On combinatorial aspects of fat Delta

TL;DR

This paper explores the structure of the fat Delta category, identifying key morphism types and relations, and demonstrating how the category is generated by these elements, aiding in understanding its combinatorial properties.

Contribution

It provides a detailed analysis of fat Delta's morphisms and relations, establishing a foundational framework for working with this modified simplex category.

Findings

Identified three types of morphisms: degenerated, standard, vertical faces.

Established six relations among these morphisms.

Proved fat Delta is generated by these morphisms and relations.

Abstract

The category fat Delta, introduced by J. Kock, is a modification of the simplex category where the degeneracies behave weakly. The objective of this note is to provide tools for working with fat Delta. In particular, we identify three types of morphisms: degenerated, standard and vertical faces, and establish six relations between these classes. We then show that fat Delta is generated by these morphisms and relations.