Filtering the linearization of the category of surjections

TL;DR

This paper constructs a filtration of the linearization of surjection categories using a module structure from injections, identifying subquotients as bimodules over finite set bijections, revealing new algebraic structures.

Contribution

It introduces a novel filtration of the $k$-linearized surjection category and characterizes its subquotients as bimodules over the category of finite set bijections.

Findings

Identification of subquotients as bimodules over $k extbf{FB}$

Construction of a primitive subcategory $k extbf{FS}^0$

Relation to $k extbf{FA}$-modules

Abstract

A filtration of the morphisms of the $k$-linearization $k \mathbf{FS}$ of the category $\mathbf{FS}$ of finite sets and surjections is constructed using a natural $k \mathbf{FI}^{op}$-module structure induced by restriction, where $\mathbf{FI}$ is the category of finite sets and injections. In particular, this yields the `primitive' subcategory $ k \mathbf{FS}^0 \subset k \mathbf{FS}$ that is of independent interest; for example, the category of $k \mathbf{FS}^0$-modules is closely related to the category of $k \mathbf{FA}$-modules, where $\mathbf{FA}$ is the category of finite sets and all maps. Working over a field of characteristic zero, the subquotients of this filtration are identified as bimodules over $k \mathbf{FB}$, where $\mathbf{FB}$ is the category of finite sets and bijections, also exhibiting and exploiting additional structure. In particular, this describes the underlying $k \mathbf{FB}$-bimodule of $k \mathbf{FS}^0$.