University of Sydney Algebra Seminar

Ross Street (Macquarie University)

Friday 15 August, 12:05-12:55pm, Place: 373

The Dold-Kan Theorem and categories of groupoid representations

This joint work with Stephen Lack began by our examining an equivalence of categories that occurs in the paper [Church-Ellenberg-Farb, ``FI-modules: a new approach to stability for \(S_n\)-representations'', arXiv:1204.4533v2]. Here \(\mathrm{FI}\) is the category of finite sets and injective functions, while an \(\mathrm{FI}\)-module is a functor \(\mathrm{FI}\to \mathrm{Mod}_R\) into a category of modules. Let \(\mathfrak{S}\) denote the symmetric groupoid: that is, the category of finite sets and bijective functions. The paper [ibid.] shows stability aspects of the representation theory of the symmetric groups can be studied profitably via \(\mathrm{FI}\)-modules. Important examples of \(\mathrm{FI}\)-modules in this story are in fact \(\mathrm{FI\#}\)-modules, where \(\mathrm{FI\#}\) is the category of finite sets and {\em injective partial} functions. We believe a vital part of the applicability of \(\mathrm{FI}\)-modules to these representations is the equivalence of categories \([\mathrm{FI\#},\mathrm{{Mod}_R}]\simeq [\mathfrak{S},\mathrm{{Mod}_R}]\), where \([\mathcal{A},\mathcal{B}]\) denotes the category of functors \(\mathcal{A}\to\mathcal{B}\) and natural transformations between them. The generalisation I will present not only gives a similar equivalence for other classical groupoids but also includes the Dold-Kan equivalence between chain complexes of \(R\)-modules and simplicial \(R\)-modules.