# Tell me something about these “component tensor” TQFT's

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?

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.