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- W4382173515 abstract "We introduce a functor calculus for functors $mathsf{FI}tomathcal{V}$, which we call $mathsf{FI}$-objects, for $mathsf{FI}$ the category of finite sets and injections and $mathcal{V}$ a stable presentable $infty$-category. We show that $n$-homogeneous $mathsf{FI}$-objects are classified by representations of $mathfrak{S}_n$ in $mathcal{V}$, allowing us to associate Taylor coefficients to an $mathsf{FI}$-object. We show that these Taylor coefficients, in aggregate, themselves carry the structure of an $mathsf{FI}$-object, and we show that, up to the vanishing of certain Tate constructions, analytic $mathsf{FI}$-objects can be recovered from their $mathsf{FI}$-object of Taylor coefficients. We then establish a close relationship between our $mathsf{FI}$-calculus and the phenomenon of representation stability for $mathsf{FI}$-modules, suggesting that $mathsf{FI}$-calculus be understood as the extension of representation stability to the $infty$-categorical setting. In this context, we show how representation-theoretic information about a representation stable $mathsf{FI}$-module can be read off from its $mathsf{FI}$-module of Taylor coefficients." @default.
- W4382173515 created "2023-06-27" @default.
- W4382173515 creator A5092273055 @default.
- W4382173515 date "2023-06-23" @default.
- W4382173515 modified "2023-09-25" @default.
- W4382173515 title "FI-calculus and representation stability" @default.
- W4382173515 doi "https://doi.org/10.48550/arxiv.2306.13597" @default.
- W4382173515 hasPublicationYear "2023" @default.
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