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Derived Categories: A Quick Tour

D. Huybrechts

in Fourier-Mukai Transforms in Algebraic Geometry

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199296866
eISBN:
9780191711329
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199296866.003.0002
Subject:
Mathematics, Geometry / Topology

This chapter briefly outlines the main steps in the construction of the derived category of an arbitrary abelian category. The homotopy category of complexes is considered as an intermediate step, ... More


Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

Friedhelm Waldhausen, Bjørn Jahren, and John Rognes

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691157757
eISBN:
9781400846528
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691157757.001.0001
Subject:
Mathematics, Geometry / Topology

Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a ... More


On simple maps

Friedhelm Waldhausen, Bjørn Jahren, and John Rognes

in Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691157757
eISBN:
9781400846528
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691157757.003.0003
Subject:
Mathematics, Geometry / Topology

This chapter deals with simple maps of finite simplicial sets, along with some of their formal properties. It begins with a discussion of simple maps of simplicial sets, presenting a proposition for ... More


Links as Sasakian Manifolds

Charles P. Boyer and Krzysztof Galicki

in Sasakian Geometry

Published in print:
2007
Published Online:
January 2008
ISBN:
9780198564959
eISBN:
9780191713712
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198564959.003.0010
Subject:
Mathematics, Geometry / Topology

This chapter is devoted to the geometry of links of isolated hypersurface singularities, as well as a review of the differential topology of homotopy spheres a la Kervaire and Milnor. The ... More


Liquid Crystals and Harmonic Maps in Polyhedral Domains

Apala Majumdar, Jonathan Robbins, and Maxim Zyskin

in Analysis and Stochastics of Growth Processes and Interface Models

Published in print:
2008
Published Online:
September 2008
ISBN:
9780199239252
eISBN:
9780191716911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199239252.003.0014
Subject:
Mathematics, Probability / Statistics, Analysis

This chapter is concerned with harmonic maps from a polyhedron to the unit two-sphere, which provide a model of nematic liquid crystals in bistable displays. This chapter looks at the Dirichlet ... More


SURGERY ON MAPS

Andrew Ranicki

in Algebraic and Geometric Surgery

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198509240
eISBN:
9780191708725
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509240.003.0010
Subject:
Mathematics, Geometry / Topology

This chapter gives the definition of a normal map. Killing elements in the relative homotopy group by surgery on a normal map is discussed. The kernel modulesand relative regular homotopy groups of ... More


Introduction

Friedhelm Waldhausen, Bjørn Jahren, and John Rognes

in Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691157757
eISBN:
9781400846528
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691157757.003.0001
Subject:
Mathematics, Geometry / Topology

This book presents a proof of the stable parametrized h-cobordism theorem, which deals with the existence of a natural homotopy equivalence for each compact CAT manifold. In this theorem, a stable ... More


The stable parametrized h-cobordism theorem

Friedhelm Waldhausen, Bjørn Jahren, and John Rognes

in Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691157757
eISBN:
9781400846528
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691157757.003.0002
Subject:
Mathematics, Geometry / Topology

This chapter deals with the stable parametrized h-cobordism theorem. It begins with a discussion of the manifold part; here DIFF is written for the category of Csuperscript infinity smooth manifolds, ... More


The non-manifold part

Friedhelm Waldhausen, Bjørn Jahren, and John Rognes

in Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691157757
eISBN:
9781400846528
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691157757.003.0004
Subject:
Mathematics, Geometry / Topology

Abstract and Keywords to be supplied.


An equivalence of categories

Ehud Hrushovski and François Loeser

in Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691161686
eISBN:
9781400881222
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161686.003.0013
Subject:
Mathematics, Geometry / Topology

This chapter deduces from Theorem 11.1.1 an equivalence of categories between a certain homotopy category of definable subsets of quasi-projective varieties over a given valued field and a suitable ... More


Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy

David Corfield

Published in print:
2020
Published Online:
March 2020
ISBN:
9780198853404
eISBN:
9780191888069
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198853404.001.0001
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

In[KF1] 1914, in an essay entitled ‘Logic as the Essence of Philosophy’, Bertrand Russell promised to revolutionize philosophy by introducing there the ‘new logic’ of Frege and Peano: “The old logic ... More


Topological Classification Of Defects

VOLOVIK GRIGORY E.

in The Universe in a Helium Droplet

Published in print:
2009
Published Online:
January 2010
ISBN:
9780199564842
eISBN:
9780191709906
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199564842.003.0013
Subject:
Physics, Condensed Matter Physics / Materials, Particle Physics / Astrophysics / Cosmology

The effective metric and effective gauge fields are simulated in superfluids by the inhomogeneity of the superfluid vacuum. In superfluids, many inhomogeneous configurations of the vacuum are stable ... More


Vortices In 3He-B

VOLOVIK GRIGORY E.

in The Universe in a Helium Droplet

Published in print:
2009
Published Online:
January 2010
ISBN:
9780199564842
eISBN:
9780191709906
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199564842.003.0014
Subject:
Physics, Condensed Matter Physics / Materials, Particle Physics / Astrophysics / Cosmology

This chapter discusses the topology of singular topological defects — defects with singular core of coherence length size — in 3He-B and in quantum chromodynamics, such as conventional mass vortices, ... More


Continuous Structures

VOLOVIK GRIGORY E.

in The Universe in a Helium Droplet

Published in print:
2009
Published Online:
January 2010
ISBN:
9780199564842
eISBN:
9780191709906
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199564842.003.0016
Subject:
Physics, Condensed Matter Physics / Materials, Particle Physics / Astrophysics / Cosmology

When several distinct energy scales are involved, the vacuum symmetry is different for different length scales: the larger the length scale, the more the symmetry is reduced. The interplay of ... More


Monopoles and Boojums

VOLOVIK GRIGORY E.

in The Universe in a Helium Droplet

Published in print:
2009
Published Online:
January 2010
ISBN:
9780199564842
eISBN:
9780191709906
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199564842.003.0017
Subject:
Physics, Condensed Matter Physics / Materials, Particle Physics / Astrophysics / Cosmology

This chapter is devoted to point defects such as hedgehogs, Dirac magnetic monopoles, 't Hooft–Polyakov monopole, and nexus. Some of these objects represent composite defects resulting from a ... More


Differential geometry, topology and fibre bundles

Reinhold A. Bertlmann

in Anomalies in Quantum Field Theory

Published in print:
2000
Published Online:
February 2010
ISBN:
9780198507628
eISBN:
9780191706400
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507628.003.0002
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces all necessary mathematical concepts. Section 2.1 briefly summarizes some topological definitions. Section 2.2 explains the homotopy of maps and the homotopy of groups. Section ... More


Chern–Simons form, homotopy operator and anomaly

Reinhold A. Bertlmann

in Anomalies in Quantum Field Theory

Published in print:
2000
Published Online:
February 2010
ISBN:
9780198507628
eISBN:
9780191706400
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507628.003.0007
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The Chern–Simons form and the homotopy operator plays an important role in connection with anomalies. In fact, the anomaly can be calculated on pure algebraic grounds from a variation of the ... More


Topological Methods for Description of Quantum Many-Body Systems

Lucjan Jacak, Piotr Sitko, Konrad Wieczorek, and Arkadiusz Wojs

in Quantum Hall systems: Braid groups, composite fermions and fractional charge

Published in print:
2003
Published Online:
January 2010
ISBN:
9780198528708
eISBN:
9780191713477
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198528708.003.0002
Subject:
Physics, Condensed Matter Physics / Materials, Theoretical, Computational, and Statistical Physics

This chapter presents a topological description of quantum many-body systems within the Feynman path integration formalism. Configuration spaces for the system of two particles on the one-dimensional ... More


Cohomology of Homotopy 2-types

Graham Ellis

in An Invitation to Computational Homotopy

Published in print:
2019
Published Online:
November 2019
ISBN:
9780198832973
eISBN:
9780191871375
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198832973.003.0005
Subject:
Mathematics, Computational Mathematics / Optimization, Geometry / Topology

This chapter introduces some of the basic ingredients in the classification of homotopy 2-types and describes datatypes and algorithms for implementing them on a computer. These are illustrated using ... More


The main theorem

Ehud Hrushovski and François Loeser

in Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691161686
eISBN:
9781400881222
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161686.003.0011
Subject:
Mathematics, Geometry / Topology

This chapter introduces the main theorem, which states: Let V be a quasi-projective variety over a valued field F and let X be a definable subset of V x Γ‎superscript Script Small l subscript ... More


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