Abstract
I discuss the axiomatic framework of (tree-level) associative open string field theory in the presence of D-branes by considering the natural extension of the case of a single boundary sector. This leads to a formulation which is intimately connected with the mathematical theory of differential graded categories. I point out that a generic string field theory as formulated within this framework is not closed under formation of D-brane composites and as such does not allow for a unitary description of D-brane dynamics. This implies that the collection of boundary sectors of a generic string field theory with D-branes must be extended by inclusion of all possible D-brane composites. I give a precise formulation of a weak unitarity constraint and show that a minimal extension which is unitary in this sense can always be obtained by promoting the original D-brane category to an enlarged category constructed by using certain generalized complexes of D-branes. I give a detailed construction of this extension and prove its closure under formation of D-brane composites. These results amount to a completely general description of D-brane composite formation within the framework of associative string field theory.