higher algebra
universal algebra
algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory
monad / (∞,1)-monad
operad / (∞,1)-operad
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
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
C-∞ algebra
E-∞ ring, E-∞ algebra
∞-module, (∞,1)-module bundle
multiplicative cohomology theory
L-∞ algebra
model structure on simplicial T-algebras / homotopy T-algebra
model structure on operads
model structure on algebras over an operad
Isbell duality
derived geometry
Deligne conjecture
delooping hypothesis
monoidal Dold-Kan correspondence
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A $C_\infty$-algebra is a commutative ring where the commutativity is relaxed up to coherent higher homotopy.
E-∞ algebra
Warning: no direct relation to C-∞ ring (unfortunately a slight terminology clash).