  The purpose of this paper is to introduce a new family of semigroups - the
free projection-generated regular $*$-semigroups - and initiate their
systematic study. Such a semigroup $PG(P)$ is constructed from a projection
algebra $P$, using the recent groupoid approach to regular $*$-semigroups. The
assignment $P\mapsto PG(P)$ is a left adjoint to the forgetful functor that
maps a regular $*$-semigroup $S$ to its projection algebra $P(S)$. In fact, the
category of projection algebras is coreflective in the category of regular
$*$-semigroups. The algebra $P(S)$ uniquely determines the biordered structure
of the idempotents $E(S)$, up to isomorphism, and this leads to a category
equivalence between projection algebras and regular $*$-biordered sets. As a
consequence, $PG(P)$ can be viewed as a quotient of the classical free
idempotent-generated (regular) semigroups $IG(E)$ and $RIG(E)$, where
$E=E(PG(P))$; this is witnessed by a number of presentations in terms of
generators and defining relations. The semigroup $PG(P)$ can also be viewed as
the fundamental groupoid of a simplicial complex explicitly constructed from
$P$. The theory is then illustrated on a number of examples. In one direction,
the free construction applied to the projection algebras of adjacency
semigroups yields a new family of graph-based path semigroups. In another, it
turns out that, remarkably, the Temperley-Lieb monoid $TL_n$ is the free
regular $*$-semigroup over its own projection algebra $P(TL_n)$.
