perfect module in nLab

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

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