# 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.