nLab
torsor with structure category

If $C$ is a small category (or even a topological category), one can define a $C$-torsor (or torsor with structure category $C$) which generalizes the torsor (principal bundle) with structure group(oid).

If $F$ is a sheaf over $X$, denote by $F_x$ its stalk over $x$ (cf. etale space).

A $C$-torsor $P$ over a space $X$ is given by a functor $P:C\to \mathrm{Shv}(X)$ such that

1. (surjectivity) every ‘total stalk’ ${\cup }_{c\in C_0}E(c{)}_x$, where $x\in X$, is nonempty;

2. (transitivity) for any two germs ‘in the same total stalk’, $\alpha \in E(c{)}_x$, $\alpha \prime \in E(c\prime {)}_x$ there is a span $u:b\to c$, $u:b\to d$ and $\xi \in E(b{)}_x$ such that $E(u)(\xi )=\alpha$ and $E(u\prime )(\xi )=\alpha \prime$;

3. (freeness) a parallel pair $u_1,u_2:c\to c\prime$ of morphisms in $C$, may induce coalescence $E(u_1)(\alpha )=E(u_2)(\alpha )$ for some $\alpha \in E(c{)}_x$ only if there is a morphism $w:b\to c$ and $\zeta \in E(b{)}_x$ such that $u_1\circ w=u_2\circ w$ and $E(w)(\zeta )=\alpha$.

The classifying space of a category $D$ classifies $D$-torsors.

Literature: I. Moerdijk, Classifying spaces and classifying topoi, Springer Lec. Notes Math. 1616 (1995)