  When studying quantum field theories and lattice models, it is often useful
to analytically continue the number of field or spin components from an integer
to a real number. In spite of this, the precise meaning of such analytic
continuations has never been fully clarified, and in particular the symmetry of
these theories is obscure. We clarify these issues using Deligne categories and
their associated Brauer algebras, and show that these provide logically
satisfactory answers to these questions. Simple objects of the Deligne category
generalize the notion of an irreducible representations, avoiding the need for
such mathematically nonsensical notions as vector spaces of non-integer
dimension. We develop a systematic theory of categorical symmetries, applying
it in both perturbative and non-perturbative contexts. A partial list of our
results is: categorical symmetries are preserved under RG flows; continuous
categorical symmetries come equipped with conserved currents; CFTs with
categorical symmetries are necessarily non-unitary.
