nLab
blob homology

Contents

Idea

The blob complex maybe thought of as a generalization of the Hochschild complex to higher categories and higher dimensional manifolds.

One thinks of the Hochschild complex as associated to a 1-category and a 1-manifold (the circle). It’s a fairly small complex, analogous to cellular homology. The blob complex for the same input data (1-category and circle) yields a quasi-isomorphic but much much larger chain complex, analogous to singular homology.

Its advantage over the Hochschild complex is that it is “local”. In higher dimensions this locality means that it is easy to (well-) define the blob complex of an n-category + n-manifold without choosing any sort of decomposition of the n-manifold.

Definition

There are two definitions. The second is more general, and is homotopy equivalent to the first when both are available. See references below for more details.

First definition

  • Start with a linear n-category CC with strong duality? – a blob n-category – (e.g. pivotal 2-category?) and an nn-manifold MM.

  • Define B 0(M)B_0(M) to be finite linear combinations of ”CC-string diagrams” drawn on MM.

  • Define B 1(M)B_1(M) to be finite linear combinations of triples (B,u,r)(B, u, r), where BMB \subset M is a ball, rr is a string diagram on MBM\setminus B, and uu is a linear combination of string diagrams on which evaluates to zero.

  • Define B 2(M)B_2(M) to be

Second definition

Examples

Blob homology has some similarities with

It should be closely related to

References

A preprint is here:

Notes from talks can be found here and here.

Some notes appeared in the context of the

See the

  • Oberwolfach report No 28, 2009, pdf

Revised on May 30, 2012 21:11:15 by phx? (77.8.218.147)