symmetric monoidal (∞,1)-category of spectra
A simplicial $C^\infty$-ring is a simplicial object in the category of C∞-rings.
When equipped with the model structure on simplicial algebras over the Lawvere theory CartSp, simplicial $C^\infty$-rings are a model for smooth (∞,1)-algebras, hence for $(\infty,1)$-algebras over CartSp regarded as an (∞,1)-algebraic theory.
Therefore, in higher analogy to how $C^\infty$-rings serve as function algebras on smooth loci in differential geometry, so simplicial $C^\infty$-rings serve as function rings on derived smooth manifolds and more general spaces in derived differential geometry.