In my work, I have often used the definition of a “simplicial line bundle”, or what is essentially equivalent, a simplicial -bundle, over a simplicial manifold. If this simplicial manifold is the nerve of , the one-object groupoid associated to a Lie group , then such an object is equivalent to a central extension of by . The construction is used by Murray and Stevenson, for instance, in their paper Higgs fields, bundle gerbes and string structures. They cite Brylinski and McLaughlin’s paper Geometry of degree-4 characteristic classes and of line bundles on loop spaces, I for the source of this construction (and, implicitly, the equivalence between the notion of the simplicial construction and the central extension).
If we go to Brylinski and McLaughlin, they have in Theorem 5.2 cited Grothendieck as the source of the equivalence between the simplicial construction and the central extension. The problem is, they cite chapter VIII of tome 1 of SGA 7, Complements sur les biextensions. Proprietes generales des biextensions des schemas en groupes. This chapter is 95 pages long. Skimming through that chapter, Grothendieck seems to cite the previous chapter as the source of the relevant theory. I might be wrong, but perhaps the citation went a bit astray? I would love to be proven wrong, and a precise numbered citation given.
Now we need to go and start reading chapter VII of SGA 7.1, Biextensions de faisceaux de groupes. It’s 90 pages long. And, as those who know the material, it’s a scanned, typewritten document, in French—and doesn’t use simplicial terminology anywhere. So there’s no text searching even for relevant key words. It’s not a huge deal, but Grothendieck is dealing with algebraic geometry, and sheaves of groups, instead of Lie groups. Worse, it’s in complete generality of arbitrary normal extensions of groups, not central extensions, so the material is made a lot more subtle and now full of irrelevant and complicating detail.
The closest I’ve come to finding what Brylinski and McLaughlin might be wanting to cite is Proposition 1.3.5 of chapter VII:
The notion of equivalence that Brylinski and McLaughlin state in their Theorem 5.2 is not that difficult, except for the implication that the definition of simplicial -bundle implies that the putative group structure on the extension group is in fact associative. The relevant condition is that some section over is equal to the canonical trivialisation of what would have been denoted by Grothendieck. The closest I can see in the proof of the above statement is the following excerpt:
The “condition d’associativité (220.127.116.11)” is the usual commutative square expressing the associativity of a group object in a category. So it’s not very clear, since the statement of the Proposition is about objects not really like what Brylinski and McLaughlin define.
I’m fully aware that Brylinski and McLaughlin may have in fact going for a citation in the sense of the classic Australian film The Castle:
It may also be that my trying to skim read 185 pages of mathematical French just missed the relevant section of SGA 7.1, chapter VIII (or possibly VII). In which case, I’d be happy to be pointed to chapter-and-verse