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Register a new userAtiyah sequence and Gauge transformations of a principal $2$-bundle over a Lie groupoid
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In this paper, a notion of a principal $2$-bundle over a Lie groupoid has been introduced. For such principal $2$-bundles, we produced a short exact sequence of VB-groupoids, namely, the Atiyah sequence. Two notions of connection structures viz. strict connections and semi-strict connections on a principal $2$-bundle arising respectively, from a retraction of the Atiyah sequence and a retraction up to a natural isomorphism have been introduced. We constructed a class of principal $mathbb{G}=[G_1rightrightarrows G_0]$-bundles and connections from a given principal $G_0$-bundle $E_0rightarrow X_0$ over $[X_1rightrightarrows X_0]$ with connection. An existence criterion for the connections on a principal $2$-bundle over a proper, etale Lie groupoid is proposed. The action of the $2$-group of gauge transformations on the category of strict and semi-strict connections has been studied. Finally we noted an extended symmetry of the category of semi-strict connections.
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Motivated by the computations done in cite{C1}, where I introduced and discussed what I called the groupoid of generalized gauge transformations, viewed as a groupoid over the objects of the category $mathsf{Bun}_{G,M}$ of principal $G$-bundles over a given manifold $M$, I develop in this paper the same ideas for the more general case of {em principal $calG$-bundles or principal bundles with structure groupoid $calG$}, where now $calG$ is a Lie groupoid in the sense of cite{Moer2}. Most of the concepts introduced in cite{C1} can be translated almost verbatim in the framework of principal bundles with structure groupoid $calG$; in particular, the key r�le for the construction of generalized gauge transformations is again played by (the equivalent in the framework of principal bundles with groupoid structure of) the division map $f_P$. Of great importance are also the generalized conjugation in a groupoid and the concept of (twisted) equivariant maps between groupoid-spaces.
In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldof from manifolds to stacks. In the process of proving our main result we simplify Schreiber and Waldorfs definition of a transport functor for principal bundles with connections over manifolds and provide a more direct proof of the correspondence between principal bundles with connections and transport functors.
We deal with the symmetries of a (2-term) graded vector space or bundle. Our first theorem shows that they define a (strict) Lie 2-groupoid in a natural way. Our second theorem explores the construction of nerves for Lie 2-categories, showing that it yields simplicial manifolds if the 2-cells are invertible. Finally, our third and main theorem shows that smooth pseudofunctors into our general linear 2-groupoid classify 2-term representations up to homotopy of Lie groupoids.
We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions.
Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we call current Lie groupoids. We then study basic differential geometry and Lie theory for these Lie groupoids of mappings. In particular, we show that certain Lie groupoid properties, like being a proper etale Lie groupoid, are inherited by the current groupoid. Furthermore, we identify the Lie algebroid of a current Lie groupoid as a current Lie algebroid (analogous to the current Lie algebra associated to a current Lie group). To establish these results, we study superposition operators given by postcomposition with a fixed function, between manifolds of $C^ell$-functions. Under natural hypotheses, these operators turn out to be a submersion (an immersion, an embedding, proper, resp., a local diffeomorphism) if so is the underlying map. These results are new in their generality and of independent interest.
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