The differential bundles of the geometric tangent category of an operad

Abstract: Affine schemes can be understood as objects of the opposite of the category of commutative and unital algebras. Similarly, $\mathscr{P}$-affine schemes can be defined as objects of the opposite of the category of algebras over an operad $\mathscr{P}$. An example is the opposite of the category of associative algebras. The category of operadic schemes of an operad carries a canonical tangent structure. This paper aims to initiate the study of the geometry of operadic affine schemes via this tangent category. For example, we expect the tangent structure over the opposite of the category of associative algebras to describe algebraic non-commutative geometry. In order to initiate such a program, the first step is to classify differential bundles, which are the analogs of vector bundles for differential geometry. In this paper, we prove that the tangent category of affine schemes of the enveloping operad $\mathscr{P}^{(A)}$ over a $\mathscr{P}$-affine scheme $A$ is precisely the slice tangent category over $A$ of $\mathscr{P}$-affine schemes. We are going to employ this result to show that differential bundles over a $\mathscr{P}$-affine scheme $A$ are precisely $A$-modules in the operadic sense.