Extending higher derivations to rings and modules of quotients

A torsion theory is called differential (higher differential) if a derivation (higher derivation) can be extended from any module to the module of quotients corresponding to the torsion theory. We study conditions equivalent to higher differentiability of a torsion theory. It is known that the Lambek, Goldie and any perfect torsion theories are differential. We show that these classes of torsion theories are higher differential as well. Then, we study conditions under which a higher derivation extended to a right module of quotients extends also to a right module of quotients with respect to a larger torsion theory. Lastly, we define and study the symmetric version of higher differential torsion theories. We prove that the symmetric versions of the results on higher differential (one-sided) torsion theories hold for higher derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie and any perfect torsion theories are higher differential

Infinitesimal 2-braidings and differential crossed modules

We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relation, leading to six categorified relations. We prove that any infinitesimal 2-braiding gives rise to a flat and fake flat 2-connection in the configuration space of $n$ particles in the complex plane, hence to a categorification of the Knizhnik-Zamolodchikov connection. We discuss infinitesimal 2-braidings in a 2-category naturally assigned to every differential crossed module, leading to the notion of a quasi-invariant tensor in a differential crossed module. Finally we prove that quasi-invariant tensors exist in the differential crossed module associated to the String Lie-2-algebra.Comment: v3 - the introduction has been expanded, overall improvements in the presentation. Final version, to appear in Adv. Mat