  We introduce monoidal width as a measure of complexity for morphisms in
monoidal categories. Inspired by well-known structural width measures for
graphs, like tree width and rank width, monoidal width is based on a notion of
syntactic decomposition: a monoidal decomposition of a morphism is an
expression in the language of monoidal categories, where operations are
monoidal products and compositions, that specifies this morphism. Monoidal
width penalises the composition operation along ``big'' objects, while it
encourages the use of monoidal products. We show that, by choosing the correct
categorical algebra for decomposing graphs, we can capture tree width and rank
width. For matrices, monoidal width is related to the rank. These examples
suggest monoidal width as a good measure for structural complexity of processes
modelled as morphisms in monoidal categories.
