An A-S TFT is a functor from $\text{Bord}_{<n−1,n>}(\mathcal{F})$ to $\text{Vect}$ where $\mathcal{F}$ denotes a set of background fields, eg a spin structure. An extended theory is a functor from $\text{Bord}_n(\mathcal{F})$ to a symmetric monoidal $(\infty,n)$-category $\mathcal{C}$ that restricts to an A-S TFT on $(n−1)$- and $n$-manifolds.

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) symmetric monoidal $\infty$-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?

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    $\begingroup$ You should say "symmetric monoidal $\infty$-groupoids" if you want to make the connection to spectra more explicit. $\endgroup$ – Qiaochu Yuan Aug 23 '14 at 4:59

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