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
associated bundle, associated ∞-bundle
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
model structure on simplicial T-algebras / homotopy T-algebra
model structure on operads
model structure on algebras over an operad
monoidal Dold-Kan correspondence
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stable homotopy theory
loop space object
stable (∞,1)-category of spectra
stable homotopy category
symmetric smash product of spectra
An E ∞-ring is a commutative monoid in the stable (∞,1)-category of spectra. Sometimes this is called a commutative ring spectrum. An E-∞ algebra in spectra.
This means that an E ∞-ring is an A-∞ ring that is commutative up to coherent higher homotopies.
Equivalently E ∞-rings may be modeled as ordinary commutative monoids with respect to the symmetric monoidal smash product of spectra.
E ∞-rings are the analoge in higher algebra of the commutative rings in ordinary algebra.