nLab blob homology

Idea

The blob complex may be 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 $C$ with strong duality? – a blob n-category – (e.g. pivotal 2-category?) and an $n$-manifold $M$.

• Define $B_0(M)$ to be finite linear combinations of “$C$-string diagrams” drawn on $M$.

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

• Define $B_2(M)$ to be …

Second definition

…

Examples

…

Related concepts

• blob n-category

Blob homology has some similarities with

• factorization algebra

• local nets in AQFT.

It should be closely related to

• topological chiral homology/factorization homology

References

A preprint is here:

• Scott Morrison, Kevin Walker, The blob complex, (arxiv/1009.5025) (website)

Notes from talks can be found here and here.

Some notes appeared in the context of the

• Oberwolfach Workshop, June 2009 – Strings, Fields, Topology

See the

• Oberwolfach report No 28, 2009, pdf