On analytic contravariant functors on free groups
Résumé
Working over a field $k$ of characteristic zero, the category of analytic contravariant functors on the category of finitely-generated free groups is shown to be equivalent to the category of representations of the $k$-linear category associated to the Lie operad. The proof uses the original Ginzburg-Kapranov approach to Koszul duality of binary quadratic operads. The equivalence is made explicit using the $k$-linear category associated to the operad encoding unital associative algebras, which provides the appropriate `twisting bimodule'.