Filtering the linearization of the category of surjections

A filtration of the morphisms of the k-linearization k FS of the category FS of finite sets and surjections is constructed using a natural k FIᵒᵖ-module structure induced by restriction, where FI is the category of finite sets and injections. This yields the `primitive' subcategory k FS⁰ ⊂ k FS that is of independent interest. Working over a field of characteristic zero, the subquotients of this filtration are identified as bimodules over k FB, where FB is the category of finite sets and bijections, also exhibiting and exploiting additional structure. In particular, this describes the underlying k FB-bimodule of k FS⁰.