Compact Lie Groups and Complex Reductive Groups

We show that the categories of compact Lie groups and complex reductive groups (not necessarily connected) are homotopy equivalent topological categories. In other words, the corresponding categories enriched in the homotopy category of topological spaces are equivalent. This can also be interpreted as an equivalence of infinity categories.
Comment: 11 pages. Version 2: minor edits. Version 3: the final journal version with minor edits