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 $\mathbf{TopGerm}_2$?

• What exactly do you want to learn about? The usual setting for this sort of thing outside the category category is with sheaves of sufficiently nice functions ($C^k$, smooth, analytic, etc.), with the germ appearing as the colimit with respect to restriction near a point; or in algebraic geometry, considering a sheaf axiomatically. – anomaly May 3 '16 at 4:47
• @anomaly, one of the first things I'd like to know is whether some analogue of the CSB theorem holds in $\mathbf{TopGerm}_1$. Nothing like this holds for topological spaces; e.g. we can inter-embed $[0,1]$ and $(0,1)$ despite that they're non-homeomorphic. Even restricting to compact Hausdorff spaces doesn't work; we can inter-embed $[0,1]$ and $[0,1] \sqcup \{*\}$, despite that they're non homeomorphic. One hopes that these sorts of difficulties will go away if we're working only in some arbitrarily small neighbourhood. – goblin May 3 '16 at 4:51
There's a brief mention of what I denote $$\mathbf{TopGerm}_1$$ above on p.39 of this article on nonstandard analysis. The idea seems to be roughly that given a set $$A \subseteq \mathbb{N}^k$$ together with a filter $$\mathcal{F}$$ on $$A$$, we get a corresponding topological space $$X$$ by defining $$X := A \cup \{\infty\}$$ as a set and declaring that $$X \setminus \{\infty\}$$ has the discrete topology, while using the filter $$\mathcal{F}$$ to give $$\infty$$ a non-trivial neighbourhood system. See the linked article for more information.