Tell me something about these "component tensor" TQFT's

Question

I noticed that there is a class of TQFT's that exists for every dimension $n\geq1$. It's probably well-known because it's quite simple, but I'm looking for a standard name or a better way to think about it.

Let $n\text{-}\mathrm{Cob}$ denote the symmetric monoidal category of (unoriented) closed $(n-1)$-manifolds $M,N$ and cobordisms $M\sqcup N=\partial B\subseteq B\;$ between them. Let $\mathrm{Cospan}_{\mathrm{Fin}}$ denote the symmetric monoidal category of finite sets $X,Y$ and cospans between them $X\to C\to Y$. In both case the monoidal product is given by disjoint union.

Consider the monoidal functor $$\pi_0: n\text{-}\mathrm{Cob}\to\mathrm{Cospan}_{\mathrm{Fin}}.$$ where $\pi_0(M)$ is the set of connected components in $M$. On morphisms it is given by $$\pi_0(M\to B\leftarrow N)=(\pi_0M\to\pi_0B\leftarrow\pi_0N).$$

For every field $k$ and natural number $r$, there's a strong monoidal functor $$\mathrm{tens}_r: \mathrm{Cospan}_{\mathrm{Fin}} \to \mathrm{Vect}_k$$ that sends a finite set $X$ to the $r^X$-dimensional vector space of order-$X$ tensors $T^{r,\ldots,r}$. For example, if $X=\{1,2\}$ then $\mathrm{tens}_r(X)\cong\mathrm{Mat}_{r\times r}\;\;$. The assignment $\mathrm{tens}_r$ is functorial via tensor multiplication, a straightforward generalization of matrix multiplication. The functor $\mathrm{tens}_r$ is strong monoidal because there is a natural isomorphism $$\mathrm{tens}_r(n)\otimes\mathrm{tens}_r(n')\cong\mathrm{tens}_r(n+n').$$

The composite $\mathrm{tens}_r\circ\pi_0\colon n\text{-}\mathrm{Cob}\to\mathrm{Vect}_k\;\;$ is a strong monoidal functor for any $r,n\in\mathbb{N}$.

Questions:

1. Is there a standard name for this sort of TQFT?
2. Does it have any universal or other nice properties?
3. Is it degenerate in any particular way?

Answer 1

The theory you describe is Dijkgraaf-Witten theory with target space a discrete set with $r$ elements. In general, if $X$ is a $\pi$-finite space (i.e., a space with finite homotopy groups, all but finitely many of which are non-trivial), then Dijkgraaf-Witten theory with target space $X$ is the functor $n\text{-}\mathrm{Cob} \to \mathrm{Vect}_k$ which associates to an $(n-1)$-manifold $M$ the vector space of functions $\pi_0\mathrm{Map}(M,X) \to k$, where $\mathrm{Map}(M,X)$ is the mapping space from $M$ to $X$ (here the assumption that $X$ is $\pi$-finite insures this mapping space is $\pi$-finite, and hence has only finitely many connected components). If $W$ is a cobordism from $M$ to $N$ and $\iota_M: M \hookrightarrow W$, $\iota_N: N \hookrightarrow W$ are the inclusions of $M,N$ in the boundary of $W$, then the associated linear map $$ T_W: k^{\pi_0\mathrm{Map}(M,X)} \to k^{\pi_0\mathrm{Map}(N,X)} $$ is defined as follows: for a function $f: \pi_0\mathrm{Map}(M,X) \to k$ define $T_W(f): \pi_0(N,X) \to k$ by the formula $$ T_W(f)(C) = \sum_{D \in \pi_0\mathrm{Map}(W,X) | \iota_N^*D = C}f(\iota_M^*D) $$ for $C \in \pi_0\mathrm{Map}(N,X)$, where $\iota_M^*: \pi_0\mathrm{Map}(W,X) \to \pi_0\mathrm{Map}(M,X)$ and $\iota_N^*: \pi_0\mathrm{Map}(W,X) \to \pi_0\mathrm{Map}(N,X)$ are the maps induced by pre-composition with $\iota_M$ and $\iota_N$ respectively. As explained in section 3 of this paper of Freed, Hopkins, Lurie and Teleman, this topological field theory should actually extend all the way down to $0$-manifolds, yielding a fully extended topological field theory. This can definitely be considered as a nice property. I'm not sure about any universal properties, but this theory is definitely related to a lot of interesting mathematics. As for degeneracy, the only form of degeneracy that comes to my mind is that the value associated to $M$ depends only on the homotopy type of $M$, so in terms of producing manifold invariant it cannot go beyond homotopy type. Note that when $X$ is a discrete set of size $r$ then $\pi_0\mathrm{Map}(M,X) = X^{\pi_0(M)}$ is a set of size $r^{\pi_0(M)}$ and the associated vector space is the space of tensors of order $\pi_0(M)$ and degree $r$, which is the case described in the question. Another case of special interest is when $X = BG$ is the classifying space of a finite group.