Lazard Lie category
From Groupprops
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
