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.