  We introduce monoidal width as a measure of the difficulty of decomposing
morphisms in monoidal categories. For graphs, we show that monoidal width and
two variations capture existing notions, namely branch width, tree width and
path width. We propose that monoidal width: (i) is a promising concept that,
while capturing known measures, can similarly be instantiated in other
settings, avoiding the need for ad-hoc domain-specific definitions and (ii)
comes with a general, formal algebraic notion of decomposition using the
language of monoidal categories.
