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

摘要

We classify, in a non-trivial amenable collection of functors, all 2-chainsup to the relation of having the same 1-shell boundary. In particular, we provethat in a rosy theory, every 1-shell of a Lascar strong type is the boundary ofsome 2-chain, hence making the 1st homology group trivial. We also show that, unlike in simple theories, in rosy theories there is noupper bound on the minimal lengths of $2$-chains whose boundary is a $1$-shell.