Topological cospans and their concatenation, by pushout, appear in the theories of tangles, ribbons, cobordisms, etc. Various algebraic invariants have been introduced for their study, which it would be interesting to link with the standard tools of Algebraic Topology, (co)homotopy and (co)homology functors.
Here we introduce collarable (and collared) cospans between topological spaces. They generalise the cospans which appear in the previous theories, as a consequence of a classical theorem on manifolds with boundary. Their interest lies in the fact that their concatenation is realised by means of homotopy pushouts. Therefore, cohomotopy functors induce `functors' from collarable cospans to spans of sets, providing - by linearisation - topological quantum field theories (TQFT) on manifolds and their cobordisms. Similarly, (co)homology and homotopy functors take collarable cospans to relations of abelian groups or (co)spans of groups, yielding other `algebraic' invariants.
This is the second paper in a series devoted to the study of cospans in Algebraic Topology. It is practically independent from the first, which deals with higher cubical cospans in abstract categories. The third article will proceed from both, studying cubical topological cospans and their collared version.
Keywords: spans, cospans, bicategory, weak double category, homotopy pushout, cohomotopy functors, cobordism, knots, links, tangles, topological quantum field theories
2000 MSC: 18D05, 55Q55, 57N70, 57M27, 57R56
Theory and Applications of Categories,
Vol. 18, 2007,
No. 19, pp 602-630.