# Lazard Lie category

## Contents

## Definition

### As the category of Lazard Lie rings

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

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

Objects | Lazard Lie rings, i.e., Lie rings for which there exists a finite such that the Lie ring has 3-local class at most and is powered over all primes . |

Morphisms | Lie ring homomorphisms between Lazard Lie rings. |

Forgetful functor to set | Sends a Lazard Lie ring to its underlying set, sends a homomorphism of Lazard 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 Lazard Lie groups

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

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

Objects | Lazard Lie groups |

Morphisms | homomorphism of groups between Lazard Lie groups. |

Forgetful functor to set | Sends a Lazard Lie group to its underlying set, sends a homomorphism of Lazard 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 Lazard Lie category`