Definition 0. By a topological germ, I mean a pointed topological space. Whenever $X$ and $Y$ denote topological germs, by a morphism of topological germs $X \rightarrow Y$, I mean a neighbourhood $U$ of $\bullet_X$, together with a basepoint-preserving continuous mapping $f : U \rightarrow Y$. Whenever $X$ and $Y$ denote topological germs, and $f,g : X \rightarrow Y$ are morphisms of topological germs, by a neighbourhood of agreement $p : f = g$, I mean a neighbourhood $U$ of $\bullet_X$ such that $f$ is defined on all of $U$, $g$ is defined on all of $U$, and $f\restriction_U = g\restriction_U$.
For example, there is a topological germ associated with $n$-dimensional Euclidean space; this is related to the concept of being "locally Euclidean" in the obvious way. More generally, let $X$ denote any topological space that is "homogeneous", in the sense that $\mathbf{Aut}(X)$ acts transitively on $X$. Then there is a germ $gX$ associated to $X$ in an obvious way.
Definition 1. Topological germs can be organized into a $2$-category by taking morphisms of topological germs as $1$-cells, and neighbourhoods of agreement as $2$-cells. (To compose $2$-cells, take their intersection.) Call this $\mathbf{TopGerm}_2$. We can truncate to a $1$-category by declaring two morphisms two be equal iff there is a $2$-cell between them. Call this $\mathbf{TopGerm}_1$.
I'd like to know more:
Question.
Where can I learn more about what I'm denoting $\mathbf{TopGerm}_1$ and/or $\mathbf{TopGerm}_2$?
Added 10/05/2016. This seems relevant.