torsor with structure category


If CC is a small category (or even a topological category), one can define a CC-torsor (or torsor with structure category CC) which generalizes the torsor (principal bundle) with structure group(oid). We present two variants in slightly different context.

Moerdijk’s definition

If FF is a sheaf over XX, denote by F xF_x its stalk over xx (cf. etale space).

A CC-torsor EE over a topological space XX is given by a functor E:CShv(X)E : C\to Shv(X) such that

  1. (surjectivity) every ‘total stalk’ cC 0E(c) x\cup_{c\in C_0} E(c)_x, where xXx\in X, is nonempty;

  2. (transitivity) for any two germs ‘in the same total stalk’, αE(c) x\alpha\in E(c)_x, αE(c) x\alpha'\in E(c')_x, there is a span cubucc\stackrel{u}\leftarrow b\stackrel{u'}\to c' and ξE(b) x\xi\in E(b)_x such that E(u)(ξ)=αE(u)(\xi)=\alpha and E(u)(ξ)=αE(u')(\xi)=\alpha';

  3. (freeness) for a parallel pair u 1,u 2:ccu_1,u_2: c\to c' of morphisms in CC, E(u 1)(α)=E(u 2)(α)E(u_1)(\alpha)=E(u_2)(\alpha) for some αE(c) x\alpha\in E(c)_x implies there is a morphism w:bcw:b\to c and ζE(b) x\zeta\in E(b)_x such that u 1w=u 2wu_1\circ w = u_2\circ w and E(w)(ζ)=αE(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 CC classifies CC-torsors.

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

Street’s definition

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

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

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

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

Revised on September 30, 2010 23:54:00 by David Roberts (