nLab
simplicial C-infinity-ring

Definition

A simplicial $C^{\mathrm{\infty }}$-ring is a simplicial object in the category of C∞-rings.

Applications

When equipped with the model structure on simplicial algebras over the Lawvere theory CartSp, simplicial $C^{\mathrm{\infty }}$-rings are a model for smooth (∞,1)-algebras, hence for $(\mathrm{\infty },1)$-algebras over CartSp regarded as an (∞,1)-algebraic theory.

Therefore, in higher analogy to how $C^{\mathrm{\infty }}$-rings serve as funcion algebras on smooth loci in differential geometry, so simplicial $C^{\mathrm{\infty }}$-rings serve as function rings on derived smooth manifolds and more general spaces in derived differential geometry.

References

• David Spivak, Derived smooth manifolds (arXiv:0810.5174)

• Dennis Borisov, Justin Noel, Simplicial approach to derived differential manifolds (arXiv:1112.0033)