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-∞ algebra

  • E-∞ ring

• L-∞ algebra

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

(∞,1)-category theory

Background

• category theory

• higher category theory

• (n,r)-category

Basic concepts

• (∞,1)-category

• hom-objects

• equivalences in/of $(\mathrm{\infty },1)$-categories

• sub-(∞,1)-category

  • reflective sub-(∞,1)-category

  • reflective localization

• opposite (∞,1)-category

• over (∞,1)-category

  • join of quasi-categories

• (∞,1)-functor

  • exact (∞,1)-functor

  • (∞,1)-category of (∞,1)-functors

  • (∞,1)-category of (∞,1)-presheaves

• fibrations

  • inner fibration

  • left/right fibration

  • Cartesian fibration

    • Cartesian morphism

Universal constructions

• limit

  • terminal object

• adjoint functors

Local presentation

• locally presentable

  • essentially small

  • locally small

  • accessible

  • idempotent-complete

Theorems

• (∞,1)-Yoneda lemma

• (∞,1)-Grothendieck construction

• adjoint (∞,1)-functor theorem

• (∞,1)-monadicity theorem

Extra stuff, structure, properties

• stable (∞,1)-category

• (∞,1)-topos

Models

• category with weak equivalences

• model category

• derivator

• quasi-category

  • model structure for quasi-categories

  • model structure for Cartesian fibrations

• relation to simplicial categories

  • homotopy coherent nerve

  • simplicial model category

  • presentable quasi-category

• Kan complex

  • model structure for Kan complexes