  In this work we investigate from a broad perspective the reduction of degrees
of freedom through serendipity techniques for polytopal methods compatible with
Hilbert complexes. We first establish an abstract framework that, given two
complexes connected by graded maps, identifies a set of properties enabling the
transfer of the homological and analytical properties from one complex to the
other. This abstract framework is designed having in mind discrete complexes,
with one of them being a reduced version of the other, such as occurring when
applying serendipity techniques to numerical methods. We then use this
framework as an overarching blueprint to design a serendipity DDR complex.
Thanks to the combined use of higher-order reconstructions and serendipity,
this complex compares favorably in terms of degrees of freedom (DOF) count to
all the other polytopal methods previously introduced and also to finite
elements on certain element geometries. The gain resulting from such a
reduction in the number of DOFs is numerically evaluated on two model problems:
a magnetostatic model, and the Stokes equations.
