher-Order Rewriting and Their Equivalence

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