A Pirashvili-type theorem for functors on non-empty finite sets

TL;DR

This paper extends Pirashvili's theorem to unpointed finite sets, analyzing morphisms and Ext groups between tensor powers of functors, revealing key differences from the pointed case relevant to higher Hochschild homology.

Contribution

It provides an unpointed analogue of Pirashvili's theorem, computing morphisms and Ext groups for functors on non-empty finite sets, and highlights their non-projectivity.

Findings

Computed morphisms between tensor powers of functors in the unpointed setting.

Calculated Ext groups, showing these functors are not projective.

Revealed differences between pointed and unpointed contexts affecting homological properties.

Abstract

Pirashvili's Dold-Kan type theorem for finite pointed sets follows from the identification in terms of surjections of the morphisms between the tensor powers of a functor playing the role of the augmentation ideal; these functors are projective. We give an unpointed analogue of this result: namely, we compute the morphisms between the tensor powers of the corresponding functor in the unpointed context. We also calculate the Ext groups between such objects, in particular showing that these functors are not projective; this is an important difference between the pointed and unpointed contexts. This work is motivated by our functorial analysis of the higher Hochschild homology of a wedge of circles.