  We consider three forms of composition of matroids, each of which extends the
category of bimatroids to a rigid monoidal category. Many well-known
constructions are functorial or defined by morphisms in these categories.
Motivating examples include: deletion and contraction, 2-sum, series and
parallel connections, the Tutte polynomial, gammoids, positroids, matroids
representable over an infinite field, M-convex sets, and matroids associated to
stable polynomials.
