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After two papers on weak cubical categories and {\it collarable} cospans,respectively, we put things together and construct a {\it weak} cubicalcategory of cubical {\it collared} cospans of topological spaces. We also builda second structure, called a {\it quasi} cubical category, formed of arbitrarycubical cospans concatenated by homotopy pushouts. This structure, simpler butweaker, has {\it lax} identities. It contains a similar framework forcobordisms of manifolds with corners and could therefore be the basis to extendthe study of TQFT's of Part II to higher cubical degree.

Cubical cospans and higher cobordisms (Cospans in algebraic topology, III)