Correspondence between Infinite-Dimensional Lie Groups and Lie 2-Groups

Definitions and mathematical statement

An infinite-dimensional Lie group is a group equipped with an infinite-dimensional smooth manifold structure for which multiplication and inversion are smooth, relative to a specified category of manifolds and a specified notion of smoothness. The supplied record does not specify these choices.

In a common groupoid formulation, a Lie $2$-group is a smooth groupoid equipped with a monoidal structure, coherent associativity and unit isomorphisms, and inverses for objects up to isomorphism. Strict formulations impose corresponding identities strictly. These descriptions explain the terminology; the source does not select a strict or weak formulation or an ambient smooth category.

The supplied research question is:

What can be learned from infinite-dimensional Lie groups about Lie $2$-groups, and conversely?

Both directions of the question are retained. The record specifies neither classes of objects to quantify over nor a particular construction, comparison, or conclusion to establish. Consequently, it determines a research direction but does not determine a precise quantified conjecture. In particular, no equivalence or correspondence theorem is asserted by this draft.

Short English statement

The question asks how the theories of infinite-dimensional Lie groups and Lie $2$-groups can inform one another. The intended smooth categories, strictness conventions, and mathematical targets remain unspecified, so a precise theorem cannot be extracted without additional information.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001487$, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record assigns the year $2010$; no individual proposer is named.
• Supplied report citation: Geometry, Quantum Fields, and Strings: Categorial Aspects (2010), Oberwolfach Reports, pp. 1485--1533, DOI 10.4171/owr/2010/25. Supplied source URL and literature reference labeled OWR 2010/25: https://doi.org/10.4171/owr/2010/25. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature assessment, checked 2026-08-21: the record labels the question open and states that integration and higher-gauge-group results address special cases without providing a general resolution. This is supplied dated curation, not an independently verified assessment. The background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement exact and reports that manual review found the extracted wording self-contained, mathematically coherent, and free of LaTeX or report-layout artifacts. These are supplied verification claims; the report has not been independently consulted for this draft.