Montgomery and Zippin de ned a group G to be approximated by Lie groups if every neighborhood of the identity contains an invariant subgroup H such that G=H is isomorphic to a Lie group. Bagley, Wu, and Yang defined a pro-Lie group, which is similar to Montgomery and Zippin's definition of being approximated by Lie groups. Covington defined a protopological group as a group G satisfying the condition that every neighborhood of the identity contain a normal subgroup N such that G=N is a topological group. In this paper, we examine these ideas from a categorical point of view. We de ne a pro-x group as a group G satisfying the condition that every neighborhood of the identity contain a normal subgroup N such that G=N has property x.

Keywords: protopological group, topological group, pro-Lie group, pro-x group