There is actually a reasonable amount of literature on the subject.

Check out Khovanov's $sl_3$-homology and the subsequent work by Marco Mackaay and Pedro Vaz, Hao Wu, and Scott Morrison. The objects are webs, the morphisms are foams. The webs have trivalent vertices and foams are modeled on the dual complex of a tetrahedron. I also liked a paper by Serge Natanzon on Network TQFT. You an find all these papers on the Arxiv.

Early on, Frank Quinn made some forays into TQFT for $2$-complexes with the idea of detecting counterexamples to the Andrews Curtis conjecture.

There are older papers of John Baez, John Barret, and then lots of papers in the physical literature. The Landau-Ginzberg approach to quantum gravity involves webs and foams.

All of this work can be understood as being about $3+1$-dimensional TQFT.

In the $1+1$ case, TQFT's turned out to be Frobenius algebras in disguise. In the $2+1$ case, TQFT's have to do with spherical categories. The corresponding correspondence for $3+1$ has yet to be understood. I would conjecture that on the way to understanding $3+1$-dimensional TQFT, a suitable and uniformly accepted theory of TQFT based on webs and foams will be established.

There is actually a reasonable amount of literature on the subject.

Check out Khovanov's $sl_3$-homology and the subsequent work by Marco Mackaay and Pedro Vaz * * *, Hao Wu, and Scott Morrison and Ari Nieh. The objects are webs, the morphisms are foams. The webs have trivalent vertices and foams are modeled on the dual complex of a tetrahedron. I also liked a paper by Serge Natanzon on Network TQFT. You an find all these papers on the Arxiv.

Early on, Frank Quinn made some forays into TQFT for $2$-complexes with the idea of detecting counterexamples to the Andrews Curtis conjecture.

There are older papers of John Baez, John Barret, and then lots of papers in the physical literature. The Landau-Ginzberg approach to quantum gravity involves webs and foams.

All of this work can be understood as being about $3+1$-dimensional TQFT.

In the $1+1$ case, TQFT's turned out to be Frobenius algebras in disguise. In the $2+1$ case, TQFT's have to do with spherical categories. The corresponding correspondence for $3+1$ has yet to be understood. I would conjecture that on the way to understanding $3+1$-dimensional TQFT, a suitable and uniformly accepted theory of TQFT based on webs and foams will be established.

There is actually a reasonable amount of literature on the subject.

Check out Khovanov's $sl_3$-homology and the subsequent work by Marco Mackaay and Pedro Vaz, Hao Wu, and Scott Morrison. The objects are webs, the morphisms are foams. The webs have trivalent vertices and foams are modeled on the dual complex of a tetrahedron. I also liked a paper by Serge Natanzon on Web TQFT. You an find all these papers on the Arxiv.

There are older papers of John Baez, John Barret, and then lots of papers in the physical literature.

There is actually a reasonable amount of literature on the subject.

Check out Khovanov's $sl_3$-homology and the subsequent work by Marco Mackaay and Pedro Vaz, Hao Wu, and Scott Morrison. The objects are webs, the morphisms are foams. The webs have trivalent vertices and foams are modeled on the dual complex of a tetrahedron. I also liked a paper by Serge Natanzon on Network TQFT. You an find all these papers on the Arxiv.

Early on, Frank Quinn made some forays into TQFT for $2$-complexes with the idea of detecting counterexamples to the Andrews Curtis conjecture.

There are older papers of John Baez, John Barret, and then lots of papers in the physical literature. The Landau-Ginzberg approach to quantum gravity involves webs and foams.

All of this work can be understood as being about $3+1$-dimensional TQFT.

In the $1+1$ case, TQFT's turned out to be Frobenius algebras in disguise. In the $2+1$ case, TQFT's have to do with spherical categories. The corresponding correspondence for $3+1$ has yet to be understood. I would conjecture that on the way to understanding $3+1$-dimensional TQFT, a suitable and uniformly accepted theory of TQFT based on webs and foams will be established.