  Contrary to previous approaches bringing together algebraic geometry and
signatures of paths, we introduce a Zariski topology on the space of paths
itself, and study path varieties consisting of all paths whose
iterated-integrals signature satisfies certain polynomial equations. Particular
emphasis lies on the role of the non-associative halfshuffle, which makes it
possible to describe varieties of paths that satisfy certain relations all
along their trajectory. Specifically, we may understand the set of paths on a
given classical algebraic variety in $\mathbb{R}^d$ starting from a fixed point
as a path variety, e.g. paths on a sphere. While the characteristic geometric
property of halfshuffle varieties is that they are stable under stopping paths
at an earlier time, we furthermore study varieties which are stable under the
natural (semi)group operation of concantenation of paths. We point out how the
notion of dimension for path varieties crucially depends on the fact that they
may be reducible into countably infinitely many subvarieties. Finally, we see
that studying halfshuffle varieties naturally leads to a generalization of
classical algebraic curves, surfaces and affine varieties in finite dimensional
space. These generalized algebraic sets, an example being the graph of the
exponential function, are now described through iterated-integral equations.
  Keywords: path variety; shuffle ideal; halfshuffle; deconcatenation
coproduct; tensor algebra; Zariski topology; concatenation of paths; Chen's
identity; subpath; tree-like equivalence; regular map; generalized variety
