nLab
Parametrized Homotopy Theory

Background

• homotopy theory

• algebraic topology, simplicial homotopy theory

• homotopy

• homotopy type

Variations

• stable homotopy theory

• proper homotopy theory

• directed homotopy theory

Definitions

• homotopy

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

• homotopy coherent category theory

  • homotopical category

    • model category

    • category of fibrant objects

    • 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

Theorems

• homotopy hypothesis-theorem

• Hurewicz theorem

• higher homotopy van Kampen theorem

• Galois theory

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

• Čech 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