A classification of 2-chains having 1-shell boundaries in rosy theories

TL;DR

This paper classifies 2-chains with 1-shell boundaries in rosy theories, proving all such 1-shells are boundaries of 2-chains and showing the unbounded minimal lengths of these chains in rosy theories.

Contribution

It provides a classification of 2-chains with 1-shell boundaries in rosy theories and establishes that the first homology group is trivial in this context.

Findings

Every 1-shell of a Lascar strong type in rosy theories is a boundary of some 2-chain.

The first homology group in rosy theories is trivial.

There is no upper bound on the minimal lengths of 2-chains with a given 1-shell boundary in rosy theories.

Abstract

We classify, in a non-trivial amenable collection of functors, all 2-chains up to the relation of having the same 1-shell boundary. In particular, we prove that in a rosy theory, every 1-shell of a Lascar strong type is the boundary of some 2-chain, hence making the 1st homology group trivial. We also show that, unlike in simple theories, in rosy theories there is no upper bound on the minimal lengths of $2$-chains whose boundary is a $1$-shell.