Cartan calculus for C∞-ringed spaces

In an earlier paper [E. Lerman, Differential forms on [Formula: see text]-ringed spaces, J. Geom. Phys. (2023) 105062, doi:10.48550/arXiv.2212.11163, doi:10.1016/j.geomphys.2023.105062.], we constructed a complex of differential forms on a local [Formula: see text]-ringed space. In this paper, we define a sheaf of vector fields (“the tangent sheaf”) on a local [Formula: see text]-ringed space, define contractions of vector fields and forms, Lie derivatives of forms with respect to vector fields, and show that the standard equations of Cartan calculus hold for vector fields and differential forms on local [Formula: see text]-ringed spaces.