nLab relaxed multicategory

Idea

Vertex algebras are intuitively analogues of associative algebras, except that certain singularities come up (in physics terminology this 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 monoid monad 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.

Literature

• Tom Leinster, Generalized enrichment for categories and multicategories, math.QA/9901139, chapter 4

• Yan Soibelman, Meromorphic tensor categories, q-alg/9709030; The meromorphic braided categroy arising in quantum affine algebras, math.QA/9901003

• Craig T. Snydal, Relaxed multi category structure of a global category of rings and modules, math.CT/9912075

• Craig T. Snydal, Equivalence of Borcherds G-vertex algebras and axiomatic vertex algebras, math.QA/9904104

• A. A. Beilinson, V. Drinfeld, Chiral Algebras, AMS 2004 (a preprint in various forms since around 1995, cf. here).