nLab Higher Algebra


This entry collects links related to the book

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.


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

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 𝒟 (𝒜)\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 𝒜𝓈𝓈 \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 ℒℳ \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 ℬℳ \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 𝔼 k\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 nn-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 𝔼 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


The book is based on the series of articles

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.