nLab
torsor with structure category

Idea

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). We present two variants in slightly different context.

Moerdijk’s definition

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

A $C$-torsor $E$ over a topological space $X$ is given by a functor $E:C\to \mathrm{Shv}(X)$ 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 $c\stackrel{u}{\leftarrow }b\stackrel{u\prime }{\to }c\prime$ and $\xi \in E(b{)}_x$ such that $E(u)(\xi )=\alpha$ and $E(u\prime )(\xi )=\alpha \prime$;

3. (freeness) for a parallel pair $u_1,u_2:c\to c\prime$ of morphisms in $C$, $E(u_1)(\alpha )=E(u_2)(\alpha )$ for some $\alpha \in E(c{)}_x$ implies 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 $E(w)(\zeta )=\alpha$.

This definition is from the monograph

• Ieke Moerdijk, Classifying spaces and classifying topoi, Springer Lec. Notes Math. 1616 (1995)

where it is shown that the classifying space of a category $C$ classifies $C$-torsors.

David Roberts: This definition should be able to be restated in terms of flat functors

Street’s definition

Suppose now $C$ is a finitely complete category with a calculus of left fractions whose morphisms are called covers.

Let $A$ be an internal category in $C$. An $A$-torsor trivialized by a cover $e:V\to U$ is a discrete fibration $A\stackrel{p}{\leftarrow }E\stackrel{q}{\to }U$ for which there exist a morphism $a:V\to A$ and a commutative diagram

in which the square is a pullback. Street says $A$-torsor at $U$ for an $A$-torsor trivialized by some cover $e:V\to U$.

• Ross Street, Combinatorial aspects of descent theory, pdf (page 25 in the file)