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 $\mathrm{FI}\mathrm{\#}$-modules, where $\mathrm{FI}\mathrm{\#}$ 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 $[\mathrm{FI}\mathrm{\#},{\mathrm{Mod}}_{\mathrm{R}}]\simeq [\mathfrak{S},{\mathrm{Mod}}_{\mathrm{R}}]$, 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.