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

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This article gives a proof/explanation of the equivalence of multiple definitions for the term Lazard Lie category
View a complete list of pages giving proofs of equivalence of definitions

Statement

Consider the following categories:

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 Lazard correspondence defines an isomorphism between the two full subcategories "over" the category of sets in the following sense: doing the 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 Lazard Lie category.

References