arXiv subjects
Publications and source records attributed to Mikala Ørsnes Jansen.
We investigate stability properties of the reductive Borel-Serre categories; these were introduced as a model for unstable algebraic K-theory in previous work. We see that they exhibit better homological stability properties than the general linear groups. We also show that they provide an explicit model for Yuan's partial algebraic K-theory.
We show that for nice enough $\mathbb{N}$-graded $\mathbb{E}_2$-algebras, a diagonal vanishing line in $\mathbb{E}_1$-homology of gives rise to slope $1$ homological stability. This is an integral version of a result by Kupers-Miller-Patzt.
We exploit the theory of $\infty$-stacks to provide some basic definitions and calculational tools regarding stratified homotopy theory of stratified topological stacks.
Let $A$ be an associative ring and $M$ a finitely generated projective $A$-module. We introduce a category $\operatorname{RBS}(M)$ and prove several theorems which show that its geometric realisation functions as a well-behaved unstable algebraic K-theory space. These categories $\operatorname{RBS}(M)$ naturally arise as generalisations of the exit path $\infty$-category of the reductive Borel-Serre compactification of a locally symmetric space, and one of our main techniques is to find purely categorical analogues of some familiar structures in these compactifications.
In 1984, Charney and Lee defined a category of stable curves and exhibited a rational homology equivalence from its geometric realisation to (the analytification of) the moduli stack of stable curves, also known as the Deligne-Mumford-Knudsen compactification. We strengthen this result by showing that, in fact, this category captures the stratified homotopy type of the moduli stack. In particular, it classifies constructible sheaves via an exodromy equivalence.
We identify the exit path $\infty$-category of the reductive Borel-Serre compactification as the nerve of a $1$-category defined purely in terms of rational parabolic subgroups and their unipotent radicals. As an immediate consequence, we identify the fundamental group of the reductive Borel-Serre compactification, recovering a result of Ji-Murty-Saper-Scherk, and we obtain a combinatorial incarnation of constructible complexes of sheaves on the reductive Borel-Serre compactification as elements in a derived functor category.