Kuranishi spaces in a category of fibrant objects and their homotopy fiber products

We study several fundamental properties of the category $\mathbf{K}\mathbf{u}\mathbf{r}$ introduced in \cite{Kim1}. By defining fibrations and weak equivalences, we show that morphisms with manifold targets admit homotopy fiber products; this is then applied to obtain a notion of generalized intersections of two submanifolds. A definition of the Weinstein category in the context of Kuranishi spaces is also presented. Finally, we establish that $\mathbf{K}\mathbf{u}\mathbf{r}$ forms a category of fibrant objects in the sense of Brown \cite{Brown}, and discuss its homotopy category $\mathrm{Ho}(\mathbf{K}\mathbf{u}\mathbf{r})$.