# TQFTs with target category of higher type than the source

In the classical version of the Cobordism Hypothesis, such as, e.g., in Jacob Lurie's On the Classification of Topological Field Theories, one considers the $\infty$-category of symmetric monoidal functors from the $(\infty,n)$-symmetric monoidal category of (framed) $n$-cobordism to some $(\infty,n)$-symmetric monoidal target category $\mathcal{C}$. One of the first things one proves is that this category of functors is actually an $(\infty,0)$-category. For instance, if $n=1$ one can take $\mathcal{C}=Vect_k$ and one sees that the datum of a 1-dimensional tqft is the choice of a finite dimensional (i.e. fully dualizable) vector space $V$, so that the datum of a natural transformation between two such tqfts is a morphism of finite dimensional vector spaces $f:V\to W$. These are the data attached to the (oriented) point. Next one looks at what happens for the data attached to 1-dimensional manifolds and sees that $f$ is constrained to be an isomorphism (the quickest test here is to notice that $f$ has to induce an isomorphism between the dimension of $V$ and the dimension of $W$, in the category whose objects are elements of $k$ and whose morphism are only the identities). As this simple example shows, what is crucial here is that 1-dimensional manifolds has come into play. This means that if we had restricted our attention to 0-dimensional manifolds instead, i.e., we would have considered a symmetric monoidal functor from the symmetric monoidal $(\infty,0)$-category of 0-dimensional cobordims to the symmetric monoidal $(\infty,1)$-category of vector spaces over $k$, we would have had complete freedom in the choice of $f$.

This suggests that in general $n$-dimensional cobordism "eats" $n$ non-invertible levels in the target, so that the $\infty$-category of of symmetric monoidal functors from the $(\infty,n)$-symmetric monoidal category of (framed) $n$-cobordism to some $(\infty,n+k)$-symmetric monoidal target category $\mathcal{C}$ is actually an $(\infty,k)$-category.

Apart from the interest in this result in itself, I'm interested into it for the following possible application to the characters of finite groups: a representation $(V,\rho)$ of a finite group $G$ can be seen as a $k$-$k[G]$-bimodule, and so as a morphism from $k$ to $k[G]$ in the $(\infty,2)$-category of algebras, bimodules, bimodule morphisms. If this naturally induces a morphism of 1-dimensional tqfts from the tqft $Z_k$ defined by assigning to the point the algebra $k$ and the tqft $Z_{k[G]}$ defined by assigning to the point the algebra $k[G]$, then we would also have a natural morphism $\varphi_\rho:Z_k(S^1)\to Z_{k[G]}(S^1)$, i.e., from $k$ to the class functions of $G$. This should be nothing but the trace of $\rho$.

Is this correct? Is this point of view already expanded in some reference?

• The idea of combining an $n$-manifold with an $(n{+}k)$-category to get a $k$-category is discussed in Example 6.2.4 of arxiv.org/pdf/1009.5025.pdf . I think there must be something similar in one of Lurie's papers, and perhaps elsewhere as well, but I don't know specifically where. – Kevin Walker Mar 24 '12 at 18:30
• Willerton and Caldararu discuss ideas similar to your last paragraph in arxiv.org/pdf/0707.2052.pdf . – Kevin Walker Mar 24 '12 at 18:34
• You could instead think of the k-mod-k[G] bimodule as giving you a boundary between the theory attached to k[G] and the theory attached to k (here it's important that your chosen 1-morphism itself satisfies appropriate dualizability conditions). I think in Jacob's paper this is discussed a little in Section 4.3. You can then think of the map you're looking for as being given by a cylinder with one theory at the top, the other theory at the bottom, and a boundary circle in the middle. – Noah Snyder Mar 24 '12 at 20:06
• Hi Kevin, hi Noah, thanks a lot for the references. Noah, you're suggesting to think of the k-mod-k[G] bimodule as a codimension-one defect separating two tqfts, right? That's a nice point of view, thanks for the suggestion. In Jacob's paper this is mentioned in Example 4.3.23, but anyway the literature on tqfts with defects is quite ample, I'll have a look there. – domenico fiorenza Mar 24 '12 at 20:49
• Domenico: some ideas that could be interested for you are discussed in arxiv.org/pdf/1502.06526v3.pdf – DamienC Sep 5 '16 at 11:28