  We define a GL-variety to be a (typically infinite dimensional) algebraic
variety equipped with an action of the infinite general linear group under
which the coordinate ring forms a polynomial representation. Such varieties
have been used to study asymptotic properties of invariants like strength and
tensor rank, and played a key role in two recent proofs of Stillman's
conjecture. We initiate a systematic study of GL-varieties, and establish a
number of foundational results about them. For example, we prove a version of
Chevalley's theorem on constructible sets in this setting.
