nLab Cospans in Algebraic Topology

A series of articles by Marco Grandis on describing extended cobordisms in terms of multi-cospans in Top

• I, Higher cospans and weak cubical categories (pdf)

• II, Collared cospans, cohomotopy and TQFT (pdf)

• III: Cubical cospans and higher cobordisms (arXiv)

Some of the crucial ideas appearing here are

• a notion of cubical multi-cospans in some category;

• the observation that extended cobordisms should be modeled by such multi-cospans (in particular in Top);

• the observation that such cospans naturally map into spans by homming them into target objects, and that such spans can be linearized such as to yield FQFTs.

  • (Grandis seems not to mention (?) the fact that groupoidification makes this interpretation of spans as linear maps very natural)

• the relevance of evaluating functors which preserve (homotopy) pushouts on (multi)-cospans.

category: reference