The category of line bundles (possibly with connection) on a smooth manifold M can be defined in two different ways: The first definition uses transition functions that satisfy a cocycle condition (possibly with additional data of a 1-form that defines a connection), and the second definition uses invertible vector bundles (possibly with connection), where vector bundles are defined algebraically as dualizable modules over the algebra of smooth functions or geometrically as vector spaces in the category of smooth submersions over M and connections are defined algebraically as certain linear maps satisfying the Leibniz identity, geometrically in terms of subbundles, or topologically as certain functorial field theories.
The above two categories (with or without connection) are equivalent to each other: There is a canonical fully faithful functor from the first category to the second one. This functor is essentially surjective and hence an equivalence of categories, even though the construction of an inverse functor requires a choice.
The first definition was categorified by Michael Murray in 1994, the result being the bicategory of bundle gerbes (with or without connection).
Can one categorify the algebro-geometric definition of a line bundle in such a way that there is an equivalence from Murray's bicategory of bundle gerbes to the bicategory of categorified line bundles, thus obtaining a “chart-free” definition of bundle gerbes?
Naïvely, one might expect that vector bundles should categorify to bundles of algebras. Just as any fiber of a line bundle is noncanonically isomorphic to the vector space of scalars, any fiber of a categorified line bundle should be noncanonically Morita-equivalent to the algebra of scalars.
Furthermore, if a line bundle over M is equipped with a connection, then any path in M gives an isomorphism between the corresponding fibers. These isomorphisms can packaged in a 1-dimensional topological field theory, and in fact a theorem by Florin Dumitrescu, Stephan Stolz, and Peter Teichner shows that any 1-TFT comes from a vector bundle with connection, thus giving an alternative definition of a vector bundle with connection. Similarly, if a bundle gerbe over M is equipped with a connection, then any path in M should give a Morita equivalence between the corresponding fibers. Moreover, a bigon in M should give an isomorphism between the corresponding Morita equivalences. Thus, one should be able to package the above parallel transports in a local 2-TFT.
Invertible morphisms between bundle gerbes should correspond to bundles of invertible bimodules (i.e., Morita equivalences) and non-invertible morphisms (introduced by Konrad Waldorf in his paper More Morphisms between Bundle Gerbes) should correspond to bundles of non-invertible bimodules.
Several papers in the literature are closely related to the above question. For example, Corollary 4.9 in the paper by Urs Schreiber and Konrad Waldorf Connections on non-abelian gerbes and their holonomy proves that the bicategory of bundle gerbes with connection over M is equivalent to the bicategory of transport functors with values in some bicategory. This statement would almost resolve the above question if not for the fact that the target bicategory has only one object, which prevents one from considering constructions like the algebra of global sections of a bundle gerbe (the analogous construction for line bundles (the vector space of global sections) plays an important rôle in geometric quantization and other areas of mathematics). In fact, the algebra of global sections of a bundle gerbe is interesting enough to warrant its own question:
Can one construct the algebra of global sections of a bundle gerbe that categorifies the vector space of global sections of a line bundle?
Finally, I am also interested in the answers to the above questions for the case of arbitrary vector bundles (respectively non-abelian bundle gerbes) and not just line bundles.
