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:**

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