nLab
Higher Algebra

This entry is about the book

• Jacob Lurie, Higher Algebra (pdf)

on higher algebra.

This book lays the basis of categorical algebra – the study of the trinity algebra, monads, operads – in the context of (∞,1)-category theory: (∞,1)-algebras, (∞,1)-monads, (∞,1)-operads.

The following is (or will eventually be) a linked list of keywords.

1. Stable $\mathrm{\infty }$-categories

• stable (∞,1)-category

  • stabilization

  • triangulated category

2. $\mathrm{\infty }$-Operads

2.1 Foundations

2.1.1 From colored operads to $\mathrm{\infty }$-operads

• Segal Gamma-category

• inert morphism

• category of operators

• (∞,1)-operad

2.1.2 Maps of $\mathrm{\infty }$-operads

• morphism of (∞,1)-operads

• fibration of (∞,1)-operads

2.1.3 Algebra objects

• algebra over an (∞,1)-operad

• $𝒪$-monoidal (∞,1)-category

2.1.4 $\mathrm{\infty }$-Preoperads

2.2 Constructions of $\mathrm{\infty }$-Operads

2.3 Disintegration and assembly

2.3.1 Disintegration and assembly

2.3.2 Generalized $\mathrm{\infty }$-operads

• generalized (∞,1)-operad

• family of (∞,1)-operads

2.3.3 Approximations to $\mathrm{\infty }$-operads

2.3.4 Disintegration of $\mathrm{\infty }$-operads

2.4 Products and coproducts

2.4.1 Cartesian symmetric monoidal structure

• cartesian monoidal (∞,1)-category

2.4.2 Monoid objects

• monoid

• Gamma space

2.4.3 CoCartesian Symmetric Monoidal Structure

2.4.4 Wreath Products

3. Algebras and modules over $\mathrm{\infty }$-operads

3.1 Free algebras

3.2 Limits and colimits of algebras

3.3 Modules over $\mathrm{\infty }$-operads

• module over an algebra over an (∞,1)-operad

3.3.1 Coherent $\mathrm{\infty }$-operads

• coherent (∞,1)-operad

3.4 General features of module $\mathrm{\infty }$-categories

4. Associative algebras and their modules

4.1 Associative algebras

4.1.1 The $\mathrm{\infty }$-Operad ${\mathrm{𝒜𝓈𝓈}}^{\otimes }$

• associative operad

• A-∞ operad

• planar (∞,1)-operad

4.1.2 Simplicial models for associative algebras

• simplicial algebra

• model structure on simplicial T-algebras

4.1.3 Monoidal Model Categories

• monoidal model category

4.1.4 Rectification of Associative Algebras

• model structure on algebras over an operad

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

4.2 Left and Right Modules

4.2.1 The $\mathrm{\infty }$-Operad ${\mathrm{ℒℳ}}^{\otimes }$

• (∞,1)-operad for (∞,1)-modules over an A-∞ algebra

• (∞,1)-module

4.2.2 Simplicial models for algebras and modules

4.2.3 Limits and colimits of algebras

4.2.4 Free modules

4.2.5 Duality in monoidal $\mathrm{\infty }$-categories

4.3 Bimodules

4.3.1 The $\mathrm{\infty }$-Operad ${\mathrm{ℬℳ}}^{\otimes }$

• operad for bimodules over algebras

• ∞-bimodule

• bitensoring

4.3.2 Simplicial models for algebras and modules

• model structure on modules over an algebra over an operad

4.3.3 Limits, Colimits, and Free Bimodules

(…)

4.3.4 Multilinear maps

• multilinear map

• bilinear map in an (∞,1)-category

4.3.5 Tensor Products and the Bar Construction

• tensor product of ∞-modules

• bar construction

4.3.6 Associtivity of the Tensor Product

• associativity

4.3.7 Duality of Bimodules

(…)

4.4 Modules over commutative algebras

5. Little cubes and factorizable sheaves

• little cubes operad

• topological chiral homology

6. Algebraic structures on $\mathrm{\infty }$-categories

• (∞,1)-monad

• symmetric monoidal (∞,1)-category of presentable (∞,1)-categories

7. Algebra in stable homotopy theory

• stable homotopy category

• stable homotopy theory

7.1 Structured ring spectra

• ring spectrum

• module spectrum

• algebra spectrum

• stable monoidal Dold-Kan correspondence

7.2 Properties of rings and modules

7.3 The cotangent complex formalism

• tangent (∞,1)-category

  • cotangent complex

  • tangent complex

7.4 Deformation theory

• deformation theory

7.5 Étale morphisms

• etale morphism

A Constructible sheaves and exit paths

• higher homotopy van Kampen theorem

B Categorical patterns

References

The book is based on the series of articles

• Jacob Lurie,

  Stable ∞-Categories (arxiv)

  Noncommutative Algebra (arXiv)

  Commutative Algebra (arXiv)

• Deformation Theory (arXiv)

• $𝔼[k]$-Algebras (pdf)

category: reference