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Pré-Publication, Document De Travail Année : 2022

Outer functors and a general operadic framework

Résumé

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

Dates et versions

hal-03550366 , version 1 (01-02-2022)

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Citer

Geoffrey Powell. Outer functors and a general operadic framework. 2022. ⟨hal-03550366⟩
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