Lifting Independence Along Functors

Given a functor $$F: \mathcal {C}\rightarrow \mathcal {D}$$ and a model-theoretic independence relation on $$\mathcal {D}$$ , we can lift that independence relation along F to $$\mathcal {C}$$ by declaring a commuting square in $$\mathcal {C}$$ to be independent if its image under F is independent. For each property of interest that an independence relation can have we give assumptions on the functor that guarantee the property to be lifted.