Cubical Cospans and Higher Cobordisms (cospans in Algebraic Topology, Iii)

After two papers on weak cubical categories and collarable cospans, respectively, we put things together and construct a weak cubical category of cubical collared cospans of topological spaces. We also build a second structure, called a quasi cubical category, formed of arbitrary cubical cospans concatenated by homotopy pushouts. This structure, simpler but weaker, has lax identities. It contains a similar framework for cobordisms of manifolds with corners and could therefore be the basis to extend the study of TQFT’s of Part II to higher cubical degree. Introduction This is a sequel to two papers, cited as Part I [6] and Part II [7]. A reference I.2, or I.2.3, relates to Section 2 or Subsection 2.3 of Part I. Similarly for Part II. In Part I we constructed a cubical structure of higher cospans Cosp∗(X) on a category X with pushouts, and abstracted from the construction the general notion of a weak cubical category. An n-cubical cospan in X is defined as a functor u : ∧n → X, where ∧ is the category ∧ : −1 → 0 ← 1 (the formal cospan). (1) These diagrams form a cubical set, equipped with compositions u +i v of iconsecutive n-cubes, for i = 1, ..., n. Such cubical compositions are computed by pushouts, and behave ‘categorically’ in a weak sense, up to suitable comparisons. To make room for the latter, the n-th component of Cosp∗(X) Cospn(X) = Cat(∧n,X), (2) is not just the set of functors u : ∧n → X (the n-cubes of the structure), but the category of such functors and their natural transformations f : u → u′ : ∧n → X (the n-maps of the structure). The comparisons are invertible n-maps; but general nmaps are also important, e.g. to define limits and colimits (I.4.6, II.1.3). Thus, a weak Work supported by a research grant of Università di Genova. Received December 4, 2007, revised May 8, 2008; published on July 20, 2008. 2000 Mathematics Subject Classification: 18D05, 55U10, 55P05, 57N70.