Baer Lie category
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