# Lazard Lie category

## 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 $c$ such that the Lie ring has 3-local class at most $c$ and is powered over all primes $\le c$.
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