  We study the combinatorial, algebraic and geometric properties of the free
product operation on matroids. After giving cryptomorphic definitions of free
product in terms of independent sets, bases, circuits, closure, flats and rank
function, we show that free product, which is a noncommutative operation, is
associative and respects matroid duality. The free product of matroids $M$ and
$N$ is maximal with respect to the weak order among matroids having $M$ as a
submatroid, with complementary contraction equal to $N$. Any minor of the free
product of $M$ and $N$ is a free product of a repeated truncation of the
corresponding minor of $M$ with a repeated Higgs lift of the corresponding
minor of $N$. We characterize, in terms of their cyclic flats, matroids that
are irreducible with respect to free product, and prove that the factorization
of a matroid into a free product of irreducibles is unique up to isomorphism.
We use these results to determine, for K a field of characteristic zero, the
structure of the minor coalgebra $\cal C$ of a family of matroids $\cal M$ that
is closed under formation of minors and free products: namely, $\cal C$ is
cofree, cogenerated by the set of irreducible matroids belonging to $\cal M$.
