Ensembles reguliers

TL;DR

This paper introduces the category of regular simplicial sets, highlighting its stability properties and cartesian closedness, contrasting it with finite simplicial sets.

Contribution

It defines the subcategory of regular simplicial sets and proves its stability and cartesian closedness, which are not present in finite simplicial sets.

Findings

The subcategory of regular simplicial sets is stable under limits and unions.

The category of finite regular simplicial sets is cartesian closed.

Finite simplicial sets are not cartesian closed.

Abstract

We define an interesting sub-category of the category of simplicial sets, $\Sr$, whose objects are called regular. Both it and the subcategory ${\cal S}_{f-{\rm reg}}$ of finite regular simplicial sets have good stability properties under limits and union. The category ${\cal S}_{f-{\rm reg}}$ is cartesian closed, in contrast to the category of finite simplicial sets which is not cartesian closed.