  We construct a category $\mathrm{HomCob}$ whose objects are {\it
homotopically 1-finitely generated} topological spaces, and whose morphisms are
{\it cofibrant cospans}.
  Given a manifold submanifold pair $(M,A)$, we prove that there exists
functors into $\mathrm{HomCob}$ from the full subgroupoid of the mapping class
groupoid $\mathrm{MCG}_{M}^{A}$, and from the full subgroupoid of the motion
groupoid $\mathrm{Mot}_{M}^{A}$, whose objects are homotopically 1-finitely
generated.
  We also construct a family of functors $\mathsf{Z}_G\colon \mathrm{HomCob}\to
\mathbf{Vect}$, one for each finite group $G$.
  These generalise topological quantum field theories previously constructed by
Yetter, and an untwisted version of Dijkgraaf-Witten.
  Given a space $X$, we prove that $\mathsf{Z}_G(X)$ can be expressed as the
$\mathbb{C}$-vector space with basis natural transformation classes of maps
$\{\pi(X,X_0)\to G\} $ for some finite representative set of points $X_0\subset
X$, demonstrating that $\mathsf{Z}_G$ is explicitly calculable.
