  The inverse Higgs phenomenon, which plays an important r\^ole in physical
systems with Goldstone bosons (such as the phonons in a crystal) involves
nonholonomic mechanical constraints. By formulating field theories with
symmetries and constraints in a general way using the language of differential
geometry, we show that many examples of constraints in inverse Higgs phenomena
fall into a special class, which we call coholonomic constraints, that are dual
(in the sense of category theory) to holonomic constraints. Just as for
holonomic constraints, systems with coholonomic constraints are equivalent to
unconstrained systems (whose degrees of freedom are known as essential
Goldstone bosons), making it easier to study their consistency and dynamics.
The remaining examples of inverse Higgs phenomena in the literature require the
dual of a slight generalisation of a holonomic constraint, which we call
(co)meronomic. Our formalism simplifies and clarifies the many ad hoc
assumptions and constructions present in the literature. In particular, it
identifies which are necessary and which are merely convenient. It also opens
the way to studying much more general dynamical examples, including systems
which have no well-defined notion of a target space.
