Mathematics Mathematical Physics

Homotopy and Cohomology in Algebraic Topology

Description

This cluster of papers focuses on the deformation quantization of Poisson manifolds, involving topics such as Lie algebroids, homotopy theory, symplectic geometry, model categories, K-theory, operads, and higher algebraic structures. It explores the application of deformation quantization in various mathematical contexts and its implications for understanding geometric and algebraic structures.

Keywords

Deformation Quantization; Poisson Manifolds; Lie Algebroids; Homotopy Theory; Symplectic Geometry; Model Categories; K-Theory; Operads; Higher Algebraic Structures; Stacks

Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to … Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorems of point-set topology, this book is perfect for comunicating complex topics and the fun nature of algebraic topology for beginners.
Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20. Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20.
Soit X une u-variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee Q est definie positive. Alors cette forme est equivalente, sur les entiers, a la … Soit X une u-variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee Q est definie positive. Alors cette forme est equivalente, sur les entiers, a la forme diagonale standard, soit dans une base: Q(u 1 ,u 2 ,...u 2 )=u 2 1 +u 2 2 +...+u 2 2
Due to the strong appeal and wide use of this monograph, it is now available in its second revised edition. The monograph gives a systematic treatment of 3-dimensional topological quantum … Due to the strong appeal and wide use of this monograph, it is now available in its second revised edition. The monograph gives a systematic treatment of 3-dimensional topological quantum field theories (TQFTs) based on the work of the author with N. Reshetikhin and O. Viro. This subject was inspired by the discovery of the Jones polynomial of knots and the Witten-Chern-Simons field theory. On the algebraic side, the study of 3-dimensional TQFTs has been influenced by the theory of braided categories and the theory of quantum groups. The book is divided into three parts. Part I presents a construction of 3-dimensional TQFTs and 2-dimensional modular functors from so-called modular categories. This gives a vast class of knot invariants and 3-manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. In Part II the technique of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFTs constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3-space. This fundamental contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with knowledge of basic algebra and topology. It is an indispensable source for everyone who wishes to enter the forefront of this fascinating area at the borderline of mathematics and physics. From the contents: Invariants of graphs in Euclidean 3-space and of closed 3-manifolds Foundations of topological quantum field theory Three-dimensional topological quantum field theory Two-dimensional modular functors 6j-symbols Simplicial state sums on 3-manifolds Shadows of manifolds and state sums on shadows Constructions of modular categories
Since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic theory. With the development of Quillen's c Since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic theory. With the development of Quillen's c
0 Introduction.- 1 The Poisson bivector and the Schouten-Nijenhuis bracket.- 1.1 The Poisson bivector.- 1.2 The Schouten-Nijenhuis bracket.- 1.3 Coordinate expressions.- 1.4 The Koszul formula and applications.- 1.5 Miscellanea.- 2 … 0 Introduction.- 1 The Poisson bivector and the Schouten-Nijenhuis bracket.- 1.1 The Poisson bivector.- 1.2 The Schouten-Nijenhuis bracket.- 1.3 Coordinate expressions.- 1.4 The Koszul formula and applications.- 1.5 Miscellanea.- 2 The symplectic foliation of a Poisson manifold.- 2.1 General distributions and foliations.- 2.2 Involutivity and integrability.- 2.3 The case of Poisson manifolds.- 3 Examples of Poisson manifolds.- 3.1 Structures on ?n. Lie-Poisson structures.- 3.2 Dirac brackets.- 3.3 Further examples.- 4 Poisson calculus.- 4.1 The bracket of 1-forms.- 4.2 The contravariant exterior differentiations.- 4.3 The regular case.- 4.4 Cofoliations.- 4.5 Contravariant derivatives on vector bundles.- 4.6 More brackets.- 5 Poisson cohomology.- 5.1 Definition and general properties.- 5.2 Straightforward and inductive computations.- 5.3 The spectral sequence of Poisson cohomology.- 5.4 Poisson homology.- 6 An introduction to quantization.- 6.1 Prequantization.- 6.2 Quantization.- 6.3 Prequantization representations.- 6.4 Deformation quantization.- 7 Poisson morphisms, coinduced structures, reduction.- 7.1 Properties of Poisson mappings.- 7.2 Reduction of Poisson structures.- 7.3 Group actions and momenta.- 7.4 Group actions and reduction.- 8 Symplectic realizations of Poisson manifolds.- 8.1 Local symplectic realizations.- 8.2 Dual pairs of Poisson manifolds.- 8.3 Isotropic realizations.- 8.4 Isotropic realizations and nets.- 9 Realizations of Poisson manifolds by symplectic groupoids.- 9.1 Realizations of Lie-Poisson structures.- 9.2 The Lie groupoid and symplectic structures of T*G.- 9.3 General symplectic groupoids.- 9.4 Lie algebroids and the integrability of Poisson manifolds.- 9.5 Further integrability results.- 10 Poisson-Lie groups.- 10.1 Poisson-Lie and biinvariant structures on Lie groups.- 10.2 Characteristic properties of Poisson-Lie groups.- 10.3 The Lie algebra of a Poisson-Lie group.- 10.4 The Yang-Baxter equations.- 10.5 Manin triples.- 10.6 Actions and dressing transformations.- References.
This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Recent developments in mathematical physics (e.g., … This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Recent developments in mathematical physics (e.g., in knot theory, gauge theory and topological quantum field theory) have led mathematicians and physicists to look for new geometric structures on manifolds and to seek a synthesis of ideas from geometry, topology and category theory. In this spirit this book develops the differential geometry associated to the topology and obstruction theory of certain fibre bundles (more precisely, associated to gerbes). The new theory is a 3-dimensional analogue of the familiar Kostant-Weil theory of line bundles. In particular the curvature now becomes a 3-form. Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, Kaehler geometry of the space of knots, Cheeger-Chern-Simons secondary characteristic classes, and group cohomology. Finally, the last chapter deals with the Dirac monopole and Dirac's quantization of the electrical charge. The book will be of interest to topologists, geometers, Lie theorists and mathematical physicists, as well as to operator algebraists. It is written for graduate students and researchers, and will be an excellent textbook. It has a self-contained introduction to the theory of sheaves and their cohomology, line bundles and geometric prequantization a la Kostant-Souriau.
Localization of model category structures: Summary of part 1 Local spaces and localization The localization model category for spaces Localization of model categories Existence of left Bousfield localizations Existence of … Localization of model category structures: Summary of part 1 Local spaces and localization The localization model category for spaces Localization of model categories Existence of left Bousfield localizations Existence of right Bousfield localizations Fiberwise localization Homotopy theory in model categories: Summary of part 2 Model categories Fibrant and cofibrant approximations Simplicial model categories Ordinals, cardinals, and transfinite composition Cofibrantly generated model categories Cellular model categories Proper model categories The classifying space of a small category The Reedy model category structure Cosimplicial and simplicial resolutions Homotopy function complexes Homotopy limits in simplicial model categories Homotopy limits in general model categories Index Bibliography.
n is said to be 1 -essential (or, for brevity, essential) if for some map into an aspherical space, /: V -» K, the induced top dimensional homomorphism on homology … n is said to be 1 -essential (or, for brevity, essential) if for some map into an aspherical space, /: V -» K, the induced top dimensional homomorphism on homology does not vanish, i.e., /JF] φ 0.Here [V] stands for the integral fundamental class, i.e., [V] e H n (V\ Z) » Z, in case the manifold Fis oriented.
These lectures provide students and specialists with preliminary and valuable information from university courses and seminars in mathematics. This set gives new proof of the h-cobordism theorem that is different … These lectures provide students and specialists with preliminary and valuable information from university courses and seminars in mathematics. This set gives new proof of the h-cobordism theorem that is different from the original proof presented by S. Smale. Originally published in 1965. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
The aim of this paper is to understand the interrelations among relations within concrete social groups. Social structure is sought, not ideal types, although the latter are relevant to interrelations … The aim of this paper is to understand the interrelations among relations within concrete social groups. Social structure is sought, not ideal types, although the latter are relevant to interrelations among relations. From a detailed social network, patterns of global relations can be extracted, within which classes of equivalently positioned individuals are delineated. The global patterns are derived algebraically through a ‘functorial’ mapping of the original pattern. Such a mapping (essentially a generalized homomorphism) allows systematically for concatenation of effects through the network. The notion of functorial mapping is of central importance in the ‘theory of categories,’ a branch of modern algebra with numerous applications to algebra, topology, logic. The paper contains analyses of two social networks, exemplifying this approach.
THEOREM 11.1. The family 9(,{, G) consists of all analytic fibre spaces Bi, '2C H'(A, f2(B#)). The notion of fibre spaces and their equivalences depends on the sheaf of structure … THEOREM 11.1. The family 9(,{, G) consists of all analytic fibre spaces Bi, '2C H'(A, f2(B#)). The notion of fibre spaces and their equivalences depends on the sheaf of structure groups.' By a (63-fibre space we mean a fibre space with a sheaf (3 of structure groups, and we say that two fibre spaces are (X-equivalent if they are equivalent as (63-fibre spaces. The fibre space Bi may be considered as an analytic fibre space, as an f2(BO)-fibre space, or as an f2(Bl)-fibre space. The cohomology class C) C H'(Ay, f(BO)) represents the f2(Bl)-equivalence class of Be. Let C& denote the fibre of BI over u c A. It is clear that
Preface1Smooth manifolds and smooth maps1Tangent spaces and derivatives2Regular values7The fundamental theorem of algebra82The theorem of Sard and Brown10Manifolds with boundary12The Brouwer fixed point theorem133Proof of Sard's theorem164The degree modulo 2 … Preface1Smooth manifolds and smooth maps1Tangent spaces and derivatives2Regular values7The fundamental theorem of algebra82The theorem of Sard and Brown10Manifolds with boundary12The Brouwer fixed point theorem133Proof of Sard's theorem164The degree modulo 2 of a mapping20Smooth homotopy and smooth isotopy205Oriented manifolds26The Brouwer degree276Vector fields and the Euler number327Framed cobordism the Pontryagin construction42The Hopf theorem508Exercises52AppClassifying 1-manifolds55Bibliography59Index63
DEFINITION. Two closed n-manifolds M, and M2 are h-cobordant1 if the disjoint sum M, + (- M2) is the boundary of some manifold W, where both M1 and (-M2) are … DEFINITION. Two closed n-manifolds M, and M2 are h-cobordant1 if the disjoint sum M, + (- M2) is the boundary of some manifold W, where both M1 and (-M2) are deformation retracts of W. It is clear that this is an equivalence relation. The connected sum of two connected n-manifolds is obtained by removing a small n-cell from each, and then pasting together the resulting boundaries. Details will be given in ? 2.
In this paper we prove a localization theorem for the K-theory of commutative rings and of schemes, Theorem 7.4, relating the K-groups of a scheme, of an open subscheme, and … In this paper we prove a localization theorem for the K-theory of commutative rings and of schemes, Theorem 7.4, relating the K-groups of a scheme, of an open subscheme, and of the category of those perfect complexes on the scheme which are acyclic on the open subscheme. The localization theorem of Quillen [Q1] for K′- or G-theory is the main support of his many results on the G-theory of noetherian schemes. The previous lack of an adequate localization theorem for K-theory has obstructed development of this theory for the fifteen years since 1973. Hence our theorem unleashes a pack of new basic results hitherto known only under very restrictive hypotheses like regularity. These new results include the “Bass fundamental theorem” 6.6, the Zariski (Nisnevich) cohomolog-ical descent spectral sequence that reduces problems to the case of local (hensel local) rings 10.3 and 19.8, the Mayer-Vietoris theorem for open covers 8.1, invariance mod ℓ under polynomial extensions 9.5, Vorst-van der Kallen theory for NK 9.12, Goodwillie and Ogle-Weibel theorems relating K-theory to cyclic cohomology 9.10, mod ℓ Mayer-Vietoris for closed covers 9.8, and mod ℓ comparison between algebraic and topological K-theory 11.5 and 11.9. Indeed most known results in K-theory can be improved by the methods of this paper, by removing now unnecessary regularity, affineness, and other hypotheses.
This work introduces a general theory of universal pseudomorphisms and develops their connection to diagrammatic coherence. The main results give hypotheses under which pseudomorphism coherence is equivalent to the coherence … This work introduces a general theory of universal pseudomorphisms and develops their connection to diagrammatic coherence. The main results give hypotheses under which pseudomorphism coherence is equivalent to the coherence theory of strict algebras. Applications include diagrammatic coherence for plain, symmetric, and braided monoidal functors. The final sections include a variety of examples. 68 pages
NULL AUTHOR_ID | Physical Review X
NULL AUTHOR_ID | Physical review. D/Physical review. D.
A sequence a_{0}<a_{1}<\cdots<a_{n} of nonnegative integers is called a Sidon sequence if the sums of pairs a_{i}+a_{j} are all different. In this paper we construct \operatorname{CAT(0)} groups and spaces from … A sequence a_{0}<a_{1}<\cdots<a_{n} of nonnegative integers is called a Sidon sequence if the sums of pairs a_{i}+a_{j} are all different. In this paper we construct \operatorname{CAT(0)} groups and spaces from Sidon sequences. The arithmetic condition of Sidon is shown to be equivalent to nonpositive curvature, and the number of ways to represent an integer as an alternating sum of triples a_{i}-a_{j}+a_{k} of integers from the Sidon sequence, is shown to determine the structure of the space of embedded flat planes in the associated \operatorname{CAT(0)} complex.
Geoffrey Powell , Christine Vespa | Bulletin de la Société mathématique de France
Abstract This note is a corrigendum to the authors' paper “Graph bundles and Ricci–flatness, Bulletin of the London Mathematical Society, 56(2), pp. 523–535, 2024”. Abstract This note is a corrigendum to the authors' paper “Graph bundles and Ricci–flatness, Bulletin of the London Mathematical Society, 56(2), pp. 523–535, 2024”.
Abstract Quantum $$L_{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> algebras are higher loop generalizations of cyclic $$L_{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> algebras. Motivated by … Abstract Quantum $$L_{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> algebras are higher loop generalizations of cyclic $$L_{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> algebras. Motivated by the problem of defining morphisms between such algebras, we construct a linear category of $$(-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -shifted symplectic vector spaces and distributional half-densities, originally proposed by Ševera. Morphisms in this category can be given both by formal half-densities and Lagrangian relations; we prove that the composition of such morphisms recovers the construction of homotopy transfer of quantum $$L_{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> algebras. Finally, using this category, we propose a new notion of a relation between quantum $$L_{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> algebras.
Abstract In this paper we describe the homotopy category of the (unital and strictly unital) $\mbox{A}_{\infty }$categories. To do that we introduce the notion of semi-free $\mbox{A}_{\infty }$category, which plays … Abstract In this paper we describe the homotopy category of the (unital and strictly unital) $\mbox{A}_{\infty }$categories. To do that we introduce the notion of semi-free $\mbox{A}_{\infty }$category, which plays the role of standard cofibration. Moreover, we define the (non-unital) $\mbox{A}_{\infty }$(resp. DG)categories with cofibrant morphisms and we prove that any (non-unital) $\mbox{A}_{\infty }$(resp. DG)category has a resolution of this kind.
| Peking University series in mathematics
| Peking University series in mathematics
| Peking University series in mathematics
| Peking University series in mathematics
Abstract We introduce higher analogs for cleavages in the context of (Kan) simplicial fibrations. We apply them to obtain geometric models for representations up to homotopy of (higher) Lie groupoids. … Abstract We introduce higher analogs for cleavages in the context of (Kan) simplicial fibrations. We apply them to obtain geometric models for representations up to homotopy of (higher) Lie groupoids. Concretely, we set an equivalence between representations up to homotopy and simplicial vector bundles endowed with a cleavage. Our result is an incarnation of the Higher Grothendieck Correspondence, it can be seen as a relative Dold–Kan Theorem and extends earlier work of Gracia-Saz and Mehta on VB-groupoids.
The focus of the workshop was on recent developments in local systems in arithmetic geometry in the broad sense, including anabelian geometry. The talks covered results in complex variations of … The focus of the workshop was on recent developments in local systems in arithmetic geometry in the broad sense, including anabelian geometry. The talks covered results in complex variations of Hodge structures, \ell -adic local systems, p -adic Hodge theory with p -adic local systems, and new insights into absolute Galois groups, with applications to rational points.
Abstract Clans are categorical representations of generalized algebraic theories that contain more information than the finite-limit categories associated to the locally finitely presentable categories of models via Gabriel–Ulmer duality. Extending … Abstract Clans are categorical representations of generalized algebraic theories that contain more information than the finite-limit categories associated to the locally finitely presentable categories of models via Gabriel–Ulmer duality. Extending Gabriel–Ulmer duality to account for this additional information, we present a duality theory between clans and locally finitely presentable categories equipped with a weak factorization system of a certain kind.
For an algebraic variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, curve class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha element-of normal upper N 1 left-parenthesis … For an algebraic variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, curve class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha element-of normal upper N 1 left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">N</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha \in \mathrm {N}_1(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g element-of double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">g \in \mathbb { N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we consider the sequence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathrm {S}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> representations obtained from the homology of the Kontsevich space of stable maps to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove script upper M With bar Subscript g comma n Baseline left-parenthesis upper X comma alpha right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\overline {\mathcal {M}}_{g,n}(X,\alpha )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Using the category of finite sets and surjections, we prove a representation stability theorem that governs the behavior of this sequence of representations for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sufficiently large.
We compare spaces of non-singular algebraic sections of ample vector bundles to spaces of continuous sections of jet bundles. Under some conditions, we provide an isomorphism in homology in a … We compare spaces of non-singular algebraic sections of ample vector bundles to spaces of continuous sections of jet bundles. Under some conditions, we provide an isomorphism in homology in a range of degrees growing with the jet ampleness. As an application, when $\mathcal{L}$ is a very ample line bundle on a smooth projective complex variety $X$, we prove that the rational cohomology of the space of non-singular algebraic sections of $\mathcal{L}^{\otimes d}$ stabilises as $d \to \infty$ and compute the stable cohomology. We also approach the integral homology using tools from stable homotopy theory.
We improve the homology stability range for the 3rd integral homology of symplectic groups over commutative local rings with infinite residue field. As an application, we show that for local … We improve the homology stability range for the 3rd integral homology of symplectic groups over commutative local rings with infinite residue field. As an application, we show that for local commutative rings containing an infinite field of characteristic not 2 the symbol map from Milnor-Witt K-theory to higher Grothendieck-Witt groups is an isomorphism in degrees 2 and 3.
We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of $\infty$-categories and we use this to prove many classical results about monads in the $\infty$-categorical … We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of $\infty$-categories and we use this to prove many classical results about monads in the $\infty$-categorical framework. Amongst other things, we show that the category of algebras for an accessible monads on a locally presentable $\infty$-category $\mathcal{E}$ is again locally presentable, and that a diagram of accessible monads on a locally presentable $\infty$-category admits a colimit. Our results also provide a new and simpler way to construct and describe monads in terms of theories.
Weakly globular double categories are a model of weak $2$-categories based on the notion of weak globularity, and they are known to be suitably equivalent to Tamsamani $2$-categories. Fair $2$-categories, … Weakly globular double categories are a model of weak $2$-categories based on the notion of weak globularity, and they are known to be suitably equivalent to Tamsamani $2$-categories. Fair $2$-categories, introduced by J. Kock, model weak $2$-categories with strictly associative compositions and weak unit laws. In this paper we establish a direct comparison between weakly globular double categories and fair $2$-categories and prove they are equivalent after localisation with respect to the $2$-equivalences. This comparison sheds new light on weakly globular double categories as encoding a strictly associative, though not strictly unital, composition, as well as the category of weak units via the weak globularity condition.