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

Part of: General theory of categories and functors Homological algebra

Published online by Cambridge University Press:  01 March 2022

Geoffrey Powell  and
Christine Vespa
Geoffrey Powell
Affiliation:
Univ Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France
Christine Vespa*
Affiliation:
Université de Strasbourg, CNRS, IRMA UMR 7501, F-67000 Strasbourg, France
*
*Corresponding author. E-mail: vespa@math.unistra.fr
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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.

Keywords

category of finite sets and surjectionscomonad on Gamma-modulesKoszul complexFIop cohomology

MSC classification

Primary: 18G15: Ext and Tor, generalizations, Künneth formula
Secondary: 18A25: Functor categories, comma categories
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 65 , Issue 1 , January 2023 , pp. 1 - 61
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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