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
E-∞ ring, E-∞ algebra
∞-module, (∞,1)-module bundle
multiplicative cohomology theory
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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The notion of smooth (∞,1)(\infty,1)-algebra is the analog in higher category theory of smooth algebra. This is the basis for the derived geometry version of differential geometry/synthetic differential geometry.
A smooth ∞\infty-algebra is an ∞-algebra over an (∞,1)-algebraic theory TT for TT the ordinary Lawvere theory of smooth algebras.
The model structure on simplicial algebras on simplicial C-∞-ring is a presentation for smooth (∞,1)(\infty,1)-algebras.
Smooth (∞,1)(\infty,1)-algebras appear as the algebras of functions in derived differential geometry, for instance on derived smooth manifolds.