Nikita Markarian’s mathblog

TQFT as a generalization of spherical homology

Posted in Mathematics by nikitamarkarian on December 15, 2009

Speculating about the question posed in the end of this confused post I recognised, that the generalization involved makes some sense. Spherical homology is generalised in two directions. Firstly 2-sphere is replaced with any compact oriented 2-manifold. Secondly a BV (f2-)algebra is replaced with a cyclic 2-category. As a result one get a structure that strongly reminds TQFT up to decorations. By decorations I mean metric, counit and so on. This conjectural structure plays for cyclic 2-categories the same role as  Hochschild homology plays for (1-)categories.

In the paper “What do DG categories form?” D. Tamarkin introduced the notion of a homotopy 2-category. Its important feature is that for any object $X$ on $\mathop{Hom}(Id_X, Id_X)$ the operad of chains of little discs acts. Below  a notion of homotopy cyclic 2-category is introduced. For it the complex $\mathop{Hom}(Id_X, Id_X)$ is acted by the operad of chains of framed little discs.