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 $E : C\to Shv(X)$ 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’, $\alpha\in E(c)_x$, $\alpha'\in E(c')_x$, there is a span $c\stackrel{u}\leftarrow b\stackrel{u'}\to c'$ and $\xi\in E(b)_x$ such that $E(u)(\xi)=\alpha$ and $E(u')(\xi)=\alpha'$;

3. (freeness) for a parallel pair $u_1,u_2: c\to c'$ 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)