Editing Tensor field (section)

== Cocycles and chain rules ==

As an advanced explanation of the ''tensor'' concept, one can interpret the [[chain rule]] in the multivariable case, as applied to coordinate changes, also as the requirement for self-consistent concepts of tensor giving rise to tensor fields.

Abstractly, we can identify the chain rule as a 1-[[Cochain (algebraic topology)|cocycle]]. It gives the consistency required to define the tangent bundle in an intrinsic way. The other vector bundles of tensors have comparable cocycles, which come from applying [[functorial]] properties of tensor constructions to the chain rule itself; this is why they also are intrinsic (read, 'natural') concepts.

What is usually spoken of as the 'classical' approach to tensors tries to read this backwards – and is therefore a heuristic, ''post hoc'' approach rather than truly a foundational one. Implicit in defining tensors by how they transform under a coordinate change is the kind of self-consistency the cocycle expresses. The construction of tensor densities is a 'twisting' at the cocycle level. Geometers have not been in any doubt about the ''geometric'' nature of tensor ''quantities''; this kind of [[descent (category theory)|descent]] argument justifies abstractly the whole theory.