I will begin by explaining the construction of a category $\operatorname{CofCos}$, whose objects are topological spaces and whose morphisms are cofibrant cospans. Here the identity cospan is chosen to be of the form $X\to X\times [0,1] \rightarrow X$, in contrast with the usual identity in the bicategory $\operatorname{Cosp}(V)$ of cospans over a category $V$. The category $\operatorname{CofCos}$ has a subcategory $\operatorname{HomCob}$ in which all spaces are homotopically $1$-finitely generated. There exist functors into $\operatorname{HomCob}$ from a number of categorical constructions which are potentially of use for modelling particle trajectories in topological phases of matter: embedded cobordism categories and motion groupoids for example. Thus, functors from $\operatorname{HomCob}$ into $\operatorname{Vect}$ give representations of the aforementioned categories.

I will also construct a family of functors $Z_G : \operatorname{HomCob} \to \operatorname{Vect}$, one for each finite group $G$, showing that topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf-Witten, generalise to functors from $\operatorname{HomCob}$. I will construct this functor in such a way that it is clear the images are finite dimensional vector spaces, and the functor is explicitly calculable.