# LCS-Lazard correspondence defines isomorphism of categories over Set between full subcategories of the category of groups and the category of Lie rings

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term LCS-Lazard Lie category

View a complete list of pages giving proofs of equivalence of definitions

## Statement

Consider the following categories:

- The full subcategory of the category of groups comprising the LCS-Lazard Lie groups. By
*full*here we mean that*every*homomorphism of groups between two LCS-Lazard Lie groups is a morphism in the category. - The full subcategory of the category of groups comprising the LCS-Lazard Lie rings. By
*full*here we mean that*every*homomorphism of Lie rings between two LCS-Lazard Lie rings is a morphism in the category.

Both categories have forgetful functors to the category of sets, obtained by restricting the corresponding forgetful functors to the category of sets from (respectively) the category of groups and the category of Lie rings.

The claim is that the LCS-Lazard correspondence defines an isomorphism between the two full subcategories "over" the category of sets in the following sense: doing the LCS-Lazard correspondence and then applying the forgetful functor to sets is equivalent to directly applying the forgetful functor to sets. In words, it means that the isomorphism of categories preserves the underlying sets.

This category is termed the LCS-Lazard Lie category.