nLab Higher Algebra

Contents

Context

Higher algebra

This entry collects links related to the book

Jacob Lurie:

Higher Algebra

( $\sim$ 2017)

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.

In particular it discusses the analogs of abelian groups and commutative rings, namely abelian ∞-groups, spectra and E-∞ rings.

Contents

1. Stable $\infty$ -categories

1.1 Foundations

1.1.1 Stability
1.1.2 The homotopy category of a stable $\infty$ -category
1.1.3 Closure properties of stable $\infty$ -categories
1.1.4 Exact functors

1.2 Stable $\infty$ -categories and homological algebra

1.2.1 t-Structures on stable $\infty$ -categories
1.2.2 Filtered objects and spectral sequences
1.2.3 The Dold-Kan correspondence
1.2.4 The $\infty$ -categorical Dold-Kan correspondence

1.3 Homological algebra and derived categories

1.3.1 Nerves of differential graded categories
1.3.2 Derived $\infty$ -categories
1.3.3 The universal property of $\mathcal{D}^-(\mathcal{A})$
1.3.4 Inverting Quasi-isomorphisms
1.3.5 Grothendieck abelian categories
1.3.6 Complexes of injective objects

1.4 Spectra and stabilization

1.4.1 The Brown representability theorem
1.4.2 Spectrum objects
1.4.3 The $\infty$ -category of spectra
1.4.4 Presentable stable $\infty$ -categories

2. $\infty$ -Operads

2.1 Foundations

2.1.1 From colored operads to $\infty$ -operads
2.1.2 Maps of $\infty$ -operads
2.1.3 Algebra objects
2.1.4 $\infty$ -Preoperads

2.2 Constructions of $\infty$ -Operads
2.3 Disintegration and assembly

2.3.1 Disintegration and assembly
2.3.2 Generalized $\infty$ -operads
2.3.3 Approximations to $\infty$ -operads
2.3.4 Disintegration of $\infty$ -operads

2.4 Products and coproducts

2.4.1 Cartesian symmetric monoidal structure
2.4.2 Monoid objects
2.4.3 CoCartesian Symmetric Monoidal Structure
2.4.4 Wreath Products

3. Algebras and modules over $\infty$ -operads

3.1 Free algebras
3.2 Limits and colimits of algebras
3.3 Modules over $\infty$ -operads

3.3.1 Coherent $\infty$ -operads

3.4 General features of module $\infty$ -categories

4. Associative algebras and their modules

4.1 Associative algebras

4.1.1 The $\infty$ -Operad $\mathcal{Ass}^\otimes$
4.1.2 Simplicial models for associative algebras
4.1.3 Monoidal Model Categories
4.1.4 Rectification of Associative Algebras

4.2 Left and Right Modules

4.2.1 The $\infty$ -Operad $\mathcal{L M}^{\otimes}$
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 $\infty$ -categories

4.3 Bimodules

4.3.1 The $\infty$ -Operad $\mathcal{BM}^{\otimes}$
4.3.2 Simplicial models for algebras and modules
4.3.3 Limits, Colimits, and Free Bimodules
4.3.4 Multilinear maps
4.3.5 Tensor Products and the Bar Construction
4.3.6 Associtivity of the Tensor Product
4.3.7 Duality of Bimodules

4.4 Modules over commutative algebras

5. Little cubes and factorizable sheaves

5.1 Definitions and basic properties
5.2 Little cubes and manifold topology

5.2.1 Embeddings of topological manifolds
5.2.2 Variations on the little cubes operad
5.2.3 Nonunital $\mathbb{E}_k$ -Algebras
5.2.4 Little cubes in a manifold

5.3 Topological chiral homology

6. Algebraic structures on $\infty$ -categories

6.1 Endomorphism objects
6.2 Monads and Barr-Beck theorem

6.2.1 Split simplicial objects
6.2.2 The Barr-Beck theorem
6.2.3 BiCartesian Fibrations
6.2.4 Descent and the Beck-Chevalley condition

7. The calculus of functors

7.1 The calculus of functors

7.1.1 $n$ -Excisive Functors
7.1.2 Taylor Tower
7.1.6 Norm maps

8. Algebra in stable homotopy theory

8.1 Structured ring spectra

8.1.1 $\mathbb{E}_1$ -rings and their modules
8.1.2 Recognition principle
8.1.3 Change of ring
8.1.4 Algebras over Commutative Rings

8.2 Properties of rings and modules

8.2.1 Free resolutions and Spectral Sequences
8.2.2 Flat and projective modules
8.2.3 Injective objects of stable $\infty$ -categories
8.2.4 Localization and Ore conditions
8.2.5 Finiteness properties of rings and modules

8.3 The cotangent complex formalism
8.4 Deformation theory
8.5 Étale morphisms

A Constructible sheaves and exit paths
B Categorical patterns
References

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

1. Stable $\infty$ -categories

1.1 Foundations

1.1.1 Stability

stable (∞,1)-category
- stabilization

1.1.2 The homotopy category of a stable $\infty$ -category

triangulated category

1.1.3 Closure properties of stable $\infty$ -categories

1.1.4 Exact functors

exact (∞,1)-functor

1.2 Stable $\infty$ -categories and homological algebra

homological algebra

1.2.1 t-Structures on stable $\infty$ -categories

1.2.2 Filtered objects and spectral sequences

1.2.3 The Dold-Kan correspondence

Dold-Kan correspondence

1.2.4 The $\infty$ -categorical Dold-Kan correspondence

1.3 Homological algebra and derived categories

1.3.1 Nerves of differential graded categories

dg-nerve

1.3.2 Derived $\infty$ -categories

derived category

1.3.3 The universal property of $\mathcal{D}^-(\mathcal{A})$

1.3.4 Inverting Quasi-isomorphisms

1.3.5 Grothendieck abelian categories

Grothendieck category

1.3.6 Complexes of injective objects

1.4 Spectra and stabilization

1.4.1 The Brown representability theorem

Brown representability theorem

1.4.2 Spectrum objects

1.4.3 The $\infty$ -category of spectra

(∞,1)-category of spectra

1.4.4 Presentable stable $\infty$ -categories

2. $\infty$ -Operads

2.1 Foundations

2.1.1 From colored operads to $\infty$ -operads

2.1.2 Maps of $\infty$ -operads

2.1.3 Algebra objects

algebra over an (∞,1)-operad
$\mathcal{O}$ -monoidal (∞,1)-category

2.1.4 $\infty$ -Preoperads

2.2 Constructions of $\infty$ -Operads

symmetric monoidal envelope

2.3 Disintegration and assembly

2.3.1 Disintegration and assembly

2.3.2 Generalized $\infty$ -operads

2.3.3 Approximations to $\infty$ -operads

2.3.4 Disintegration of $\infty$ -operads

2.4 Products and coproducts

2.4.1 Cartesian symmetric monoidal structure

cartesian monoidal (∞,1)-category

2.4.2 Monoid objects

2.4.3 CoCartesian Symmetric Monoidal Structure

cocartesian monoidal (∞,1)-category

2.4.4 Wreath Products

3. Algebras and modules over $\infty$ -operads

3.1 Free algebras

3.2 Limits and colimits of algebras

3.3 Modules over $\infty$ -operads

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

3.3.1 Coherent $\infty$ -operads

coherent (∞,1)-operad

3.4 General features of module $\infty$ -categories

4. Associative algebras and their modules

4.1 Associative algebras

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

4.1.2 Simplicial models for associative algebras

4.1.3 Monoidal Model Categories

monoidal model category

4.1.4 Rectification of Associative Algebras

4.2 Left and Right Modules

4.2.1 The $\infty$ -Operad $\mathcal{L M}^{\otimes}$

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 $\infty$ -categories

4.3 Bimodules

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

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

4.3.5 Tensor Products and the 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

5.1 Definitions and basic properties

5.2 Little cubes and manifold topology

5.2.1 Embeddings of topological manifolds

5.2.2 Variations on the little cubes operad

5.2.3 Nonunital $\mathbb{E}_k$ -Algebras

5.2.4 Little cubes in a manifold

5.3 Topological chiral homology

topological chiral homology

6. Algebraic structures on $\infty$ -categories

6.1 Endomorphism objects

6.2 Monads and Barr-Beck theorem

6.2.1 Split simplicial objects

6.2.2 The Barr-Beck theorem

Barr-Beck theorem

6.2.3 BiCartesian Fibrations

6.2.4 Descent and the Beck-Chevalley condition

higher monadic descent

7. The calculus of functors

Goodwillie calculus

7.1 The calculus of functors

7.1.1 $n$ -Excisive Functors

7.1.2 Taylor Tower

Goodwillie-Taylor tower

7.1.6 Norm maps

8. Algebra in stable homotopy theory

8.1 Structured ring spectra

8.1.1 $\mathbb{E}_1$ -rings and their modules

8.1.2 Recognition principle

8.1.3 Change of ring

connective cover

8.1.4 Algebras over Commutative Rings

8.2 Properties of rings and modules

8.2.1 Free resolutions and Spectral Sequences

8.2.2 Flat and projective modules

8.2.3 Injective objects of stable $\infty$ -categories

8.2.4 Localization and Ore conditions

8.2.5 Finiteness properties of rings and modules

8.3 The cotangent complex formalism

tangent (∞,1)-category
- cotangent complex
- tangent complex

8.4 Deformation theory

deformation theory

8.5 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)
$\mathbb{E}[k]$ -Algebras (pdf)

category: reference

Last revised on May 4, 2024 at 20:13:00. See the history of this page for a list of all contributions to it.