Combinatorics arising from lax colimits of posets

TL;DR

This paper explores the combinatorial structures arising from lax colimits of posets, revealing connections to familiar objects like Dyck paths and Kreweras walks through the study of maximal chains in constructed lattices.

Contribution

It introduces a novel approach to studying combinatorial objects via lax colimits of posets, linking higher-dimensional lattice constructions to classical combinatorial paths.

Findings

Maximal chains correspond to known combinatorial objects.

Lattices constructed from chains via lax colimits exhibit familiar path structures.

Connections between higher-dimensional lattice theory and classical combinatorics.

Abstract

In this paper we study maximal chains in certain lattices constructed from powers of chains by iterated lax colimits in the $2$-category of posets. Such a study is motivated by the fact that in lower dimensions, we get some familiar combinatorial objects such as Dyck paths and Kreweras walks.