turkmath.org

Let FI be the category of finite sets and injections. A functor $V: FI \rightarrow$ Abelian groups, i.e. an FI-module, encodes a sequence ${V_n}$ of $Sym(n)$-representations for various $n$ and transition maps between them. Homological finiteness properties of $V$ then strongly restricts the eventual behavior of the sequence. I will discuss several such homological invariants of $V$ whose vanishing in a range of degrees manifest in concrete stabilization patterns for the sequence, with examples from topology. I will discuss improvements to these "stable ranges" of $V$, some valid very generally, and some as a result of V possessing extra functorial structure extending the FI-action. Let FI be the category of finite sets and injections. A functor $V: FI \rightarrow$ Abelian groups, i.e. an FI-module, encodes a sequence ${V_n}$ of $Sym(n)$-representations for various $n$ and transition maps between them. Homological finiteness properties of $V$ then strongly restricts the eventual behavior of the sequence. I will discuss several such homological invariants of $V$ whose vanishing in a range of degrees manifest in concrete stabilization patterns for the sequence, with examples from topology. I will discuss improvements to these "stable ranges" of $V$, some valid very generally, and some as a result of $V$ possessing extra functorial structure extending the FI-action.