Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospans

TL;DR

This paper extends additivity theorems for cobordism categories to locally (co)Cartesian fibrations, broadening their applicability, and computes the classifying space difference for cospans of finite sets.

Contribution

It generalizes Steimle's additivity theorem to a wider class of fibrations and calculates the classifying space difference for cospans of finite sets.

Findings

Weakened conditions in additivity theorem to locally (co)Cartesian fibrations.

Computed the classifying space difference between cospans of finite sets and its homotopy category.

Abstract

We show that the conditions in Steimle's 'additivity theorem for cobordism categories' can be weakened to only require \emph{locally} (co)Cartesian fibrations, making it applicable to a larger class of functors. As an application we compute the difference in classifying spaces between the infinity category of cospans of finite sets and its homotopy category.