nLab contents of contents

Contents of Contents

These are the various “contents” pages on the nLab, generated from the database on 2011-08-10 (not in any particular order).

Mathematics

• mathematics

  • math resources

  • history of mathematics

• Structural Foundations

  • logic

    • internal language
    • classical mathematics
    • constructive mathematics
    • predicative mathematics
    • category:foundational axiom

  • set theory

    • structural set theory

  • category theory

    • Categories and Sheaves

    • topos theory

      • Sheaves in Geometry and Logic

  • higher category theory

    • higher topos theory

      • (∞,1)-topos theory
        • models for ∞-stack (∞,1)-toposes
        • cohomology

    • homotopy theory

      • stable homotopy theory
      • rational homotopy theory

• Topology and Geometry

  • geometry (general list), topology (general list)
  • general topology
  • differential topology
  • differential geometry
    • synthetic differential geometry
    • symplectic geometry
  • algebraic geometry
  • noncommutative algebraic geometry
  • noncommutative geometry (general flavour)
  • higher geometry

• Algebra

  • universal algebra

  • higher algebra

  • homological algebra

  • group theory, ring theory

    • representation theory

  • algebraic approaches to differential calculus

  • counterexamples in algebra

• analysis

  • nonstandard analysis

  • functional analysis

    • operator algebras

  • Fourier transform

• Lie theory

  • higher Lie theory

• probability theory

• discrete mathematics

Cohomology

cohomology

• cocycle, coboundary, coefficient

• homology

  • chain, cycle, boundary

• characteristic class

  • universal characteristic class

  • secondary characteristic class

  • differential characteristic class

• fiber sequence/long exact sequence in cohomology

• fiber ∞-bundle, principal ∞-bundle, associated ∞-bundle,

  twisted ∞-bundle

• ∞-group extension

• obstruction

Special and general types

• cochain cohomology

• ordinary cohomology, singular cohomology

• group cohomology, nonabelian group cohomology, Lie group cohomology

• Galois cohomology

• groupoid cohomology, nonabelian groupoid cohomology

• generalized (Eilenberg-Steenrod) cohomology

  • cobordism cohomology theory

  • integral cohomology

  • K-theory

  • elliptic cohomology, tmf

  • taf

• abelian sheaf cohomology

  • Deligne cohomology

    • de Rham cohomology

    • Dolbeault cohomology

  • etale cohomology

    • group of units, Picard group, Brauer group

  • crystalline cohomology

  • syntomic cohomology

  • motivic cohomology

• cohomology of operads

• Hochschild cohomology, cyclic cohomology

  • string topology

• nonabelian cohomology

  • principal ∞-bundle

    • universal principal ∞-bundle, groupal model for universal principal ∞-bundles

    • principal bundle, Atiyah Lie groupoid

    • principal 2-bundle/gerbe

  • covering ∞-bundle/local system

  • (∞,1)-vector bundle / (∞,n)-vector bundle

  • quantum anomaly

    • orientation, Spin structure, Spin^c structure, String structure, Fivebrane structure

• cohomology with constant coefficients / with a local system of coefficients

• ∞-Lie algebra cohomology

  • Lie algebra cohomology, nonabelian Lie algebra cohomology, Lie algebra extensions, Gelfand-Fuks cohomology,

• bialgebra cohomology

Special notions

• ?ech cohomology

• hypercohomology

Variants

• equivariant cohomology

  • equivariant homotopy theory

  • Bredon cohomology

• twisted cohomology

  • twisted bundle

    • twisted K-theory, twisted spin structure, twisted spin^c structure

  • twisted differential c-structures

    • twisted differential string structure, twisted differential fivebrane structure

• differential cohomology

  • differential generalized (Eilenberg-Steenrod) cohomology

    • differential cobordism cohomology

    • Deligne cohomology

    • differential K-theory

    • differential elliptic cohomology

  • differential cohomology in a cohesive topos

    • Chern-Weil theory

    • ∞-Chern-Weil theory

• relative cohomology

Extra structure

• Hodge structure

• orientation, in generalized cohomology

Operations

• cohomology operations

• cup product

• connecting homomorphism, Bockstein homomorphism

• fiber integration, transgression

• cohomology localization

Theorems

• universal coefficient theorem

• Künneth theorem

• de Rham theorem, Poincare lemma, Stokes theorem

• Hodge theory, Hodge theorem

  nonabelian Hodge theory, noncommutative Hodge theory

• Brown representability theorem

• hypercovering theorem

• Eckmann-Hilton-Fuks duality

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Differential Graded Objects

differential graded objects

and

rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)

dg-Algebra

• graded vector space

• differential graded vector space

• differential graded algebra

  • model structure on dgc-algebras

  • model structure on equivariant dgc-algebras

• differential graded coalgebra

  • model structure on dg-coalgebras

• differential graded Lie algebra

  • model structure on dg-Lie algebras

• L-∞ algebra

  • model structure for L-∞ algebras

• differential graded Hopf algebra

• bar and cobar construction

Rational spaces

• nilpotent space

• finite type

• rational space

  • formal space

• rationalization

PL de Rham complex

• differential forms on simplices

• PL de Rham complex

• Sullivan construction

  • Lie integration

• fundamental theorem of dg-algebraic rational homotopy theory

Sullivan models

• minimal$\,$Sullivan model

Examples of Sullivan models in rational homotopy theory:

• Sullivan model of an n-sphere

• Sullivan model of a spherical fibration

• Sullivan model of complex projective space

• Sullivan model of a classifying space

• Sullivan model of mapping space

• Sullivan model of free loop space

• Sullivan model of a suspension

• Sullivan model of a finite G-quotient

Related topics

• homotopy theory

• stable homotopy theory

• homological algebra

• ∞-Lie theory

Higher Geometry

higher geometry / derived geometry

Ingredients

• higher topos theory

• higher algebra

Concepts

• geometric little (∞,1)-toposes

  • structured (∞,1)-topos

  • geometry (for structured (∞,1)-toposes)

  • generalized scheme

• geometric big (∞,1)-toposes

  • cohesive (∞,1)-topos

• function algebras on ∞-stacks

  • geometric ∞-stacks

Constructions

• loop space object, free loop space object

• fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos / of a locally ∞-connected (∞,1)-topos

Examples

• derived algebraic geometry

  • étale (∞,1)-site, Hochschild cohomology of dg-algebras

  • dg-geometry

    • dg-scheme

  • schematic homotopy type

• derived noncommutative geometry

  • noncommutative geometry

• derived smooth geometry

  • differential geometry, differential topology

  • derived smooth manifold, dg-manifold

  • smooth ∞-groupoid, ∞-Lie algebroid

• higher symplectic geometry

• higher Klein geometry

• higher Cartan geometry

Theorems

• Isbell duality

• Jones' theorem, Deligne-Kontsevich conjecture

• Tannaka duality for geometric stacks

(infinity,1)-topos

(∞,1)-topos theory

Background

• sheaf and topos theory

• (∞,1)-category

• (∞,1)-functor

• (∞,1)-presheaf

• (∞,1)-category of (∞,1)-presheaves

Definitions

• elementary (∞,1)-topos

• (∞,1)-site

• reflective sub-(∞,1)-category

  • localization of an (∞,1)-category

  • topological localization

  • hypercompletion

• (∞,1)-category of (∞,1)-sheaves

  • (∞,1)-sheaf/∞-stack/derived stack

• (∞,1)-topos

• (n,1)-topos, n-topos

  • n-truncated object

  • n-connected object

  • (1,1)-topos

    • presheaf

    • sheaf

  • (2,1)-topos, 2-topos

    • (2,1)-presheaf

• (∞,1)-quasitopos

  • separated (∞,1)-presheaf

  • quasitopos

    • separated presheaf

  • (2,1)-quasitopos?

    • separated (2,1)-presheaf

• (∞,2)-topos

• (∞,n)-topos

Characterization

• universal colimits

• object classifier

• groupoid object in an (∞,1)-topos

  • effective epimorphism

Morphisms

• (∞,1)-geometric morphism

• (∞,1)Topos

• Lawvere distribution

Extra stuff, structure and property

• hypercomplete (∞,1)-topos

  • hypercomplete object

  • Whitehead theorem

• over-(∞,1)-topos

• n-localic (∞,1)-topos

• locally n-connected (n,1)-topos

• structured (∞,1)-topos

  • geometry (for structured (∞,1)-toposes)

• locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos

• local (∞,1)-topos

  • concrete (∞,1)-sheaf

• cohesive (∞,1)-topos

Models

• models for ∞-stack (∞,1)-toposes

  • model category

  • model structure on functors

  • model site/sSet-site

  • model structure on simplicial presheaves

  • descent for simplicial presheaves

  • descent for presheaves with values in strict ∞-groupoids

Constructions

structures in a cohesive (∞,1)-topos

• shape / coshape

• cohomology

• homotopy

  • fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos/of a locally ∞-connected (∞,1)-topos

  • categorical/geometric homotopy groups

  • Postnikov tower

  • Whitehead tower

• rational homotopy

• dimension

  • homotopy dimension

  • cohomological dimension

  • covering dimension

  • Heyting dimension

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Higher Algebra

higher algebra

universal algebra

Algebraic theories

• algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory

• monad / (∞,1)-monad

• operad / (∞,1)-operad

Algebras and modules

• algebra over a monad

  ∞-algebra over an (∞,1)-monad

• algebra over an algebraic theory

  ∞-algebra over an (∞,1)-algebraic theory

• algebra over an operad

  ∞-algebra over an (∞,1)-operad

• action, ∞-action

• representation, ∞-representation

• module, ∞-module

• associated bundle, associated ∞-bundle

Higher algebras

• monoidal (∞,1)-category

  • symmetric monoidal (∞,1)-category

  • monoid in an (∞,1)-category

  • commutative monoid in an (∞,1)-category

• symmetric monoidal (∞,1)-category of spectra

  • smash product of spectra

  • symmetric monoidal smash product of spectra

  • ring spectrum, module spectrum, algebra spectrum

• A-∞ algebra

  • A-∞ ring, A-∞ space

• C-∞ algebra

• E-∞ ring, E-∞ algebra

  • ∞-module, (∞,1)-module bundle

  • multiplicative cohomology theory

• L-∞ algebra

  • deformation theory

Model category presentations

• model structure on simplicial T-algebras / homotopy T-algebra

• model structure on operads

  model structure on algebras over an operad

Geometry on formal duals of algebras

• Isbell duality

• derived geometry

Theorems

• Deligne conjecture

• delooping hypothesis

• monoidal Dold-Kan correspondence

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Higher Category Theory

higher category theory

• category theory
• homotopy theory

Basic concepts

• k-morphism, coherence
• looping and delooping
• looping and suspension

Basic theorems

• homotopy hypothesis-theorem

• delooping hypothesis-theorem

• periodic table

• stabilization hypothesis-theorem

• exactness hypothesis

• holographic principle

Applications

• applications of (higher) category theory

• higher category theory and physics

Models

• (n,r)-category
  • Theta-space
  • ∞-category/∞-category
  • (∞,n)-category
    • n-fold complete Segal space
  • (∞,2)-category
  • (∞,1)-category
    • quasi-category
      • algebraic quasi-category
    • simplicially enriched category
    • complete Segal space
    • model category
  • (∞,0)-category/∞-groupoid
    • Kan complex
      • algebraic Kan complex
      • simplicial T-complex
  • (∞,Z)-category
  • n-category = (n,n)-category
    • 2-category, (2,1)-category
    • 1-category
    • 0-category
    • (-1)-category
    • (-2)-category
  • n-poset = (n-1,n)-category
    • poset = (0,1)-category
    • 2-poset = (1,2)-category
  • n-groupoid = (n,0)-category
    • 2-groupoid, 3-groupoid
• categorification/decategorification
• geometric definition of higher category
  • Kan complex
  • quasi-category
  • simplicial model for weak ∞-categories
    • complicial set
    • weak complicial set
• algebraic definition of higher category
  • bicategory
  • bigroupoid
  • tricategory
  • tetracategory
  • strict ∞-category
  • Batanin ∞-category
  • Trimble ∞-category
  • Grothendieck-Maltsiniotis ∞-categories
• stable homotopy theory
  • symmetric monoidal category
  • symmetric monoidal (∞,1)-category
  • stable (∞,1)-category
    • dg-category
    • A-∞ category
    • triangulated category

Morphisms

• k-morphism
  • 2-morphism
• transfor
  • natural transformation
  • modification

Functors

• functor
• 2-functor
  • pseudofunctor
  • lax functor
• (∞,1)-functor

Universal constructions

• 2-limit
• (∞,1)-adjunction
• (∞,1)-Kan extension
  • (∞,1)-limit
• (∞,1)-Grothendieck construction

Extra properties and structure

• cosmic cube
  • k-tuply monoidal n-category
  • strict ∞-category, strict ∞-groupoid
• stable (∞,1)-category
• (∞,1)-topos

1-categorical presentations

• homotopical category
• model category theory
• enriched category theory

Stable Homotopy Theory

stable homotopy theory

• homological algebra, higher algebra

Introduction

Ingredients

• homotopy theory

Contents

• loop space object

• suspension object

• looping and delooping

• stable (∞,1)-category

  • stabilization

    • spectrum object

  • stable derivator

  • triangulated category

• stable (∞,1)-category of spectra

  • spectrum

  • stable homotopy category

• smash product of spectra

  • symmetric smash product of spectra

  • Spanier-Whitehead duality

  • A-∞ ring

  • E-∞ ring

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Category Theory

category theory

Concepts

• category

• functor

• natural transformation

• Cat

Universal constructions

• universal construction

  • representable functor

  • adjoint functor

  • limit/colimit

  • weighted limit

  • end/coend

  • Kan extension

Theorems

• Yoneda lemma

• Isbell duality

• Grothendieck construction

• adjoint functor theorem

• monadicity theorem

• adjoint lifting theorem

• Tannaka duality

• Gabriel-Ulmer duality

• small object argument

• Freyd-Mitchell embedding theorem

• relation between type theory and category theory

Extensions

• sheaf and topos theory

• enriched category theory

• higher category theory

Applications

• applications of (higher) category theory

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Topos Theory

topos theory

• Toposes

Background

• category theory

  • category

  • functor

Toposes

• (0,1)-topos, Heyting algebra, locale

• pretopos

• topos

  • Grothendieck topos

    • category of presheaves

      • presheaf

      • representable presheaf

    • category of sheaves

      • site

        • sieve

        • coverage, pretopology, topology

      • sheaf

      • sheafification

• quasitopos

• base topos, indexed topos

Internal Logic

• categorical semantics

• internal logic

  • subobject classifier

  • natural numbers object

Topos morphisms

• logical morphism

• geometric morphism

  • direct image/inverse image

  • global sections

  • geometric embedding

  • surjective geometric morphism

  • essential geometric morphism

    • locally connected geometric morphism

    • connected geometric morphism

    • totally connected geometric morphism

  • étale geometric morphism

  • open geometric morphism

  • proper geometric morphism, compact topos

  • separated geometric morphism, Hausdorff topos

  • local geometric morphism

  • bounded geometric morphism

  • base change

  • localic geometric morphism

  • hyperconnected geometric morphism

  • atomic geometric morphism

Extra stuff, structure, properties

• topological locale

• localic topos

• petit topos/gros topos

• locally connected topos, connected topos, totally connected topos, strongly connected topos

• local topos

• cohesive topos

• classifying topos

• smooth topos

Cohomology and homotopy

• cohomology

• homotopy

• abelian sheaf cohomology

• model structure on simplicial presheaves

In higher category theory

• higher topos theory

• (0,1)-topos

  • (0,1)-site

• 2-topos

  • 2-site

  • 2-sheaf, stack

• (∞,1)-topos

  • (∞,1)-site

  • (∞,1)-sheaf, ∞-stack, derived stack

Theorems

• Diaconescu's theorem

• Barr's theorem

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Functorial Quantum Field Theory

functorial quantum field theory

Contents

• cobordism category

  • cobordism

  • extended cobordism

  • (∞,n)-category of cobordisms

  • Riemannian bordism category

• cobordism hypothesis

  • generalized tangle hypothesis

  • classification of TQFTs

• functorial field theory

  • unitary functorial field theory

  • extended functorial field theory

  • CFT

    • vertex operator algebra

  • TQFT

    • Reshetikhin-Turaev model / Chern-Simons theory

    • HQFT

    • TCFT

      • A-model, B-model, Gromov-Witten theory

      • homological mirror symmetry

• FQFT and cohomology

  • (1,1)-dimensional Euclidean field theories and K-theory

  • (2,1)-dimensional Euclidean field theory

  • geometric models for tmf

• holographic principle of higher category theory

  • holographic principle

  • AdS/CFT correspondence

  • quantization via the A-model

Supergeometry

superalgebra and (synthetic ) supergeometry

Background

• algebra

• geometry

• graded object

Introductions

• geometry of physics – superalgebra

• geometry of physics – supergeometry

Superalgebra

• super commutative monoid

• super abelian group

• super ring

  • supercommutative ring

  • exterior ring

  • Clifford ring

• super module

• super vector space, SVect

• super algebra

  • supercommutative algebra

  • exterior algebra

  • Clifford algebra

• superdeterminant

• super Lie algebra

  • super Poincare Lie algebra

• chain complex of super vector spaces (model structure)

• differential graded-commutative superalgebra (model structure)

• super L-infinity algebra

Supergeometry

• superpoint

• super Cartesian space

• supermanifold, SDiff

  • NQ-supermanifold

  • super vector bundle

  • complex supermanifold

  • Euclidean supermanifold

• super spacetime

  • super Minkowski spacetime

• integration over supermanifolds

  • Berezin integral

• super Lie group

  • super translation group

  • super Euclidean group

• super ∞-groupoid

• super formal smooth ∞-groupoid

• super line 2-bundle

Supersymmetry

supersymmetry

• division algebra and supersymmetry

• super Poincare Lie algebra

• supermultiplet

• BPS state

• M-theory super Lie algebra, type II super Lie algebra

• supergravity Lie 3-algebra, supergravity Lie 6-algebra

Supersymmetric field theory

• superfield

• supersymmetric quantum mechanics

  • superparticle

  • adinkra

• super Yang-Mills theory

• supergravity

  • gauged supergravity

• superstring theory

  • type II string theory

  • heterotic string theory

Applications

• geometric model for elliptic cohomology

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Synthetic Differential Geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

• differentiation, chain rule

• differentiable function

• infinitesimal space, infinitesimally thickened point, amazing right adjoint

V-manifolds

• differentiable manifold, coordinate chart, atlas

• smooth manifold, smooth structure, exotic smooth structure

• analytic manifold, complex manifold

• formal smooth manifold, derived smooth manifold

smooth space

• diffeological space, Frölicher space

• manifold structure of mapping spaces

Tangency

• tangent bundle, frame bundle

  • vector field, multivector field, tangent Lie algebroid;

  • differential forms, de Rham complex, Dolbeault complex

    • pullback of differential forms, invariant differential form, Maurer-Cartan form, horizontal differential form,

    • cogerm differential form

    • integration of differential forms

• local diffeomorphism, formally étale morphism

• submersion, formally smooth morphism,

• immersion, formally unramified morphism,

• de Rham space, crystal

• infinitesimal disk bundle

The magic algebraic facts

• embedding of smooth manifolds into formal duals of R-algebras

• smooth Serre-Swan theorem

• derivations of smooth functions are vector fields

Theorems

• Hadamard lemma

• Borel's theorem

• Boman's theorem

• Whitney extension theorem

• Steenrod-Wockel approximation theorem

• Whitney embedding theorem

• Poincare lemma

• Stokes theorem

• de Rham theorem

• Hochschild-Kostant-Rosenberg theorem

• differential cohomology hexagon

Axiomatics

• Kock-Lawvere axiom

  • smooth topos, super smooth topos

  • microlinear space

  • integration axiom

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

  $(\esh \dashv \flat \dashv \sharp )$

  • discrete object, codiscrete object, concrete object

  • points-to-pieces transform

  • structures in cohesion

• dR-shape modality$\dashv$ dR-flat modality

  $\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

• classical modality

tangent cohesion

• differential cohomology diagram

differential cohesion

• (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

  $(\Re \dashv \Im \dashv \&)$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

graded differential cohesion

• fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality

  $(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

singular cohesion

$$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$$

Models

• Models for Smooth Infinitesimal Analysis

  • smooth algebra ($C^\infty$-ring)

  • smooth locus

  • Fermat theory

• Cahiers topos

• smooth ∞-groupoid

• formal smooth ∞-groupoid

• super formal smooth ∞-groupoid

Lie theory, ∞-Lie theory

• Lie algebra, Lie n-algebra, L-∞ algebra

• Lie group, Lie 2-group, smooth ∞-group

differential equations, variational calculus

• D-geometry, D-module

• jet bundle

• variational bicomplex, Euler-Lagrange complex

• Euler-Lagrange equation, de Donder-Weyl formalism,

• phase space

Chern-Weil theory, ∞-Chern-Weil theory

• connection on a bundle, connection on an ∞-bundle

• differential cohomology

  • ordinary differential cohomology, Deligne complex

  • differential K-theory

  • differential cobordism cohomology

• parallel transport, higher parallel transport, fiber integration in differential cohomology

• holonomy, higher holonomy

• gauge theory, higher gauge theory

• Wilson line, Wilson surface

Cartan geometry (super, higher)

• Klein geometry, (higher)

• G-structure, torsion of a G-structure

• Euclidean geometry, hyperbolic geometry, elliptic geometry

• (pseudo-)Riemannian geometry

  • orthogonal structure

  • isometry, Killing vector field, Killing spinor

  • spacetime, super-spacetime

• complex geometry

• symplectic geometry

• conformal geometry

Homotopy

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

• Introduction to Basic Homotopy Theory

• Introduction to Abstract Homotopy Theory

• geometry of physics – homotopy types

Definitions

• homotopy, higher homotopy

• homotopy type

• Pi-algebra, spherical object and Pi(A)-algebra

• homotopy coherent category theory

  • homotopical category

    • model category

    • category of fibrant objects, cofibration category

    • Waldhausen category

  • homotopy category

    • Ho(Top)

• (∞,1)-category

  • homotopy category of an (∞,1)-category

Paths and cylinders

• left homotopy

  • cylinder object

  • mapping cone

• right homotopy

  • path object

  • mapping cocone

  • universal bundle

• interval object

  • homotopy localization

  • infinitesimal interval object

Homotopy groups

• homotopy group

  • fundamental group

    • fundamental group of a topos

  • Brown-Grossman homotopy group

  • categorical homotopy groups in an (∞,1)-topos

  • geometric homotopy groups in an (∞,1)-topos

• fundamental ∞-groupoid

  • fundamental groupoid

    • path groupoid

  • fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos

  • fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos

• fundamental (∞,1)-category

  • fundamental category

Basic facts

• fundamental group of the circle is the integers

Theorems

• fundamental theorem of covering spaces

• Freudenthal suspension theorem

• Blakers-Massey theorem

• higher homotopy van Kampen theorem

• nerve theorem

• Whitehead's theorem

• Hurewicz theorem

• Galois theory

• homotopy hypothesis-theorem

Compact Object

objects $d \in C$ such that $C(d,-)$ commutes with certain colimits

• coproduct-preserving representable

• connected object

• compact object

  • compact object in a (0,1)-category, compact object in an (∞,1)-category

  • finite object

  • small object

• tiny object

• atomic object

  • amazing right adjoint

• accessible category

  • presentable category

Models

• compact topos

• compact topological space

Relative version

• proper geometric morphism

• proper map

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Model Category Theory

model category, model $\infty$-category

Definitions

• category with weak equivalences

  (relative category, homotopical category)

• fibration, cofibration

• weak factorization system

• resolution

• cell complex

• small object argument

• homotopy

• homotopy category$\;$of a model category

Morphisms

• Quillen adjunction

  • Quillen equivalence

  • Quillen bifunctor

  • derived functor

Universal constructions

• homotopy Kan extension

• homotopy limit/homotopy colimit

  homotopy weighted (co)limit

  homotopy (co)end

• Bousfield-Kan map

Refinements

• monoidal model category

  • monoidal Quillen adjunction

• enriched model category

  • enriched Quillen adjunction

• monoidal enriched model category

• simplicial model category

  • simplicial Quillen adjunction

  • simplicial monoidal model category

• cofibrantly generated model category

  • combinatorial model category

  • cellular model category

• algebraic model category

• compactly generated model category

• proper model category

• cartesian closed model category, locally cartesian closed model category

• stable model category

Producing new model structures

• on functor categories (global)

  • Reedy model structure

• on slice categories

• Bousfield localization

• transferred model structure

  • on algebraic fibrant objects

• Grothendieck construction for model categories

Presentation of $(\infty,1)$-categories

• (∞,1)-category

• simplicial localization

• (∞,1)-categorical hom-space

• presentable (∞,1)-category

Model structures

• Cisinski model structure

for $\infty$-groupoids

for ∞-groupoids

• on topological spaces

  • classical model structure

  • on compactly generated spaces

  • on Delta-generated spaces

  • on diffeological spaces

  • Strøm model structure

• Thomason model structure

• model structure on presheaves over a test category

• on simplicial sets, on semi-simplicial sets

  • classical model structure

  • constructive model structure

  • for right/left fibrations

• model structure on simplicial groupoids

• on cubical sets

• on strict ∞-groupoids, on groupoids

• on chain complexes/model structure on cosimplicial abelian groups

  related by the Dold-Kan correspondence

• model structure on cosimplicial simplicial sets

for equivariant $\infty$-groupoids

• fine model structure on topological G-spaces

• coarse model structure on topological G-spaces

  (Borel model structure)

for rational $\infty$-groupoids

• model structure on dgc-algebras

for rational equivariant $\infty$-groupoids

• model structure on equivariant chain complexes

• model structure on equivariant dgc-algebras

for $n$-groupoids

• for n-groupoids/for n-types

• for 1-groupoids

for $\infty$-groups

• model structure on simplicial groups

• model structure on reduced simplicial sets

for $\infty$-algebras

general $\infty$-algebras

• on monoids

• on simplicial T-algebras, on homotopy T-algebras

• on algebas over a monad

• on algebras over an operad,

  on modules over an algebra over an operad

specific $\infty$-algebras

• model structure on differential-graded commutative algebras

• model structure on differential graded-commutative superalgebras

• on dg-algebras over an operad

  • on dg-algebras and on on simplicial rings/on cosimplicial rings

    related by the monoidal Dold-Kan correspondence

  • for L-∞ algebras: on dg-Lie algebras, on dg-coalgebras, on simplicial Lie algebras

• model structure on dg-modules

for stable/spectrum objects

• model structure on spectra

• model structure on ring spectra

• model structure on parameterized spectra

• model structure on presheaves of spectra

for $(\infty,1)$-categories

• on categories with weak equivalences

• Joyal model for quasi-categories (and its cubical version)

• on sSet-categories

• for complete Segal spaces

• for Cartesian fibrations

for stable $(\infty,1)$-categories

• on dg-categories

for $(\infty,1)$-operads

• on operads, for Segal operads

  on algebras over an operad,

  on modules over an algebra over an operad

• on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations

for $(n,r)$-categories

• for (n,r)-categories as ∞-spaces

• for weak ∞-categories as weak complicial sets

• on cellular sets

• on higher categories in general

• on strict ∞-categories

for $(\infty,1)$-sheaves / $\infty$-stacks

• on homotopical presheaves

  • on simplicial presheaves

    global model structure/Cech model structure/local model structure

    on simplicial sheaves

    on presheaves of simplicial groupoids

    on sSet-enriched presheaves

• model structure for (2,1)-sheaves/for stacks

Categorification

categorification

Background

• category theory

• higher category theory

Contents

• vertical categorification

• horizontal categorification

• (∞,1)-categorification

• decategorification, Grothendieck group

Examples

• categorification in representation theory

• Khovanov homology

• Kazhdan-Lusztig theory

• categorification via groupoid schemes

• The geometric Langlands program

• geometric Satake

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nLab

• Content

  • Mathematics
  • Physics
  • Philosophy

• Resources

  • Mathematics Resources
  • Physics Resources
  • Philosophy Resources

• Community

  • nForum

    • Latest Changes

  • HowTo

  • FAQ

  • Contributors?

  • Uploaded Files

  • Sandbox

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Infinity-lie Theory

∞-Lie theory (higher geometry)

Background

Smooth structure

• generalized smooth space

  • smooth manifold

  • diffeological space

  • Frölicher space

• smooth topos

  • Cahiers topos

• smooth ∞-groupoid, concrete smooth ∞-groupoid

• synthetic differential ∞-groupoid

Higher groupoids

• ∞-groupoid

  • groupoid

  • 2-groupoid

  • strict ∞-groupoid

    • crossed complex

• ∞-group

  • simplicial group

Lie theory

• Lie theory

  • Lie integration, Lie differentiation

  • Lie's three theorems

  • Lie theory for stacky Lie groupoids

∞-Lie groupoids

• ∞-Lie groupoid

  • strict ∞-Lie groupoid

  • Lie groupoid

    • differentiable stack

    • orbifold

• ∞-Lie group

  • Lie group

    • simple Lie group, semisimple Lie group

  • Lie 2-group

∞-Lie algebroids

• ∞-Lie algebroid

  • Lie algebroid

• Lie ∞-algebroid representation

• L-∞-algebra

  • model structure for L-∞ algebras: on dg-Lie algebras, on dg-coalgebras, on simplicial Lie algebras

  • Lie algebra

    • semisimple Lie algebra, compact Lie algebra

  • Lie 2-algebra

    • strict Lie 2-algebra

    • differential crossed module

• Lie 3-algebra

  • differential 2-crossed module

• dg-Lie algebra, simplicial Lie algebra

• super L-∞ algebra

  • super Lie algebra

Formal Lie groupoids

• formal group, formal groupoid

Cohomology

• Lie algebra cohomology

  • Chevalley-Eilenberg algebra

  • Weil algebra

  • invariant polynomial

    • Killing form

• nonabelian Lie algebra cohomology

Homotopy

• homotopy groups of a Lie groupoid

Related topics

• ∞-Chern-Weil theory

Examples

$\infty$-Lie groupoids

• Atiyah Lie groupoid

• fundamental ∞-groupoid

  • path groupoid

  • path n-groupoid

• smooth principal ∞-bundle

$\infty$-Lie groups

• orthogonal group

  • special orthogonal group

  • spin group

  • string 2-group

  • fivebrane 6-group

• unitary group

  • special unitary group

• circle Lie n-group

  • circle group

$\infty$-Lie algebroids

• tangent Lie algebroid

• action Lie algebroid

• Atiyah Lie algebroid

• symplectic Lie n-algebroid

  • symplectic manifold

  • Poisson Lie algebroid

  • Courant Lie algebroid

    • generalized complex geometry

$\infty$-Lie algebras

• general linear Lie algebra

• orthogonal Lie algebra, special orthogonal Lie algebra

• endomorphism L-∞ algebra

• automorphism ∞-Lie algebra

• string Lie 2-algebra

• fivebrane Lie 6-algebra

• supergravity Lie 3-algebra

• supergravity Lie 6-algebra

• line Lie n-algebra

Infinity-limits

limits and colimits

1-Categorical

• limit and colimit

  • limits and colimits by example

  • commutativity of limits and colimits

  • small limit

  • filtered colimit

    • directed colimit

      • sequential colimit

  • sifted colimit

  • connected limit, wide pullback

  • preserved limit, reflected limit, created limit

  • product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum

  • finite limit

    • exact functor

• Kan extension

  • Yoneda extension

• weighted limit

• end and coend

• fibered limit

2-Categorical

• 2-limit

  • inserter

  • isoinserter

  • equifier

  • inverter

  • PIE-limit

• 2-pullback, comma object

(∞,1)-Categorical

• (∞,1)-limit

  • (∞,1)-pullback

    • fiber sequence

Model-categorical

• homotopy Kan extension

• homotopy limit

  • homotopy product

  • homotopy equalizer

  • homotopy fiber

  • mapping cone

  • homotopy pullback

  • homotopy totalization

  • homotopy end

• homotopy colimit

  • homotopy coproduct

  • homotopy coequalizer

  • homotopy cofiber

  • mapping cocone

  • homotopy pushout

  • homotopy realization

  • homotopy coend

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Categorytheory

Category Theory

Contents

• Item 1

  • Sub-Item

• Item 2

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Type Theory

natural deduction metalanguage, practical foundations

• judgement

• hypothetical judgement, sequent

  • antecedents$\vdash$ consequent, succedents

1. type formation rule
2. term introduction rule
3. term elimination rule
4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

• calculus of constructions

syntax object language

• theory, axiom

• proposition/type (propositions as types)

• definition/proof/program (proofs as programs)

• theorem

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

homotopy levels

• type theory

• 2-type theory, 2-categorical logic

• homotopy type theory, homotopy type theory - contents

  • homotopy type

  • univalence, function extensionality, internal logic of an (∞,1)-topos

  • cohesive homotopy type theory

  • directed homotopy type theory

  • HoTT methods for homotopy theorists

semantics

• internal logic, categorical semantics

  • display map

• internal logic of a topos

  • Mitchell-Benabou language

  • Kripke-Joyal semantics

• internal logic of an (∞,1)-topos

  • type-theoretic model category

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Quasi-category Theory

(∞,1)-category theory

Background

• category theory

• higher category theory

• (n,r)-category

Basic concepts

• (∞,1)-category

• hom-objects

• equivalences in/of $(\infty,1)$-categories

• sub-(∞,1)-category

  • reflective sub-(∞,1)-category

  • reflective localization

• opposite (∞,1)-category

• over (∞,1)-category

  • join of quasi-categories

• (∞,1)-functor

  • exact (∞,1)-functor

  • (∞,1)-category of (∞,1)-functors

  • (∞,1)-category of (∞,1)-presheaves

• fibrations

  • inner fibration

  • left/right fibration

  • Cartesian fibration

    • Cartesian morphism

Universal constructions

• limit

  • terminal object

• adjoint functors

Local presentation

• locally presentable

  • essentially small

  • locally small

  • accessible

  • idempotent-complete

Theorems

• (∞,1)-Yoneda lemma

• (∞,1)-Grothendieck construction

• adjoint (∞,1)-functor theorem

• (∞,1)-monadicity theorem

Extra stuff, structure, properties

• stable (∞,1)-category

• (∞,1)-topos

Models

• category with weak equivalences

• model category

• derivator

• quasi-category

  • model structure for quasi-categories

  • model structure for Cartesian fibrations

• relation to simplicial categories

  • homotopy coherent nerve

  • simplicial model category

  • presentable quasi-category

• Kan complex

  • model structure for Kan complexes

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

• open subset, closed subset, neighbourhood

• topological space, locale

• base for the topology, neighbourhood base

• finer/coarser topology

• closure, interior, boundary

• separation, sobriety

• continuous function, homeomorphism

• uniformly continuous function

• embedding

• open map, closed map

• sequence, net, sub-net, filter

• convergence

• categoryTop

  • convenient category of topological spaces

Universal constructions

• initial topology, final topology

• subspace, quotient space,

• fiber space, space attachment

• product space, disjoint union space

• mapping cylinder, mapping cocylinder

• mapping cone, mapping cocone

• mapping telescope

• colimits of normal spaces

Extra stuff, structure, properties

• nice topological space

• metric space, metric topology, metrisable space

• Kolmogorov space, Hausdorff space, regular space, normal space

• sober space

• compact space, proper map

  sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact

• compactly generated space

• second-countable space, first-countable space

• contractible space, locally contractible space

• connected space, locally connected space

• simply-connected space, locally simply-connected space

• cell complex, CW-complex

• pointed space

• topological vector space, Banach space, Hilbert space

• topological group

• topological vector bundle, topological K-theory

• topological manifold

Examples

• empty space, point space

• discrete space, codiscrete space

• Sierpinski space

• order topology, specialization topology, Scott topology

• Euclidean space

  • real line, plane

• cylinder, cone

• sphere, ball

• circle, torus, annulus, Moebius strip

• polytope, polyhedron

• projective space (real, complex)

• classifying space

• configuration space

• path, loop

• mapping spaces: compact-open topology, topology of uniform convergence

  • loop space, path space

• Zariski topology

• Cantor space, Mandelbrot space

• Peano curve

• line with two origins, long line, Sorgenfrey line

• K-topology, Dowker space

• Warsaw circle, Hawaiian earring space

Basic statements

• Hausdorff spaces are sober

• schemes are sober

• continuous images of compact spaces are compact

• closed subspaces of compact Hausdorff spaces are equivalently compact subspaces

• open subspaces of compact Hausdorff spaces are locally compact

• quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff

• compact spaces equivalently have converging subnet of every net

  • Lebesgue number lemma

  • sequentially compact metric spaces are equivalently compact metric spaces

  • compact spaces equivalently have converging subnet of every net

  • sequentially compact metric spaces are totally bounded

• continuous metric space valued function on compact metric space is uniformly continuous

• paracompact Hausdorff spaces are normal

• paracompact Hausdorff spaces equivalently admit subordinate partitions of unity

• closed injections are embeddings

• proper maps to locally compact spaces are closed

• injective proper maps to locally compact spaces are equivalently the closed embeddings

• locally compact and sigma-compact spaces are paracompact

• locally compact and second-countable spaces are sigma-compact

• second-countable regular spaces are paracompact

• CW-complexes are paracompact Hausdorff spaces

Theorems

• Urysohn's lemma

• Tietze extension theorem

• Tychonoff theorem

• tube lemma

• Michael's theorem

• Brouwer's fixed point theorem

• topological invariance of dimension

• Jordan curve theorem

Analysis Theorems

• Heine-Borel theorem

• intermediate value theorem

• extreme value theorem

topological homotopy theory

• left homotopy, right homotopy

• homotopy equivalence, deformation retract

• fundamental group, covering space

• fundamental theorem of covering spaces

• homotopy group

• weak homotopy equivalence

• Whitehead's theorem

• Freudenthal suspension theorem

• nerve theorem

• homotopy extension property, Hurewicz cofibration

• cofiber sequence

• Strøm model category

• classical model structure on topological spaces

Functional Analysis

• Functional Analysis

Overview diagrams

• topological vector spaces

• locally convex topological vector spaces

Basic concepts

• topological vector space

  • locally convex topological vector space

  • Banach Spaces

    • reflexive

  • Smith Spaces

  • Hilbert Spaces, Fréchet Spaces, Sobolev spaces, Lebesgue Spaces

  • Bornological Vector Spaces

  • Barrelled Vector Spaces

• linear operator

  • bounded, unbounded, self-adjoint, compact, Fredholm

  • spectrum of an operator

• operator algebras

  • functional calculus

Theorems

• Stone-Weierstrass theorem

• spectral theory

  • spectral theorem

  • Gelfand duality

• functional calculus

• Riesz representation theorem

• measure theory

Topics in Functional Analysis

• Bases

• Algebraic Theories in Functional Analysis

• An Elementary Treatment of Hilbert Spaces

• When are two Banach spaces isomorphic?

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Monoidal Categories

monoidal categories

• enriched monoidal category, tensor category

• string diagram, tensor network

With braiding

• braided monoidal category

• balanced monoidal category

  • twist

• symmetric monoidal category

With duals for objects

• category with duals (list of them)

• dualizable object (what they have)

• rigid monoidal category, a.k.a. autonomous category

• pivotal category

• spherical category

• ribbon category, a.k.a. tortile category

• compact closed category

With duals for morphisms

• monoidal dagger-category?

• symmetric monoidal dagger-category

• dagger compact category

With traces

• trace

• traced monoidal category

Closed structure

• closed monoidal category

• cartesian closed category

• closed category

• star-autonomous category

Special sorts of products

• cartesian monoidal category

• semicartesian monoidal category

• monoidal category with diagonals

• multicategory

Semisimplicity

• semisimple category

• fusion category

• modular tensor category

Morphisms

• monoidal functor

  (lax, oplax, strong bilax, Frobenius)

• braided monoidal functor

• symmetric monoidal functor

Internal monoids

• monoid in a monoidal category

• commutative monoid in a symmetric monoidal category

• module over a monoid

Examples

• tensor product

• closed monoidal structure on presheaves

• Day convolution

Theorems

• coherence theorem for monoidal categories

• monoidal Dold-Kan correspondence

In higher category theory

• monoidal 2-category

  • braided monoidal 2-category

• monoidal bicategory

  • cartesian bicategory

• k-tuply monoidal n-category

  • little cubes operad

• monoidal (∞,1)-category

  • symmetric monoidal (∞,1)-category

• compact double category

2-category Theory

2-category theory

Definitions

• 2-category

• strict 2-category

• bicategory

• enriched bicategory

Transfors between 2-categories

• 2-functor

  • pseudofunctor

  • lax functor

  • equivalence of 2-categories

• 2-natural transformation

  • lax natural transformation

  • icon

• modification

• Yoneda lemma for bicategories

Morphisms in 2-categories

• fully faithful morphism

• faithful morphism

• conservative morphism

• pseudomonic morphism

• discrete morphism

• eso morphism

Structures in 2-categories

• adjunction

• mate

• monad

• cartesian object

• fibration in a 2-category

• codiscrete cofibration

Limits in 2-categories

• 2-limit

• 2-pullback

• comma object

• inserter

• inverter

• equifier

Structures on 2-categories

• 2-monad

  • lax-idempotent 2-monad

  • pseudomonad

  • pseudoalgebra for a 2-monad

• monoidal 2-category

  • cartesian bicategory

• Gray tensor product

• proarrow equipment

Yoneda Lemma

Yoneda lemma

Ingredients

• category

• functor

• natural transformation

• presheaf

• category of presheaves

• representable presheaf

• Yoneda embedding

Incarnations

• Yoneda lemma

• enriched Yoneda lemma

• co-Yoneda lemma

• Yoneda reduction

Properties

• free cocompletion

• Yoneda extension

Universal aspects

• representable functor

• universal construction

• universal element

Classification

• classifying space, classifying stack

• moduli space, moduli stack, derived moduli space

• classifying topos

• subobject classifier

• universal principal bundle, universal principal ∞-bundle

• classifying morphism

Induced theorems

• Tannaka duality

…

In higher category theory

Yoneda lemma for higher categories:

• Yoneda lemma for (∞,1)-categories

• Yoneda lemma for bicategories

• Yoneda lemma for tricategories

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Enriched Category Theory

enriched category theory

Background

• category theory

• monoidal category, closed monoidal category

• cosmos, multicategory, bicategory, double category, virtual double category

Basic concepts

• enriched category

• enriched functor, profunctor

• enriched natural transformation

• enriched adjoint functor

• enriched product category

• enriched functor category

Universal constructions

• weighted limit

• end, coend

Extra stuff, structure, property

• copowering (tensoring)

  powering (cotensoring)

• monoidal enriched category

  • cartesian closed enriched category

  • locally cartesian closed enriched category

Homotopical enrichment

• enriched homotopical category

• enriched model category

• model structure on homotopical presheaves

Aqft And Operator Algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

• classical, pre-quantum, quantum, perturbative quantum

• relativistic, Euclidean, thermal

Lagrangian field theory

• field (physics)

  • field bundle

  • field history

  • space of field histories

• Lagrangian density

  • Euler-Lagrange form, presymplectic current

  • Euler-Lagrangeequations of motion

• locally variational field theory

• covariant phase space

  • Peierls-Poisson bracket

    • advanced and retarded propagator,

    • causal propagator

quantization

• geometric quantizationof symplectic groupoids

• algebraic deformation quantization, star algebra

quantum mechanical system, quantum probability

• subsystem

• observables

  • field observables

  • local observables

  • polynomial observables

  • microcausal observables

  • operator algebra, C*-algebra, von Neumann algebra

• local net of observables

  • causal locality

  • Haag-Kastler axioms

  • Wightman axioms

  • field net

  • conformal net

• state on a star-algebra, expectation value

  • pure state

    wave function

    collapse of the wave function/conditional expectation value

  • mixed state, density matrix

  • space of quantum states

  • vacuum state

    • quasi-free state,

    • Hadamard state

    • Wightman propagator

• picture of quantum mechanics

free field quantization

• star algebra, Moyal deformation quantization

• Wick algebra

  • canonical commutation relations, Weyl relations

  • normal ordered product

• Fock space

gauge theories

• gauge symmetry

• BRST complex, BV-BRST formalism

• local BV-BRST complex

• BV-operator

• quantum master equation

• master Ward identity

• gauge anomaly

interacting field quantization

• causal perturbation theory, perturbative AQFT

• interaction

• S-matrix, scattering amplitude

  • causal additivity

  • time-ordered product, Feynman propagator

  • Feynman diagram, Feynman perturbation series

  • effective action

  • vacuum stability

• interacting field algebra

  • Bogoliubov's formula

  • quantum Møller operator

• adiabatic limit

  • infrared divergence

  • interacting vacuum

renormalization

• ("re-")normalization scheme

  • extension of distributions

  • ("re"-)normalization condition

  • quantum anomaly

• renormalization group

  • interaction vertex redefinition

  • Stückelberg-Petermann renormalization group

  • renormalization group flow/running coupling constants

• effective quantum field theory

  • UV cutoff

  • counterterms

  • relative effective action

  • Wilsonian RG, Polchinski flow equation

Theorems

States and observables

• order-theoretic structure in quantum mechanics

  • Alfsen-Shultz theorem

  • Harding-Döring-Hamhalter theorem

• Kochen-Specker theorem

• Bell's theorem

• Fell's theorem

• Gleason's theorem

• Wigner theorem

• Bub-Clifton theorem

• Kadison-Singer problem

Operator algebra

• Wick's theorem

• GNS construction

  • cyclic vector, separating vector

• modular theory

• Fell's theorem

• Stone-von Neumann theorem

• Haag's theorem

Local QFT

• Reeh-Schlieder theorem

• Bisognano-Wichmann theorem

• PCT theorem

• spin-statistics theorem

• DHR superselection theory

• Osterwalder-Schrader theorem (Wick rotation)

Perturbative QFT

• Schwinger-Dyson equation

• main theorem of perturbative renormalization

Manifolds And Cobordisms

manifolds and cobordisms

cobordism theory, Introduction

Definitions

• locally Euclidean space

  • coordinate chart, coordinate transformation

  • atlas,

  • smooth structure

• manifold

  • topological manifold

  • differentiable manifold, ,smooth manifold

• infinite dimensional manifold

  • Banach manifold, Hilbert manifold, ILH manifold, Frechet manifold, convenient manifold

• tangent bundle

• normal bundle

• G-structure, torsion of a G-structure

  • orientation, spin structure, string structure, fivebrane structure

• Cartan geometry:

  • Riemannian manifold

  • complex manifold

  • symplectic manifold

• cobordism

  • B-bordism

  • extended cobordism

  • cobordism category

  • (∞,n)-category of cobordisms

    • functorial quantum field theory

  • Thom spectrum

    • cobordism ring

    • genus

Genera and invariants

• signature genus, Kervaire invariant

• A-hat genus, Witten genus

Classification

• 2-manifolds/surfaces

  • genus of a surface

• 3-manifolds

  • Kirby calculus

• 4-manifolds

  • Dehn surgery

  • exotic smooth structure

Theorems

• Whitney embedding theorem

• Thom's transversality theorem

• Pontrjagin-Thom construction

• Galatius-Tillmann-Madsen-Weiss theorem

• geometrization conjecture,

  • Poincaré conjecture

  • elliptization conjecture

• cobordism hypothesis-theorem

Homological Algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

• additive and abelian categories

  • Ab-enriched category

  • pre-additive category

  • additive category

  • pre-abelian category

  • abelian category

  • Grothendieck category

• abelian sheaves

• semi-abelian category

Basic definitions

• kernel, cokernel

• complex

  • differential

  • homology

• category of chain complexes

  • chain complex

  • chain map

  • chain homotopy

• chain homology and cohomology

  • quasi-isomorphism

  • homological resolution

  • simplicial homology

  • generalized homology

• exact sequence,

  • short exact sequence, long exact sequence, split exact sequence

• injective object, projective object

  • injective resolution, projective resolution

  • flat resolution

Stable homotopy theory notions

• derived category

• triangulated category, enhanced triangulated category

• stable (∞,1)-category

  • stable model category

  • pretriangulated dg-category

  • A-∞-category

• (∞,1)-category of chain complexes

• derived functor, derived functor in homological algebra

  • Tor, Ext

  • homotopy limit, homotopy colimit

  • lim^1 and Milnor sequences

  • fr-code

• abelian sheaf cohomology

Constructions

• double complex

• Koszul-Tate resolution, BRST-BV complex

• spectral sequence

  • spectral sequence of a filtered complex

  • spectral sequence of a double complex

  • Grothendieck spectral sequence

    • Leray spectral sequence

    • Serre spectral sequence

    • Hochschild-Serre spectral sequence

Lemmas

diagram chasing

• 3x3 lemma

• four lemma, five lemma

• snake lemma, connecting homomorphism

• horseshoe lemma

• Baer's criterion

Schanuel's lemma

Homology theories

• singular homology

• cyclic homology

Theorems

• Dold-Kan correspondence / monoidal, operadic

  • Moore complex, Alexander-Whitney map, Eilenberg-Zilber map

• Eilenberg-Zilber theorem

  • cochain on a simplicial set

• universal coefficient theorem

• Künneth theorem

Differential Cohomology

differential cohomology

Ingredients

• cohomology

• differential geometry

Connections on bundles

• connection on a bundle

  • parallel transport, holonomy

• curvature

  • curvature characteristic form, Chern character

• Chern-Weil theory

  • Chern-Weil homomorphism

  • secondary characteristic class

  • differential characteristic class

Higher abelian differential cohomology

• differential function complex

• differential orientation

• ordinary differential cohomology

  • differential Thom class

  • differential characters,

  • Deligne cohomology

  • circle n-bundle with connection,

  • bundle gerbe with connection

• differential K-theory

• differential elliptic cohomology

• differential cobordism cohomology

Higher nonabelian differential cohomology

• principal 2-bundle, principal ∞-bundle

• connection on a 2-bundle, connection on an ∞-bundle

• Chern-Weil theory in Smooth∞Grpd

  • ∞-Lie algebra cohomology

• ∞-Chern-Simons theory

Fiber integration

• higher holonomy

• fiber integration in differential cohomology

  • fiber integration in ordinary differential cohomology

  • fiber integration in differential K-theory

Application to gauge theory

• gauge theory

• gauge field

  • electromagnetic field

  • Yang-Mills field

  • Kalb-Ramond field/B-field

  • RR-field

  • supergravity C-field

  • supergravity

• quantum anomaly

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Infinity-chern-weil Theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

• Chern-Weil theory

• ∞-Chern-Weil theory introduction

Ingredients

• cohomology

  • differential cohomology

• ∞-Lie theory

  • Lie integration

• ∞-Lie algebra cohomology

  • Chevalley-Eilenberg algebra, Weil algebra, invariant polynomial

Connection

• ∞-Lie algebroid valued differential forms

• ∞-connection on a principal ∞-bundle

Curvature

• curvature

  • Bianchi identity

  • curvature characteristic form

  • covariant derivative

• secondary characteristic class

• Chern-Simons form

Theorems

• Chern-Weil homomorphism

• Chern-Gauss-Bonnet theorem

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Gravity

gravity, supergravity

Formalism

• Poincaré Lie algebra

  • super Poincaré Lie algebra, supergravity Lie 3-algebra, supergravity Lie 6-algebra

• differential geometry

• (super-)Cartan geometry

  • pseudo-Riemannian manifold

  • spacetime

Definition

• Einstein-Hilbert action, Einstein's equations, general relativity

  • first-order formulation of gravity, D'Auria-Fre formulation of supergravity

  • gravity as a BF-theory, Plebanski formulation of gravity, teleparallel gravity

Spacetime configurations

• spacetime

• Minkowski spacetime

• black hole spacetime

  • Schwarzschild spacetime

  • Kerr spacetime

Properties

• Kaluza-Klein mechanism

• Bekenstein-Hawking entropy

• generalized second law of thermodynamics

• cosmic censorship hypothesis

• weak gravity conjecture

Spacetimes

• standard model of cosmology

• Minkowski spacetime

black hole spacetimes	vanishing angular momentum	positive angular momentum
vanishing charge	Schwarzschild spacetime	Kerr spacetime
positive charge	Reissner-Nordstrom spacetime	Kerr-Newman spacetime

• asymptotically flat spacetime

• gravitational wave

• anti de Sitter spacetime

• FRW spacetime

• Taub-NUT spacetime

• Einstein manifold

• pp-wave spacetime

• KK-monopole

• Malament–Hogarth spacetime

• super spacetime

  • super Minkowski spacetime

• black brane

Quantum theory

• quantum gravity

  • graviton, gravitino

  • string theory

Knot Theory

knot theory

knot, link

• isotopy

• knot complement

• knot diagrams, chord diagram

• Reidemeister move

Examples/classes:

• trefoil knot

• torus knot

• singular knot

• hyperbolic knot

• Borromean link

• Whitehead link

• Hopf link

Types

• prime knot

• mutant knot

knot invariants

• crossing number

• bridge number

• unknotting number

• colorability

• knot group

• knot genus?

• polynomial knot invariants

  (observables of non-perturbative Chern-Simons theory)

  • Jones polynomial

  • HOMFLY polynomial

  • Alexander polynomial

  • Reshetikhin-Turaev invariants

• Vassiliev knot invariants

  (observables of pertrubative Chern-Simons theory)

• Khovanov homology

• Kauffman bracket?

link invariants

• Milnor mu-bar invariants

• linking number

Related concepts:

• Vassiliev skein relation

• Seifert surface

• virtual knot theory

• Dehn surgery, Kirby calculus

• volume conjecture

• arithmetic topology

category: knot theory

Riemannian Geometry

Riemannian geometry

Basic definitions

• Riemannian manifold

  • metric, isometry, isometry group

• moduli space of Riemannian metrics

• pseudo-Riemannian manifold

  • Lorentzian manifold, spacetime

• geodesic

  • geodesic convexity

  • geodesic flow

• Levi-Civita connection

  • Riemann curvature, torsion of a metric connection

Further concepts

• Hodge inner product, Hodge star operator

• gradient, gradient flow

Theorems

• Poincaré conjecture-theorem

Applications

• gravity

  • Einstein-Hilbert action, Einstein equations, supergravity

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Infinity-chern-simons Theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

• quantum field theory

• circle n-bundle with connection

• ∞-Chern-Weil theory

  • ∞-Lie algebroid cohomology

  • Chern-Simons element

Definition

• level (Chern-Simons theory)

Examples

• For semisimple Lie algebra targets

  • Chern-Simons theory

• For discrete group targets

  • Dijkgraaf-Witten theory

• For discrete 2-group targets

  • Crane-Yetter model

• For Lie 2-algebra targets

  • BF-theory coupled to topological Yang-Mills theory

• For targets extending the super Poincare Lie algebra

  (such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)

  • Chern-Simons gravity

  • Chern-Simons-supergravity

• 1-dimensional Chern-Simons theory

• for higher abelian targets

  • higher dimensional Chern-Simons theory

  • higher electric background charge coupling

• for symplectic Lie n-algebroid targets

  • AKSZ theory

    • Poisson sigma model

      • A-model, B-model

    • Courant sigma model

      • Chern-Simons theory

    • topological Yang-Mills theory

• for the $L_\infty$-structure on the BRST complex of the closed string:

  • string field theory

• higher dimensional Chern-Simons theory

  • 1d Chern-Simons theory

    • Wilson loop

  • 2d Chern-Simons theory

    • Poisson sigma-model,

    • A-model, B-model topological string

  • 3d Chern-Simons theory

    • Chern-Simons theory

    • Dijkgraaf-Witten theory

    • Courant sigma-model

    • 3d quantum gravity

  • 4d Chern-Simons theory

    • Yetter model

    • BF-theory

  • 5d Chern-Simons theory

  • 6d Chern-Simons theory

  • 7d Chern-Simons theory

    • topological AdS7/CFT6-sector

    • topological M-theory

  • 11d Chern-Simons theory

    • RR-field

• ∞-Chern-Simons theory

  • ∞-Dijkgraaf-Witten theory

  • string field theory

  • AKSZ sigma-model

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Foundations

foundations

The basis of it all

• mathematical logic

• deduction system, natural deduction, sequent calculus, lambda-calculus, judgment

• type theory, simple type theory, dependent type theory

• collection, object, type, term, set, element

• equality, judgmental equality, typal equality

• universe, size issues

• higher-order logic

 Set theory

set theory

• fundamentals of set theory
  • propositional logic
  • first-order logic
  • typed predicate logic
  • membership relation
  • propositional equality
  • set, element, function, relation
  • universe, small set, large set
• material set theory
  • membership relation, propositional equality, axiom of extensionality
  • pairing structure, axiom of pairing
  • union structure, axiom of union
  • powerset structure, axiom of power sets
  • natural numbers structure, axiom of infinity
• presentations of set theory
  • first-order set theory
  • unsorted set theory
  • simply sorted set theory
    • one-sorted set theory
    • two-sorted set theory
    • three-sorted set theory
  • dependently sorted set theory
  • structurally presented set theory
• structuralism in set theory
  • material set theory
    • ZFC
    • ZFA
    • Mostowski set theory
    • New Foundations
  • structural set theory
    • categorical set theory
      • ETCS
        • fully formal ETCS
        • ETCS with elements
        • Trimble on ETCS I
        • Trimble on ETCS II
        • Trimble on ETCS III
      • structural ZFC
    • allegorical set theory
      • SEAR
• class-set theory
  • class, proper class
  • universal class, universe
  • category of classes
  • category with class structure
• constructive set theory
• algebraic set theory

Foundational axioms

foundational axioms

• basic constructions:

  • axiom of cartesian products
  • axiom of disjoint unions
  • axiom of the empty set
  • axiom of fullness
  • axiom of function sets
  • axiom of power sets
  • axiom of quotient sets

• material axioms:

  • axiom of extensionality
  • axiom of foundation
  • axiom of anti-foundation
  • Mostowski's axiom
  • axiom of pairing
  • axiom of transitive closure
  • axiom of union

• structural axioms:

  • axiom of materialization

• type theoretic axioms:

  • axioms of set truncation
    • uniqueness of identity proofs
    • axiom K
    • boundary separation
    • equality reflection
    • axiom of circle type localization
  • homotopy type theoretic axioms:
    • univalence axiom
    • Whitehead's principle

• axioms of choice:

  • axiom of choice
  • axiom of countable choice
  • axiom of dependent choice
  • axiom of excluded middle
  • axiom of existence
  • axiom of multiple choice
  • Markov's axiom
  • presentation axiom
  • small cardinality selection axiom
  • axiom of small violations of choice
  • axiom of weakly initial sets of covers

• large cardinal axioms:

  • axiom of infinity
  • axiom of universes
  • regular extension axiom
  • inaccessible cardinal
  • measurable cardinal
  • elementary embedding
  • supercompact cardinal
  • Vopěnka's principle

• strong axioms

  • axiom of separation
  • axiom of replacement

• further

  • reflection principle

• axiom of inequality spaces

Removing axioms

• constructive mathematics
• predicative mathematics

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Cohesive Infinity-toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

• (∞,1)-topos theory

• higher geometry

• motivation for cohesive toposes

Definition

• locally connected topos / locally ∞-connected (∞,1)-topos

  • connected topos / ∞-connected (∞,1)-topos

  • strongly connected topos / strongly ∞-connected (∞,1)-topos

  • totally connected topos / totally ∞-connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos

Presentation over a site

• locally connected site / locally ∞-connected site

  • connected site / ∞-connected site

  • strongly connected site / strongly ∞-connected site

  • totally connected site / totally ∞-connected site

• local site / ∞-local site

• cohesive site, ∞-cohesive site

Models

• discrete ∞-groupoid

  • discrete space, discrete group, discrete groupoid

• D-topological ∞-groupoid

  • topological space, topological groupoid, topological group

• smooth ∞-groupoid

  • smooth space, diffeological space, smooth manifold, Lie group, Lie groupoid, differentiable stack, Lie 2-group, Lie 2-groupoid, smooth ∞-groupoid

• synthetic differential ∞-groupoid

  • Lie algebra, Lie algebroid, ∞-Lie algebra, ∞-Lie algebroid

• super ∞-groupoid

  • superalgebra, super Lie algebra, super L-∞ algebra

• smooth super ∞-groupoid

  • superpoint, supergeometry, supermanifold, super Lie group, super ∞-groupoid, synthetic differential super ∞-groupoid

• cohesion of global- over G-equivariant homotopy theory

Group Theory

group theory

• group, ∞-group
• group object, group object in an (∞,1)-category
• abelian group, spectrum
• super abelian group
• group action, ∞-action
• representation, ∞-representation
• progroup
• homogeneous space

Classical groups

• general linear group

• unitary group

  • special unitary group. projective unitary group

• orthogonal group

  • special orthogonal group

• symplectic group

Finite groups

• finite group

• symmetric group, cyclic group, braid group

• classification of finite simple groups

• sporadic finite simple groups

  • Monster group, Mathieu group

Group schemes

• algebraic group

• abelian variety

Topological groups

• topological group

• compact topological group, locally compact topological group

• maximal compact subgroup

• string group

Lie groups

• Lie group

• compact Lie group

• Kac-Moody group

Super-Lie groups

• super Lie group

• super Euclidean group

Higher groups

• 2-group

  • crossed module, strict 2-group

• n-group

• ∞-group

  • simplicial group

  • crossed complex

  • k-tuply groupal n-groupoid

  • spectrum

• circle n-group, string 2-group, fivebrane Lie 6-group

Cohomology and Extensions

• group cohomology

• group extension,

• ∞-group extension, Ext-group

Related concepts

• quantum group

Variational Calculus

variational calculus

Differential geometric version

• variational bicomplex

• local Lagrangian, local action functional

  • Euler-Lagrange form, source form

  • Lepage form

• principle of extremal action, Euler-Lagrange equations, de Donder-Weyl formalism

• covariant phase space

• Noether's theorem

  • conserved current, charge

  • symmetry

Derived differential geometric version

• BV-BRST complex

• D-module

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Mapping Space

internal hom/mapping space

General abstract

• hom-set, hom-object, internal hom, exponential object, derived hom-space

• loop space object, free loop space object, derived loop space

Topology

• mapping space (compact-open topology)

• topology of mapping spaces

• evaluation fibration of mapping spaces

• loop space, free loop space

  • free loop space of a classifying space

Simplicial homotopy theory

• simplicial mapping complex

• inertia groupoid

Differential topology

• differential topology of mapping spaces

  • C-k topology

• manifold structure of mapping spaces

  • tangent spaces of mapping spaces

• smooth loop space

Stable homotopy theory

• mapping spectrum

Geometric homotopy theory

• mapping stack

• inertia stack

• free loop stack

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Measure Theory

measure theory

probability theory

(quantum probability)

Measure theory

• measurable space, measurable locale

• measure, measure space

• von Neumann algebra

• geometric measure theory

Probability theory

• probability space

• probability distribution

• state

  • in AQFT and operator algebra

  • GNS construction

  • Fell's theorem

• entropy, relative entropy

Information geometry

• information geometry

• information metric

• Wasserstein metric

Thermodynamics

• thermodynamics

• second law of thermodynamics, generalized second law

• ergodic theory

Theorems

• Riesz representation theorem

• de Finetti's theorem

• law of large numbers

• Kolmogorov extension theorem

Applications

• machine learning, neural networks

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

• (higher) category theory and physics

• geometry of physics

• books and reviews, physics resources

---

theory (physics), model (physics)

experiment, measurement, computable physics

• mechanics

  • mass, charge, momentum, angular momentum, moment of inertia

  • dynamics on Lie groups

    • rigid body dynamics

• field (physics)

  • Lagrangian mechanics

    • configuration space, state

    • action functional, Lagrangian

    • covariant phase space, Euler-Lagrange equations

  • Hamiltonian mechanics

    • phase space

    • symplectic geometry

      • Poisson manifold

      • symplectic manifold

      • symplectic groupoid

    • multisymplectic geometry

      • n-symplectic manifold

  • spacetime

    • smooth Lorentzian manifold

    • special relativity

    • general relativity

    • gravity

      • supergravity, dilaton gravity

      • black hole

• Classical field theory

  • classical physics

    • classical mechanics
    • waves and optics
    • thermodynamics

• Quantum Mechanics

  • in terms of ∞-compact categories

  • quantum information

  • Hamiltonian operator

  • density matrix

  • Kochen-Specker theorem

  • Bell's theorem

  • Gleason's theorem

• Quantization

  • geometric quantization

  • deformation quantization

  • path integral quantization

  • semiclassical approximation

• Quantum Field Theory

  • Axiomatizations

    • algebraic QFT

      • Wightman axioms

      • Haag-Kastler axioms

        • operator algebra

        • local net

      • conformal net

      • Reeh-Schlieder theorem

      • Osterwalder-Schrader theorem

      • PCT theorem

      • Bisognano-Wichmann theorem

        • modular theory

      • spin-statistics theorem

        • boson, fermion

    • functorial QFT

      • cobordism

      • (∞,n)-category of cobordisms

      • cobordism hypothesis-theorem

      • extended topological quantum field theory

  • Tools

    • perturbative quantum field theory, vacuum

    • effective quantum field theory

    • renormalization

    • BV-BRST formalism

    • geometric ∞-function theory

  • particle physics

    • phenomenology

    • models

      • standard model of particle physics

      • fields and quanta

      • Grand Unified Theories, MSSM

    • scattering amplitude

      • on-shell recursion, KLT relations

  • Structural phenomena

    • universality class

    • quantum anomaly

      • Green-Schwarz mechanism

    • instanton

    • spontaneously broken symmetry

    • Kaluza-Klein mechanism

    • integrable systems

    • holonomic quantum fields

  • Types of quantum field thories

    • TQFT

      • 2d TQFT

      • Dijkgraaf-Witten theory

      • Chern-Simons theory

    • TCFT

      • A-model, B-model

      • homological mirror symmetry

    • QFT with defects

    • conformal field theory

    • (1,1)-dimensional Euclidean field theories and K-theory

    • (2,1)-dimensional Euclidean field theory and elliptic cohomology

    • CFT

      • WZW model

      • 6d (2,0)-supersymmetric QFT

    • gauge theory

      • field strength

      • gauge group, gauge transformation, gauge fixing

      • examples

        • electromagnetic field, QED

      • electric charge

      • magnetic charge

        • Yang-Mills field, QCD

      • Yang-Mills theory

      • spinors in Yang-Mills theory

      • topological Yang-Mills theory

        • Kalb-Ramond field
        • supergravity C-field
        • RR field
        • first-order formulation of gravity

      • general covariance

      • supergravity

      • D'Auria-Fre formulation of supergravity

      • gravity as a BF-theory

    • sigma-model

      • particle, relativistic particle, fundamental particle, spinning particle, superparticle

      • string, spinning string, superstring

      • membrane

      • AKSZ theory

• String Theory

  • string theory results applied elsewhere

• number theory and physics

  • Riemann hypothesis and physics

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category: meta