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

• inductive type
• higher inductive type
• coinductive type

Rules

• induction, coinduction
• recursion, corecursion

Categorical semantics

• initial algebra of an endofunctor, terminal coalgebra of an endofunctor

• initial algebra of a presentable ∞-monad

Examples

• natural numbers object, list,

• identity type

• W-type