  We present a framework for studying the dynamics of equivariant vector fields
near relative equilibria. To overcome the lack of linearization at a relative
equilibrium or the possible non-smoothness of the orbit space, we categorify
the space of equivariant vector fields. A category where the objects are
equivariant vector fields was first introduced by Hepworth in the context of
smooth stacks [Hepworth, Theory Appl. Categ. 22 (2009), 542-587]. Central to
our approach is the ensuing notion of isomorphic equivariant vector fields. The
idea is that considering equivariant vector fields, and their corresponding
dynamics, up to isomorphism is a way to take into account the symmetries of the
group action without passing to the orbit space. In particular, the category of
equivariant vector fields near a relative equilibrium is equivalent to the
category of equivariant vector fields on the slice representation of the
relative equilibrium. In this paper we apply this to bifurcations to and from
relative equilibria, and to the genericity of conditions for equivariant
bifurcation from relative equilibria.
