  Tangent categories provide an axiomatic framework for understanding various
tangent bundles and differential operations that occur in differential
geometry, algebraic geometry, abstract homotopy theory, and computer science.
Previous work has shown that one can formulate and prove a wide variety of
definitions and results from differential geometry in an arbitrary tangent
category, including generalizations of vector fields and their Lie bracket,
vector bundles, and connections.
  In this paper we investigate differential and sector forms in tangent
categories. We show that sector forms in any tangent category have a rich
structure: they form a symmetric cosimplicial object. This appears to be a new
result in differential geometry, even for smooth manifolds. In the category of
smooth manifolds, the resulting complex of sector forms has a subcomplex
isomorphic to the de Rham complex of differential forms, which may be
identified with alternating sector forms. Further, the symmetric cosimplicial
structure on sector forms arises naturally through a new equational
presentation of symmetric cosimplicial objects, which we develop herein.
