On the fibrewise effective Burnside $\infty$-category

Semigroup actions on sets and the Burnside ring

In this paper we discuss some enlargements of the category of sets with semigroup actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category …

简单$\mathcal{C}$-集的伯恩赛德环

The Burnside ring of simple $\mathcal{C}$-sets

本文引入有限范畴$\mathcal{C}$的简单伯恩赛德环$B^S(\mathcal{C})$，它由简单$\mathcal{C}$-集构造，推广了有限群的经典伯恩赛德环且秩有限。研究其基础理论与结构性质，包括确定到$\mathbb{Z}$的同态、描述素谱及得到分解定理。

Tambarization of a Mackey functor and its application to the Witt-Burnside construction

For an arbitrary group $G$, a (semi-)Mackey functor is a pair of covariant and contravariant functors from the category of $G$-sets, and is regarded as a $G$-bivariant analog of a commutative (semi-)group. In this view, a $G$-bivariant analog of a (semi-)ring should be a (semi-)Tambara functor. A Tambara functor is firstly defined by Tambara, which he called a TNR-functor, when $G$ is finite. As shown by Brun, a Tambara functor plays a natural role in the Witt-Burnside construction. It will be a natural question if there exist sufficiently many examples of Tambara functors, compared to the wid…

Categorified groupoid-sets and their Burnside ring

We explore the category of internal categories in the usual category of (right) group-sets, whose objects are referred to as categorified group-sets. More precisely, we develop a new Burnside theory, where the equivalence relation between two categorified group-sets is given by a particular equivalence between the underlying categories. We also exhibit some of the differences between the old Burnside theory and the new one. Lastly, we briefly explain how to extend these new techniques and concepts to the context of groupoids, employing the categories of (right) groupoid-sets, aiming by this to…

Idempotents of double Burnside algebras, L-enriched bisets, and decomposition of p-biset functors

Let R be a (unital) commutative ring, and G be a finite group with order invertible in R. We introduce new idempotents in the double Burnside algebra RB(G,G), indexed by conjugacy classes of minimal sections of G, i.e. pairs (T,S) of subgroups of G, where S is a normal subgroup of T contained in the Frattini subgroup of T. These idempotents are orthogonal, and their sum is equal to the identity. It follows that for any biset functor F over R, the evaluation F (G) splits as a direct sum of specific R-modules indexed by minimal sections of G, up to conjugation. The restriction of these construct…

Categorified algebra and equivariant homotopy theory

This dissertation comprises three collections of results, all united by a common theme. The theme is the study of categories via algebraic techniques, considering categories themselves as algebraic objects. This algebraic approach to category theory is central to noncommutative algebraic geometry, as realized by recent advances in the study of noncommutative motives. We have success proving algebraic results in the general setting of symmetric monoidal and semiring $\infty$-categories, which categorify abelian groups and rings, respectively. For example, we prove that modules over the semiring…