nLab
smooth (infinity,1)-algebra

Idea

The notion of smooth $(\mathrm{\infty },1)$-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.

Definition

A smooth $\mathrm{\infty }$-algebra is an ∞-algebra over an (∞,1)-algebraic theory $T$ for $T$ the ordinary Lawvere theory of smooth algebras.

Properties

The model structure on simplicial algebras on simplicial C-∞-ring is a presentation for smooth $(\mathrm{\infty },1)$-algebras.

Applications

Smooth $(\mathrm{\infty },1)$-algebras appear as the algebras of functions in derived differential geometry, for instance onderived smooth manifolds.