# Baer Lie category

From Groupprops

## Contents

## Definition

### As the category of Baer Lie rings

The **Baer Lie category** can be defined as the following concrete category:

Item | Value |
---|---|

Objects | Baer Lie rings, i.e., Lie rings that are uniquely 2-divisible and have class at most two |

Morphisms | Lie ring homomorphisms between Baer Lie rings. Note that a Lie ring homomorphism automatically preserves the unique 2-division. |

Forgetful functor to set | Sends a Baer Lie ring to its underlying set, sends a homomorphism of Baer Lie rings to the map of underlying sets. |

Viewed this way, the category is a full subcategory of the category of Lie rings.

### As the category of Baer Lie groups

The **Baer Lie category** can be defined as the following concrete category:

Item | Value |
---|---|

Objects | Baer Lie groups, i.e., groups that are uniquely 2-divisible and have class at most two |

Morphisms | homomorphism of groups between Baer Lie groups. This automatically preserves the unique 2-division. |

Forgetful functor to set | Sends a Baer Lie group to its underlying set, sends a homomorphism of Baer Lie groups to the map of underlying sets. |

Viewed this way, the category is a full subcategory of the category of groups.

### Equivalence of definitions

`Further information: equivalence of definitions of Baer Lie category`