Vertex algebras are intuitively analogues of associative algebras, except that certain singularities come up (in physics terminology thsi comes from OPE’s).

Relaxed multicategories can be thought of, according to Borcherds, as multicategories in which the morphisms may have some sort of singularity. In the language of pseudotensor categories (a variant of coloured operads) this is studied by Beilinson and Drinfeld. In connection to quantum groups this has been studied by Soibelman.

Sketch of a definition

Let $T$ be the free monoidmonad on Set and $T^+$ the monad induced by the adjunction of the forgetful functor from $T$-multicategories to $T$-graphs. Leinster defines a particular $T^+$-multicategory (see generalized multicategory) $V$. A relaxed multicategory is any $V$-enriched$T$-multicategory.