Monoidal Width: Capturing Rank Width

Monoidal width was recently introduced by the authors as a measure of complexity decomposing morphisms in monoidal categories. We have shown that category cospans graphs, and its variants can be used to capture tree width, path branch width. In this paper we study matrices, an extension different open where connectivity information is handled with matrix algebra graphs are composed along edges instead vertices. show here captures rank width: graph has received much attention recent years.