LCS-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	LCS-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 $\leq c$.
Morphisms	Lie ring homomorphisms between Lazard Lie rings.
Forgetful functor to set	Sends a LCS-Lazard Lie ring to its underlying set, sends a homomorphism of LCS-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 LCS-Lazard Lie groups

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

Item	Value
Objects	LCS-Lazard Lie groups
Morphisms	homomorphism of groups between LCS-Lazard Lie groups.
Forgetful functor to set	Sends a LCS-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 LCS-Lazard Lie category