  Pastures are a class of field-like algebraic objects which include both
partial fields hyperfields and have nice categorical properties. We prove
several lift theorems for representations of matroids over pastures, including
a generalization of Pendavingh and van Zwam's Lift Theorem for partial fields.
By embedding the earlier theory into a more general framework, we are able to
establish new results even in the case of lifts of partial fields, for example
the conjecture of Pendavingh--van Zwam that their lift construction is
idempotent. We give numerous applications to matroid representations, e.g. we
show that, up to projective equivalence, every pair consisting of a hexagonal
representation and an orientation lifts uniquely to a near-regular
representation. The proofs are different from the arguments used by Pendavingh
and van Zwam, relying instead on a result of Gelfand-Rybnikov-Stone inspired by
Tutte's homotopy theorem.
