  Connection matrices are one of the central tools in Conley's approach to the
study of dynamical systems, as they provide information on the existence of
connecting orbits in Morse decompositions. They may be considered a
generalisation of the boundary operator in the Morse complex in Morse theory.
Their computability has recently been addressed by Harker, Mischaikow, and
Spendlove in the context of lattice filtered chain complexes. In the current
paper, we extend the recently introduced Conley theory for combinatorial vector
and multivector fields on Lefschetz complexes by transferring the concept of
connection matrix to this setting. This is accomplished by the notion of
connection matrix for arbitrary poset filtered chain complexes, as well as an
associated equivalence, which allows for changes in the underlying posets. We
show that for the special case of gradient combinatorial vector fields in the
sense of Forman, connection matrices are necessarily unique. Thus, the
classical results of Reineck have a natural analogue in the combinatorial
setting.
