  According to the algebraic approach to spacetime, a thoroughgoing dynamicism,
physical fields exist without an underlying manifold. This view is usually
implemented by postulating an algebraic structure (e.g., commutative ring) of
scalar-valued functions, which can be interpreted as representing a scalar
field, and deriving other structures from it. In this work, we point out that
this leads to the unjustified primacy of an undetermined scalar field. Instead,
we propose to consider algebraic structures in which all (and only) physical
fields are primitive. We explain how the theory of \emph{natural operations} in
differential geometry -- the modern formalism behind classifying
diffeomorphism-invariant constructions -- can be used to obtain concrete
implementations of this idea for any given collection of fields.
  For concrete examples, we illustrate how our approach applies to a number of
particular physical fields, including electrodynamics coupled to a Weyl spinor.
