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A cohomological description of connections and curvature over posets

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John E. Roberts and Giuseppe Ruzzi

What remains of a geometrical notion like that of a principal bundle when the base space is not
a manifold but a coarse graining of it, like the poset formed by a base for the topology
ordered under inclusion? Motivated by the search for a geometrical framework for developing
gauge theories in algebraic quantum field theory, we give, in the present paper, a first answer
to this question. The notions of transition function, connection form and curvature form find a
nice description in terms of cohomology, in general non-Abelian, of a poset with values in a
group G. Interpreting a 1-cocycle as a principal bundle, a connection turns out to be a
1-cochain associated in a suitable way with this 1-cocycle; the curvature of a connection
turns out to be its 2-coboundary. We show the existence of nonflat connections, and relate
flat connections to homomorphisms of the fundamental group of the poset into G. We discuss
holonomy and prove an analogue of the Ambrose-Singer theorem.

2000 MSC:
53C05, 18D05, 05E25

*Theory and Applications of Categories,*
Vol. 16, 2006,
No. 30, pp 855-895.

http://www.tac.mta.ca/tac/volumes/16/30/16-30.dvi

http://www.tac.mta.ca/tac/volumes/16/30/16-30.ps

http://www.tac.mta.ca/tac/volumes/16/30/16-30.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/30/16-30.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/30/16-30.ps

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