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Abstract

We study finiteness properties, especially the noetherian property, the Krull dimension and a variation of finite presentation, in categories of polynomial functors (notion introduced by Djament and Vespa) from a small symmetric monoidal category whose unit is an initial object to an abelian category. We prove in particular that the category of polynomial functors from the category of free abelian groups $\mathbb {Z}^n$ with split monomorphisms to abelian groups is “almost” locally noetherian. We also give an application to functors related to automorphisms of free groups.