For a topological field theory to be a true “extension” of an Atiyah-Segal theory, the top two levels of its target (*ie* its $(n-1)^{\text{st}}$ loop space) must look like $\text{Vect}$. What other (physical) considerations constrain the choice of target category? The targets of invertible field theories are (by definition) $\infty$-groupoids and (therefore) can be represented as spectra. What constraints can we impose on target spectra of invertible theories; in particular, on their homotopy groups?

This post imported from StackExchange Physics at 2014-08-03 09:26 (UCT), posted by SE-user user151696