### Abstract

A Lie 2-group G is a category internal to the category of Lie groups. Consequently it is a monoidal category and a Lie groupoid. The Lie groupoid structure on G gives rise to the Lie 2-algebra X(G) of multiplicative vector fields, see [2]. The monoidal structure on G gives rise to a left action of the 2-group G on the Lie groupoid G, hence to an action of G on the Lie 2-algebra X(G). As a result we get the Lie 2-algebra X(G)^{G} of left-invariant multiplicative vector fields. On the other hand there is a well-known construction that associates a Lie 2-algebra g to a Lie 2-group G: apply the functor Lie: LieGp → LieAlg to the structure maps of the category G. We show that the Lie 2-algebra g is isomorphic to the Lie 2-algebra X(G)^{G} of left invariant multiplicative vector fields.

Original language | English (US) |
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Article number | 21 |

Pages (from-to) | 604-634 |

Number of pages | 31 |

Journal | Theory and Applications of Categories |

Volume | 34 |

State | Published - Jan 1 2019 |

### Keywords

- 2 limit
- Invariant vector fields
- Lie 2-algebra
- Lie 2-group

### ASJC Scopus subject areas

- Mathematics (miscellaneous)

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## Cite this

*Theory and Applications of Categories*,

*34*, 604-634. [21].