# Homotopy quantum field theories as functors

A homotopy quantum field theory is a symmetric monoidal functor $\tau:\mathrm{HCobord}(n,X)\to\mathrm{Vect}_{\mathbb{K}}$, with $X$ a path connected space with basepoint $\ast$. There is the following diagram (see page 15 of this ArXiv article): \begin{multline} \begin{matrix} \mathrm{Cobord}(n) &\rightleftarrows\space \mathrm{HCobord}(n,X) &\longleftarrow& \pi_1\mathrm{Map}(S^n,X)\\ \searrow & \downarrow & \swarrow & \\ & \mathrm{Vect}_{\mathbb{K}}\\ &\downarrow&\\& \mathrm{Set} \end{matrix} \end{multline} A homotopy QFT restricted to domain $\pi_1\mathrm{Map}(S^n,X)$ is a contravariant functor $\pi_1\mathrm{Map}(S^n,X)\to\mathrm{Vect}_{\mathbb{K}}\hookrightarrow\mathrm{Set}$, which is a locally constant sheaf on $\pi_1\mathrm{Map}(S^n,X)$, i.e., a sheaf of sections of a covering space of $\pi_1\mathrm{Map}(S^n,X)$. Let us denote by $\mathrm{LSheaf}\pi_1\mathrm{Map}(S^n,X)$ the category of locally constant sheaves on $\pi_1\mathrm{Map}(S^n,X)$. Then, $\mathrm{LSheaf}\pi_1\mathrm{Map}(S^n,X)\hookrightarrow \mathrm{Sh}\pi_1\mathrm{Map}(S^n,X)\hookrightarrow \mathrm{PSh}\pi_1\mathrm{Map}(S^n,X)$.

Proposition: Loops on $\mathrm{Map}(S^n,X)$ are $X$-cobordisms, since $\pi_1\mathrm{Map}(S^n,X)\hookrightarrow\mathrm{HCobord}(n,X)$.

Is the above proposition and the above argument correct?

In the following article, it is written that a string connection assigns to a loop a vector space and assigns to cobordisms between loops, linear transformations. Hence,

Does it follow that string connections defined on the associated vector bundle over $\pi_1\mathrm{Map}(S^n,X)$ as defined in the article above are all homotopy quantum field theories restricted to domain $\pi_1\mathrm{Map}(S^n,X)$? What exactly would it mean for a string connection to be a HQFT?

A loop on $\text{Map}(S^n,X)$ is just a map $S^{n+1}\to X$. So it is an $X$-cobordism, though one without boundary, i.e. a cobordism between $\emptyset$ and $\emptyset$. An $n$-dimensional HQFT assigns the base field $k$ to $\emptyset$ and so multiplication by an element of $k$ to such a loop. On the other hand an $n+1$-HQFT will assign a vector space to such a loop seen as an $X$-manifold. So this latter is the case relevant to your second article. In a rough sense your second question then follows, although I see no reason a string connection should extend to an HQFT in general.