higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
For $\{U_i \to X\}$ a cover of a space $X$, the corresponding Čech groupoid is the internal groupoid
whose set of objects is the disjoint union $\coprod_i U_i$ of the covering patches, and the set of morphisms is the disjoint union of the intersections $U_i \cap U_j$ of these patches.
This is the $2$-coskeleton of the full Čech nerve. See there for more details.
If we speak about generalized points of the $U_i$ (which are often just ordinary points, in applications), then
an object of $C(\{U_i\})$ is a pair $(x,i)$ where $x$ is a point in $U_i$;
there is a unique morphism $(x,i) \to (x,j)$ for all pairs of objects labeled by the same $x$ such that $x \in U_i \cap U_j$;
hence the composition of morphism is of the form
For $X$ a smooth manifold and $\{U_i \to X\}$ an atlas by coordinate charts, the Čech groupoid is a Lie groupoid which is equivalent to $X$ as a Lie groupoid: $C(\{U_i\}) \stackrel{\simeq}{\to} X$
For $\mathbf{B}G$ the Lie groupoid with one object coming from a Lie group $G$ morphisms of Lie groupoids of the form
are also called anafunctors from $X$ to $\mathbf{B}G$. They correspond to smooth $G$-principal bundles on $X$.