Outer functors and a general operadic framework

Abstract

For O an operad in -vector spaces, F O is the category of -linear functors from the PROP associated to O to -vector spaces. Given µ ∈ O(2) that satisfies a right Leibniz condition, the full subcategory F µ O ⊂ F O is introduced here and its properties studied. This is motivated by the case of the Lie operad Lie, where µ is taken to be the generator. F Lie is equivalent to the category of analytic functors on the opposite of the category gr of finitely-generated free groups. The main result shows that F µ Lie identifies with the category of outer analytic functors on gr op , as introduced in earlier work of the author with Vespa.

Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology

“…We hope to return to this point in future work. Christine Vespa informed us that the Lie structure described here features in recent work of Geoffrey Powell [Pow21,Pow22]. Briefly, Powell constructs an equivalence of categories, one exhibiting compatible sequences of Out(F g )-representations and the other consisting of certain modules over the Lie operad.…”

On the functors associated with beaded open Jacobi diagrams

“…Lie be the full subcategory of F Lie of functors such that µ F = 0. By [Powb,Theorem 4.16], under the equivalence of categories F ω (gr op ; K) ≃ F Lie , the full subcategory…”

Baby bead representations

“…Lie . To fix notation, this subsection reviews some material from [Pow21] and [Pow22b]; the reader should consult these references for details. (2) F Lie the category of representations of Cat Lie, i.e., -linear functors from Cat Lie to -vector spaces;…”

“…(3) F µ Lie ⊂ F Lie the full subcategory of representations on which the appropriate generalization of the adjoint action vanishes (see [Pow22b] for details).…”

“…As for the case of Lie, one can form the categories F Lie ≤2 and F µ Lie ≤2 ; in particular, the general theory of [Pow22b] applies to the operad Lie ≤2 , yielding: Proposition 4.8. There is a commutative diagram of inclusions of full abelian subcategories:…”