nLab
pseudo-connection

Context

\infty-Chern-Weil theory

Differential cohomology

Contents

Idea

A connection on a bundle may be understood as a construction that serves to interpolate between a principal bundle and its characteristic class in real cohomology, represented by curvature characteristic forms of the connection.

If one allows (and it makes good sense to allow this) curvature characteristic forms to be represented not necessarily by globally defined forms, but by cocycles in the abelian sheaf cohomology of the truncated de Rham complex Ω 1()d dRΩ 2()Ω cl n(X)\Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \to \cdots \to \Omega^n_{cl}(X) then a weaker notion than that of a connection serves the same purpose: that of a pseudo-connection .

Effectively a pseudo-connection on a principal bundle is locally a collection of Lie-algebra valued 1-forms, that do not have to satisfy the cocycle condition familiar from connections.

That makes pseudo-connections rather empty structures. And in a precise sense this is actually the point of them: there is a (non-concrete) Lie groupoid BG diff\mathbf{B}G_{diff} such that Cech cocycles with values in it are those for GG-principal bundles with pseudo-connections. This groupoid is in fact weakly equivalent to the delooping groupoid BG\mathbf{B}G that serves to classify just bare smooth GG-principal bundles

BG diffBG. \mathbf{B}G_{diff} \stackrel{\simeq}{\to} \mathbf{B}G \,.

The purpose of pseudo-connections is precisely to form this resolution of BG\mathbf{B}G, which allows to represent an intrinsic curvature characteristic class to be modeled by an anafunctor with tip BG diff\mathbf{B}G_{diff}.

While for ordinary GG-principal bundles the notion of pseudo-connection can be (and traditionally is) circumvented, it does become crucial for gerbes and principal 2-bundles and more generally in infinity-Chern-Weil theory. For instance an equivariant gerbe with connection is not an equivariant object in the 2-category of gerbes with ordinary connections, but a certain constrained equivariant gerbe with pseudo-connection.

Details

For the moment see infinity-Chern-Weil theory.

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References

The term pseudo-connection for generalized connections on bundle gerbes apparently first appeared in

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Not under this name, but being the same concept, they are considered on bundle gerbes in some detail in

Created on September 8, 2010 23:31:10 by Urs Schreiber (188.20.66.18)